
\documentclass[10pt]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{sw20elba}

%TCIDATA{OutputFilter=LATEX.DLL}
%TCIDATA{Version=5.50.0.2890}
%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
%TCIDATA{BibliographyScheme=BibTeX}
%TCIDATA{Created=Wednesday, July 31, 2013 08:28:26}
%TCIDATA{LastRevised=Friday, August 23, 2013 08:10:30}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="DocumentShell" CONTENT="Articles\SW\mrvl">}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=LaTeX article (bright).cst}

\newtheorem{theorem}{Theorem}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{{\hfill $\Box$ \\}}
\input{tcilatex}
\addtolength{\textheight}{30pt}
\addtolength{\textwidth}{140pt}
\addtolength{\oddsidemargin}{-1in}
\addtolength{\evensidemargin}{-1in}

\begin{document}

\title{The nilpotent Lie rings of order $p^{k}$ for $k\leq 7$}
\author{Michael Vaughan-Lee}
\date{August 2013}
\maketitle

\section{Introduction}

We give presentations for the nilpotent Lie rings of order $p^{k}$ for $%
k\leq 7$, valid for all $p>3$. The presentations give a complete
classification of these Lie rings. The class 2 presentations also give a
classification for the class 2 Lie rings of order $3^{k}$ for $k\leq 7$. (On
a historical note, the classification was originally obtained with the aim
of classifying the groups of order $p^{k}$ for $k\leq 7$ via the
Baker-Campbell-Hausdorff formula. See \cite{newobvl} and \cite{obrienvl2}.
The Baker-Campbell-Hausdorff formula only applies for $p$-groups of class at
most $p-1$, and so computing the nilpotent Lie rings of order $2^{k}$ was of
no use in this project, and computing the nilpotent Lie rings of order $%
3^{k} $ was only useful in this project for classifying $3$-groups of
nilpotency class 2.)

Our method of classifying nilpotent Lie rings of order $p^{n}$ closely
follows the $p$-group generation algorithm (see Newman \cite{Newman77} and
O'Brien \cite{OBrien90}). A Lie ring $L$ is an abelian group under $+$
(addition) together with a bilinear product which satisfies 
\begin{eqnarray*}
aa &=&0\text{ for all }a\in L, \\
(ab)c+(bc)a+(ca)b &=&0\text{ for all }a,b,c\in L.
\end{eqnarray*}%
(We use $ab$ to denote the Lie product of $a$ and $b$, rather than the more
familiar $[a,b]$.) Note that the axiom $aa=0$ together with bilinearity
implies that $ba=-ab$. The identity $(ab)c+(bc)a+(ca)b=0$ is the Jacobi
identity. Since the Lie product is not associative the bracketing of a
product is significant. We adopt the left-normed convention whereby 
\[
a_{1}a_{2}\ldots a_{n}=(\ldots ((a_{1}a_{2})a_{3})\ldots a_{n-1})a_{n}. 
\]%
For a Lie ring $L$, we define the \textit{lower }$p$-\textit{central series} 
\[
L=L_{1}\geq L_{2}\geq L_{3}\geq \ldots \geq L_{c}\geq \ldots 
\]%
in an analogous way to groups. We set $L_{1}=L$, $L_{2}=L^{2}+pL$, and for $%
c>1$ we set $L_{c+1}=L_{c}L+pL_{c}$. (Here $L^{2}=\langle ab\,|\,a,b\in
L\rangle $, and $L_{c}L=\langle ab\,|\,a\in L_{c},\,b\in L\rangle $.) The
ideal $L_{c}$ consists of all linear combinations of terms of the form 
\[
a_{1}a_{2}\ldots a_{c},\,pa_{1}a_{2}\ldots a_{c-1},\,p^{2}a_{1}a_{2}\ldots
a_{c-2},\ldots ,\,p^{c-1}a_{1}, 
\]%
with $a_{i}\in L$. We say that $L$ has $p$-class $c$ if $L_{c+1}=\{0\}$, $%
L_{c}\neq \{0\}$. If $L$ is a nilpotent Lie ring with finite order $p^{n}$
for some prime $p$, then $L_{c+1}$ will equal $\{0\}$ for some $c$. In fact
if $L$ is nilpotent of class $k$, and if the exponent of $L$ as a finite
abelian group is $p^{m}$ then $L$ has $p$-class $c$ for some $c$ with $k\leq
c<k+m$.

If $L$ and $M$ are two finite nilpotent Lie rings with prime-power order,
then $L$ is a \textit{descendant} of $M$ if $L/L_c\cong M$ for some $c\geq 2$%
. If $L/L_c\cong M$ and $L$ has $p$-class $c$ (so that $L_c\neq \{0\}$, $%
L_{c+1}=\{0\}$) then $L$ is an \textit{immediate descendant} of $M$. Note
that if $L$ is a descendant of $M$ then $L/L_2\cong M/M_2$, so that $L$ and $%
M$ have the same generator number.

The key idea for calculating the nilpotent Lie rings $L$ with order $p^{n}$
is as follows. If $L$ has $p$-class 1 then $L$ is the direct sum of $n$
copies of $\mathbb{Z}_{p}$; call $L$ \textit{elementary abelian}. If $L$ has 
$p$-class $c>1$, then $L$ is an immediate descendant of $L/L_{c}$, which is
a nilpotent Lie ring with order $p^{m}$ for some $m<n$. To compute the
nilpotent Lie rings of order $p^{n}$, first compute the nilpotent Lie rings
of order $p^{k}$ for $k<n$, and for each of these Lie rings compute their
immediate descendants of order $p^{n}$. A method for doing this is described
in \cite{newobvl}. Our presentations are grouped according to their
immediate ancestors.

In their simplest form, the presentations below are fairly standard. For
example, here is a presentation for one of the nilpotent Lie rings of order $%
p^{4}$:

\begin{equation}
\langle a,b\,|\,bab,\,pa,\,pb-baa,\,\text{class }3\rangle .  \tag{4.11}
\end{equation}%
Note that we specify the lower $p$-central class of the Lie ring as well as
giving generators and relations. So if $L$ is the Lie ring with presentation 
$\langle a,b\,|\,bab,\,pa,\,pb-baa\rangle $, algebra 4.11 is $L/L_{4}$. One
key property of these presentations is that the prime $p$ is symbolic. If we
substitute the prime $5$ for $p$ in this presentation then we get a
presentation for a nilpotent Lie ring of order $5^{4}$. And if we substitute 
$7$ for $p$ in this presentation, then we get a presentation for a nilpotent
Lie ring of order $7^{4}$. However many of the presentations are more
complicated than this, and involve parameters. For example algebra 4.12 has
presentation

\begin{equation}
\langle a,b\,|\,bab,\,pa,\,pb-\omega baa,\,\text{class }3\rangle . 
\tag{4.12}
\end{equation}%
The parameter $\omega $ occurs in hundreds of the presentations below, and
is assumed throughout to be a primitive element modulo $p$. If you want to
use the presentations to generate a complete and irredundant list of the
nilpotent Lie rings of order $p^{k}$ for some $k\leq 7$ and some specific
prime $p$, then it is important that you choose a primitive element modulo $%
p $ at the outset, and use this one value for $\omega $ throughout.
Actually, this is not strictly necessary, but if you use different values
for $\omega $ in different presentations, then no guarantees are given!

It is perhaps worthwhile to take a moment to explore algebras 4.11 and 4.12
in more detail. Consider a presentation%
\[
\langle a,b\,|\,bab,\,pa,\,pb-xbaa,\,\text{class }3\rangle 
\]%
where $x$ is an integer. Then (for any given $p$) this presentation defines
a nilpotent Lie ring $L_{x}$ of order $p^{4}$. Since $L_{x}$ has class 3,
and since $baa\in L_{3}$, the element $baa$ has additive order $p$, and so $%
L_{x}$ is isomorphic to $L_{y}$ whenever $x=y\func{mod}p$. It turns out (in
this example) that if $x$ and $y$ are non-zero modulo $p$, then $L_{x}$ is
isomorphic to $L_{y}$ if and only if $\frac{x}{y}$ is a square modulo $p$.
So (for $p>2$) there are two isomorphism classes of algebras $L_{x}$ with $%
x\neq 0\func{mod}p$: one where $x$ is a square modulo $p$, and one where $x$
is not a square modulo $p$. Taking $x=1,\omega $ we obtain representatives
for these two isomorphism classes. (In this case, $\omega $ does not need to
be a primitive element modulo $p$, and any integer which is not a square
modulo $p$ would do.)

Many presentations involve parameters other than $\omega $. Here are two
simple examples for Lie rings of order $p^{6}$ and $p^{7}$:%
\begin{equation}
\langle a,b,c\,|\,pa-ba,\,pb-cb,\,pc-xba-ca,\,\text{class }2\rangle \;(\text{%
all }x)  \tag{6.114}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,baab-baaab,babb-baaab,pa-xbaaab,pb-ybaaab,\,\text{%
class }5\rangle \;(\text{all }x,y)  \tag{7.351}
\end{equation}%
Presentation 6.114 has a single parameter $x$, and since $x$ appears as the
coefficient of $ba$, and since the algebra has class 2, it is clear that two
parameters $x$ and $x^{\prime }$ give isomorphic algebras if $x=x^{\prime }$
mod $p$. The somewhat cryptic comment \textquotedblleft all $x$%
\textquotedblright\ signifies that this is the only restriction on the
parameter $x$, and that there are $p$ distinct isomorphism classes of
algebras with presentation 6.114, with $x$ and $x^{\prime }$ giving
isomorphic algebras if and only if $x=x^{\prime }\func{mod}p$. This means
that you can obtain $p$ non-isomorphic algebras by taking $x=0,1,\ldots ,p-1$%
. However, sometimes you might want to make a different choice of
parameters. For example, it might be that the algebra is terminal unless $%
x=-1\func{mod}p$. In that case you might want to take $x=-1,0,1,\ldots ,p-2$
as a representative set of parameters. Similarly, the comment
\textquotedblleft all $x,y$\textquotedblright\ in presentation 7.351
indicates that this family of presentations gives $p^{2}$ non-isomorphic
algebras. Parameters $x,y$ and $x^{\prime },y^{\prime }$ give isomorphic
algebras if and only if $x=x^{\prime }\func{mod}p$ and $y=y^{\prime }\func{%
mod}p$. Would that things were always this simple! Often the parameters are
given a limited range, such as \textquotedblleft $x\neq 0$%
\textquotedblright\ or \textquotedblleft all $x$ with $1+4x$ not a square
mod $p$\textquotedblright . Here $x\neq 0$ is to be understood to mean $%
x\neq 0\func{mod}p$. More generally, if we have a one parameter family of
Lie rings $L_{x}$ then we define an equivalence relation $\sim $ on the
parameters $x$ by setting $x\sim y$ if $L_{x}$ is isomorphic to $L_{y}$. It
is to be understood that $x\sim y$ whenever $x=y\func{mod}p$, but the
comments associated with the presentations below often define slightly more
complex equivalence relations. To save space the comments are rather
cryptic. Here are some examples:%
\[
x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}\func{mod}p, 
\]%
\[
x\sim x^{\prime }\text{ if }x^{4}=x^{\prime 4}\func{mod}p, 
\]%
\[
x\sim -x, 
\]%
\[
x\neq 0,\;x\sim x^{-1}. 
\]%
The first two examples are clear enough. In the third example the cryptic
comment $x\sim -x$ is intended to mean that $x\sim y$ if and only if $x=\pm y%
\func{mod}p$. In the fourth example, we restrict to parameters $x$ with $%
x\neq 0\func{mod}p$, and $x\sim y$ if and only if $x=y\func{mod}p$ or $xy=1%
\func{mod}p$. The simplest way of thinking about these equivalence relations
is to think of them as being defined on subsets of the field $\mathbb{Z}_{p}$%
, and if you if you want to compute a set of representatives for the
equivalence classes for a particular prime $p$, it is often easiest to
compute in $\mathbb{Z}_{p}$. But the nevertheless, the parameters take
values in the integers. This can be confusing, but the same applies in the $%
p $-group generation algorithm --- the isomorphism problem for the immediate
descendants of a finite $p$-group is solved over $\mathbb{Z}_{p}$, but the
presentations for the descendants have integer exponents.

The presentations for the nilpotent Lie rings of order $p^{k}$ for $k\leq 4$
do not involve parameters, apart from the parameter $\omega $ which is taken
to be a primitive element modulo $p$ whenever it appears. The presentations
for the nilpotent Lie rings of order $p^{5}$ involve at most one parameter
other than $\omega $. But many of the presentations for the nilpotent Lie
rings of order $p^{6}$ and $p^{7}$ involve two or more parameters other than 
$\omega $. Just as with a single parameter, we associate a comment with
these presentations to describe the restrictions on the values that the
parameters can take, and to describe the equivalence relation $\sim $
associated with the isomorphism problem. Here are a few examples:%
\[
\text{all }x,\,y,\,(x,\,y)\sim (y,\,x), 
\]%
\[
x\neq 0,\,y\neq 0,\,1,-1,\;y\sim y^{-1}, 
\]%
\[
x\neq 0,\;\text{all }y,z,\;x\sim x^{\prime }\text{ if }x^{4}=x^{\prime 4}%
\func{mod}p. 
\]%
In the last of these examples we have a presentation with three parameters $%
x,y,z$, and for any choice of $y,z$ the algebra defined by parameters $x,y,z$
is isomorphic to the algebra defined by parameters $x^{\prime },y,z$ if $%
x^{4}=x^{\prime 4}\func{mod}p$.

In a few cases I have been unable to give simply defined equivalence
relations on the parameter sets, and I need to give more complicated
definitions. See for example algebra 6.62 below.

There is one further type of comment attached to many of the presentations
--- some of the presentations only apply for certain residue classes modulo $%
p$. A comment \textquotedblleft $p=1\func{mod}3$\textquotedblright , for
example, implies that the presentation should only be included for the prime 
$p$ if $p=1\func{mod}3$.

The classification given here has been criticized on the grounds that to
produce a complete and irredundant set of Lie rings for any given $p$ you
have to do some computations \textquotedblleft on the fly\textquotedblright
. Now, even with a one parameter presentation with restriction $x\neq 0$, $%
x\sim x^{-1}$, if you want a complete and irredundant set of representatives
for the equivalence classes of the parameter $x$ for a given prime $p$, you
have to do a computation on the fly. And one parameter presentations like
that are about as good as it gets! So I think the criticism is that there
are some presentations with several parameters, and quite complicated
equivalence relations. There are%
\[
3p^{2}+39p+344+24\gcd (p-1,3)+11\gcd (p-1,4)+2\gcd (p-1,5) 
\]%
nilpotent Lie rings of order $p^{6}$ for every prime $p\geq 5$, and so there
have to be some presentations for the Lie rings of order $p^{6}$ involving
at least two parameters (as well as possibly involving the parameter $\omega 
$). The presentations for the Lie rings of order $p^{6}$ involve at most 2
parameters (other than $\omega $) in all \ except one case. The one
exception is algebra 6.178 which has a presentation involving four
parameters. Actually one of the four parameters can be restricted to values $%
0,1$, and so this four parameter presentation quickly reduces to two three
parameter presentations. For $p\geq 5$ the number of nilpotent Lie rings of
order $p^{7}$ is

\begin{eqnarray*}
&&\allowbreak 3p^{5}+12p^{4}+44p^{3}+\allowbreak 170p^{2}+707p+2455 \\
&&+(4p^{2}+44p+291)\gcd (p-1,3)+\allowbreak (p^{2}+19p+135)\gcd (p-1,4) \\
&&+(3p+31)\gcd (p-1,5)+4\gcd (p-1,7)+5\gcd (p-1,8)+\gcd (p-1,9),
\end{eqnarray*}%
and so there have to be presentations for Lie rings of order $p^{7}$
involving at least 5 parameters. The ideal, I suppose, would be a number of
presentations with at most five parameters, and with no restrictions on the
parameters. There is no reason to suppose that this ideal is achievable, and
even if it were it would still take $p^{5}$ work to list all the different
algebras given by a five parameter presentation. If you had a five parameter
presentation with anything but the simplest restrictions on the parameters,
and anything but the simplest equivalence relation solving the isomorphism
problem, it could easily take more than $p^{5}$ work to list them all.

So the real issue is the amount of work required to produce a complete list
of algebras of order $p^{7}$ for a given $p$. Ideally we would not need more
than $p^{5}$ work, and perhaps that is achievable. But I have been unable to
reach that goal in every case. The amount of work required to produce a
complete list of algebras for any given prime $p$ for any of the
parametrized presentations in the classification is of order at most $p^{5}$%
, except in one case. The one exception is presentation 7.3285. This is a 12
parameter presentation for (some) immediate descendants of algebra 4.1. The
equivalence relation defined on these parameters to solve the isomorphism
problem is the most complicated of all the conditions appearing in this
classification. It is fairly easy to see that there are 11 orbits for the
values of the first 6 parameters, so that the problem of finding a set of
representatives for the 12 parameters quickly reduces to 11 problems with 6
parameters. I have a program which computes a set of representatives for the
orbits with work of order somewhere between $p^{6}$ and $p^{7}$. For primes
up to 23 the time taken to run the program is roughly proportional to $%
p^{6.2}$. It would be nice to do better than this, as this particular
presentation creates a serious bottleneck in producing a complete list of
Lie rings of order $p^{7}$ for any $p$ greater than $5$ or $7$.

\section{Order $p$}

\begin{equation}
\langle a\,|\,\text{class }1\rangle  \tag{1.0}
\end{equation}

\section{Order $p^{2}$}

\begin{equation}
\langle a\,|\,\text{class }2\rangle  \tag{2.0}
\end{equation}

\begin{equation}
\langle a,b\,|\,\text{class }1\rangle  \tag{2.1}
\end{equation}

\section{Order $p^{3}$}

\begin{equation}
\langle a\,|\,\text{class }3\rangle  \tag{3.0}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,\text{class }1\rangle  \tag{3.1}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa,\,pb,\,\text{class }2\rangle  \tag{3.2}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-ba,\,pb,\,\text{class }2\rangle  \tag{3.3}
\end{equation}

\begin{equation}
\langle a,b\,|\,ba,\,pb,\,\text{class }2\rangle  \tag{3.4}
\end{equation}

\section{Order $p^{4}$}

\begin{equation}
\langle a\,|\,\text{class }4\rangle  \tag{4.0}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,\text{class }1\rangle  \tag{4.1}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba\,,ca,\,cb,\,pb,\,pc,\,\text{class }2\rangle  \tag{4.2}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa,\,pb,\,pc,\,\text{class }2\rangle  \tag{4.3}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa-ba,\,pb,\,pc,\,\text{class }2\rangle 
\tag{4.4}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa,\,pb,\,pc-ba,\,\text{class }2\rangle 
\tag{4.5}
\end{equation}

\begin{equation}
\langle a,b\,|\,ba,\,\text{class }2\rangle  \tag{4.6}
\end{equation}

\begin{equation}
\langle a,b\,|\,pb,\,\text{class }2\rangle  \tag{4.7}
\end{equation}

\begin{equation}
\langle a,b\,|\,pb-ba,\,\text{class }2\rangle  \tag{4.8}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,pa,\,pb,\,\text{class }3\rangle  \tag{4.9}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,pa-baa,\,pb,\,\text{class }3\rangle  \tag{4.10}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,pa,\,pb-baa,\,\text{class }3\rangle  \tag{4.11}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,pa,\,pb-\omega baa,\,\text{class }3\rangle  \tag{4.12}
\end{equation}

\begin{equation}
\langle a,b\,|\,ba,\,\,pb,\,\text{class }3\rangle  \tag{4.13}
\end{equation}

\begin{equation}
\langle a,b\,|\,ba-p^{2}a,\,\,pb,\,\text{class }3\rangle  \tag{4.14}
\end{equation}

\section{Order $p^{5}$}

There are $2p+61+2\gcd (p-1,3)+\gcd (p-1,4)$ nilpotent Lie rings of order $%
p^{5}$ for $p\geq 5$. We first list the one generator and five generator Lie
rings of order $p^{5}$, and then we group the rest according to their
immediate ancestors.%
\begin{equation}
\langle a\,|\,\text{class }5\rangle  \tag{5.0}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,\text{class }1\rangle .  \tag{5.1}
\end{equation}

\subsection{Descendants of 4.1}

\begin{equation}
\langle a,b,c,d\,|\,ba,ca,da,cb,db,dc,pb,pc,pd,\,\text{class }2\rangle . 
\tag{5.2}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,dc,pa,pb,pc,pd,\,class2\rangle  \tag{5.3}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,dc,pa-ba,pb,pc,pd,\,\text{class }2\rangle . 
\tag{5.4}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,dc,pa,pb,pc-ba,pd,\,\text{class }2\rangle . 
\tag{5.5}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,dc-ba,pa,pb,pc,pd,\,\text{class }2\rangle , 
\tag{5.6}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,dc-ba,pa-ba,pb,pc,pd,\,\text{class }2\rangle
,  \tag{5.7}
\end{equation}

\subsection{Descendants of 3.1}

\begin{equation}
\langle a,b,c\,|\,ba,\,ca,\,cb,\,pc,\,\text{class }2\rangle .  \tag{5.8}
\end{equation}
\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pb,\,pc,\,\text{class }2\rangle ,  \tag{5.9}
\end{equation}%
\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pb-ba,\,pc,\,\text{class }2\rangle .  \tag{5.10}
\end{equation}
\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pb,\,pc-ba,\,\text{class }2\rangle .  \tag{5.11}
\end{equation}
\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa,\,pb,\,\text{class }2\rangle ,  \tag{5.12}
\end{equation}%
\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa-ba,\,pb,\,\text{class }2\rangle .  \tag{5.13}
\end{equation}%
\begin{equation}
\langle a,b,c\,|\,cb,\,pa,\,pb,\,pc,\,\text{class }2\rangle ,  \tag{5.14}
\end{equation}%
\begin{equation}
\langle a,b,c\,|\,cb,\,pa-ba,\,pb,\,pc,\,\text{class }2\rangle .  \tag{5.15}
\end{equation}
\begin{equation}
\langle a,b,c\,|\,cb,\,pa,\,pb-ba,\,pc,\,\text{class }2\rangle ,  \tag{5.16}
\end{equation}%
\begin{equation}
\langle a,b,c\,|\,cb,\,pa-ca,\,pb-ba,\,pc,\,\text{class }2\rangle , 
\tag{5.17}
\end{equation}%
\begin{equation}
\langle a,b,c\,|\,cb,\,pa,\,pb-ca,\,pc,\,\text{class }2\rangle ,  \tag{5.18}
\end{equation}%
\begin{equation}
\langle a,b,c\,|\,cb,\,pa-ba,\,pb-ca,\,pc,\,\text{class }2\rangle . 
\tag{5.19}
\end{equation}%
\begin{equation}
\langle a,b,c\,|\,cb,\,pa,\,pb-ba,\,pc-xca,\,\text{class }2\rangle \;(x\neq
0,\;x\sim x^{-1}),  \tag{5.20}
\end{equation}%
\begin{equation}
\langle a,b,c\,|\,cb,\,pa,\,pb-ba-ca,\,pc-ca,\,\text{class }2\rangle ; 
\tag{5.21}
\end{equation}%
\begin{equation}
\langle a,b,c\,|\,cb,\,pa,\,pb-\omega ca,\,pc-ba,\,\text{class }2\rangle ; 
\tag{5.22}
\end{equation}%
\begin{equation}
\langle a,b,c\,|\,cb,\,pa,\,pb-xca,\,pc-ba-ca,\,\text{class }2\rangle \;(%
\text{all }x\text{ with }1+4x\text{ not a square }\func{mod}p),  \tag{5.23}
\end{equation}

\subsection{Descendants of 4.2}

\begin{equation}
\langle a,b,c\,|\,ba,\,ca,\,cb,\,pb,\,pc,\,\text{class }3\rangle . 
\tag{5.24}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba,\,ca,\,cb-p^{2}a,\,pb,\,pc,\,\text{class }3\rangle . 
\tag{5.25}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba-p^{2}a,\,ca,\,cb,\,pb,\,pc,\,\text{class }3\rangle . 
\tag{5.26}
\end{equation}

\subsection{Descendants of 4.3}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,bab,\,pa,\,pb,\,pc,\,\text{class }3\rangle , 
\tag{5.27}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,bab,\,pa-baa,\,pb,\,pc,\,\text{class }3\rangle ,
\tag{5.28}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,bab,\,pa,\,pb-baa,\,pc,\,\text{class }3\rangle ,
\tag{5.29}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,bab,\,pa,\,pb-\omega baa,\,pc,\,\text{class }%
3\rangle .  \tag{5.30}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,bab,\,pa,\,pb,\,pc-baa,\,\text{class }3\rangle .
\tag{5.31}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb-baa,\,bab,\,pa,\,pb,\,pc,\,\text{class }3\rangle ,
\tag{5.32}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb-baa,\,bab,\,pa-baa,\,pb,\,pc,\,\text{class }%
3\rangle ,  \tag{5.33}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb-baa,\,bab,\,pa,\,pb-baa,\,pc,\,\text{class }%
3\rangle ,  \tag{5.34}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb-baa,\,bab,\,pa,\,pb-\omega baa,\,pc,\,\text{class }%
3\rangle .  \tag{5.35}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb-baa,\,bab,\,pa,\,pb,\,pc-baa,\,\text{class }%
3\rangle .  \tag{5.36}
\end{equation}

\subsection{Descendants of 4.4}

Algebra 4.4 is terminal.

\subsection{Descendants of 4.5}

Algebra 4.5 is terminal.

\subsection{Descendants of 2.1}

\begin{equation}
\langle a,b\,|\,\text{class }2\rangle .  \tag{5.37}
\end{equation}

\subsection{Descendants of 3.2}

\begin{equation}
\langle a,b\,|\,pa,\,pb,\,\text{class }3\rangle .  \tag{5.38}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-bab,\,pb,\,\text{class }3\rangle ,  \tag{5.39}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-\omega bab,\,pb,\,\text{class }3\rangle ,  \tag{5.40}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-baa,\,pb,\,\text{class }3\rangle .  \tag{5.41}
\end{equation}%
\begin{equation}
\langle a,b\,|\,pa-baa,\,pb-xbab,\,\text{class }3\rangle \;(x\neq 0,\;x\sim
x^{-1}),  \tag{5.42}
\end{equation}
\begin{equation}
\langle a,b\,|\,pa-baa-bab,\,pb-bab,\,\text{class }3\rangle ,  \tag{5.43}
\end{equation}%
\begin{equation}
\langle a,b\,|\,pa-baa-\omega bab,\,pb-bab,\,\text{class }3\rangle , 
\tag{5.44}
\end{equation}%
\begin{equation}
\langle a,b\,|\,pa-\omega bab,\,pb-baa,\,\text{class }3\rangle ;  \tag{5.45}
\end{equation}%
\begin{equation}
\langle a,b\,|\,pa-xbab,\,pb-baa-bab,\,\text{class }3\rangle \;(\text{all }x%
\text{ with }1+4x\text{ not a square }\func{mod}p),  \tag{5.46}
\end{equation}

\subsection{Descendants of 3.3}

Algebra 3.3 is terminal.

\subsection{Descendants of 3.4}

Algebra 3.4 is terminal.

\subsection{Descendants of 4.6}

\begin{equation}
\langle a,b\,|\,ba,\,p^{2}b,\,\text{class }3\rangle ,  \tag{5.47}
\end{equation}

\begin{equation}
\langle a,b\,|\,ba-p^{2}a,\,p^{2}b,\,\text{class }3\rangle .  \tag{5.48}
\end{equation}

\subsection{Descendants of 4.7}

\begin{equation}
\langle a,b\,|\,baa,\,bab,\,pb,\,\text{class }3\rangle .  \tag{5.49}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,p^{2}a,\,pb,\,\text{class }3\rangle ,  \tag{5.50}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,p^{2}a,\,pb-baa,\,\text{class }3\rangle ,  \tag{5.51}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,p^{2}a,\,pb-\omega baa,\,\text{class }3\rangle , 
\tag{5.52}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,p^{2}a-baa,\,pb,\,\text{class }3\rangle .  \tag{5.53}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,\,p^{2}a,\,pb,\,\text{class }3\rangle ,  \tag{5.54}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,\,p^{2}a,\,pb-bab,\,\text{class }3\rangle ,  \tag{5.55}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,\,p^{2}a-bab,\,pb,\,\text{class }3\rangle ,  \tag{5.56}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,\,p^{2}a-\omega bab,\,pb,\,\text{class }3\rangle . 
\tag{5.57}
\end{equation}

\subsection{Descendants of 4.8}

\begin{equation}
\langle a,b\,|\,baa,\,pb-ba,\,\text{class }3\rangle .  \tag{5.58}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a,\,pb-ba,\,\text{class }3\rangle .  \tag{5.59}
\end{equation}

\subsection{Descendants of 4.9}

\begin{equation}
\langle a,b\,|\,bab,\,pa,\,pb,\,\text{class }4\rangle ,  \tag{5.60}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,pa-baaa,\,pb,\,\text{class }4\rangle .  \tag{5.61}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,pa,\,pb-baaa,\,\text{class }4\rangle ,  \tag{5.62}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,pa,\,pb-\omega baaa,\,\text{class }4\rangle \;(p=1%
\func{mod}3),  \tag{5.63}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,pa,\,pb-\omega ^{2}baaa,\,\text{class }4\rangle \;(p=1%
\func{mod}3).  \tag{5.64}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,pa,\,pb,\,\text{class }4\rangle ,  \tag{5.65}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,pa-baaa,\,pb,\,\text{class }4\rangle ,  \tag{5.66}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,pa-\omega baaa,\,pb,\,\text{class }4\rangle . 
\tag{5.67}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,pa-\omega ^{2}baaa,\,pb,\,\text{class }4\rangle
\;(p=1\func{mod}4),  \tag{5.68}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,pa-\omega ^{3}baaa,\,pb,\,\text{class }4\rangle
\;(p=1\func{mod}4).  \tag{5.69}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,pa,\,pb-baaa,\,\text{class }4\rangle ,  \tag{5.70}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,pa,\,pb-\omega baaa,\,\text{class }4\rangle \;(p=1%
\func{mod}3),  \tag{5.71}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,pa,\,pb-\omega ^{2}baaa,\,\text{class }4\rangle
\;(p=1\func{mod}3).  \tag{5.72}
\end{equation}

\subsection{Descendants of 4.10}

Algebra 4.10 is terminal.

\subsection{Descendants of 4.11 and 4.12}

Algebras 4.11 and 4.12 are terminal.

\subsection{Descendants of 4.13}

\begin{equation}
\langle a,b\,|\,ba,\,pb,\,\text{class }4\rangle ,  \tag{5.73}
\end{equation}

\begin{equation}
\langle a,b\,|\,ba-p^{3}a,\,pb,\,\text{class }4\rangle .  \tag{5.74}
\end{equation}

\subsection{Descendants of 4.14}

Algebra 4.14 is terminal.

\section{Order $p^6$}

There are%
\[
3p^{2}+39p+344+24\gcd (p-1,3)+11\gcd (p-1,4)+2\gcd (p-1,5) 
\]%
nilpotent Lie rings of order $p^{6}$ for every prime $p\geq 5$.

The following table lists the nilpotent Lie rings of order at most $p^{5}$
which have immediate descendants of order $p^{6}$.

{\ }

{\small 
\[
\begin{tabular}{|l|l|}
\hline
3.1 & $\langle a,b,c\;|\;\text{ class }1\rangle $ \\ \hline
4.1 & $\langle a,b,c,d\;|\;\text{ class }1\rangle $ \\ \hline
4.3 & $\langle a,b,c\;|\;ca,cb,pa,pb,pc,\text{ class }2\rangle $ \\ \hline
4.6 & $\langle a,b\;|\;ba,\text{ class }2\rangle $ \\ \hline
4.7 & $\langle a,b\;|\;pb,\text{ class }2\rangle $ \\ \hline
4.8 & $\langle a,b\;|\;pb-ba,\text{ class }2\rangle $ \\ \hline
5.0 & $\langle a\;|\;\text{ class }5\rangle $ \\ \hline
5.1 & $\langle a,b,c,d,e\;|\;\text{ class }1\rangle $ \\ \hline
5.2 & $\langle a,b,c,d\;|\;ba,ca,da,cb,db,dc,pb,pc,pd,\text{ class }2\rangle 
$ \\ \hline
5.3 & $\langle a,b,c,d\;|\;ca,cb,da,db,dc,pa,pb,pc,pd,\text{ class }2\rangle 
$ \\ \hline
5.8 & $\langle a,b,c\;|\;ba,ca,cb,pc,\text{ class }2\rangle $ \\ \hline
5.9 & $\langle a,b,c\;|\;ca,cb,pb,pc,\text{ class }2\rangle $ \\ \hline
5.10 & $\langle a,b,c\;|\;ca,cb,pb-ba,pc,\text{ class }2\rangle $ \\ \hline
5.11 & $\langle a,b,c\;|\;ca,cb,pb,pc-ba,\text{ class }2\rangle $ \\ \hline
5.12 & $\langle a,b,c\;|\;ca,cb,pa,pb,\text{ class }2\rangle $ \\ \hline
5.13 & $\langle a,b,c\;|\;ca,cb,pa-ba,pb,\text{ class }2\rangle $ \\ \hline
5.14 & $\langle a,b,c\;|\;cb,pa,pb,pc,\text{ class }2\rangle $ \\ \hline
5.15 & $\langle a,b,c\;|\;cb,pa-ba,pb,pc,\text{ class }2\rangle $ \\ \hline
5.16 & $\langle a,b,c\;|\;cb,pa,pb-ba,pc,\text{ class }2\rangle $ \\ \hline
5.18 & $\langle a,b,c\;|\;cb,pa,pb-ca,pc,\text{ class }2\rangle $ \\ \hline
5.19 & $\langle a,b,c\;|\;cb,pa-ba,pb-ca,pc,\text{ class }2\rangle $ \\ 
\hline
5.24 & $\langle a,b,c\;|\;ba,ca,cb,pb,pc,\text{ class }3\rangle $ \\ \hline
5.27 & $\langle a,b,c\;|\;bab,ca,cb,pa,pb,pc,\text{ class }3\rangle $ \\ 
\hline
5.32 & $\langle a,b,c\;|\;bab,ca,cb-baa,pa,pb,pc,\text{ class }3\rangle $ \\ 
\hline
5.37 & $\langle a,b\;|\;\text{ class }2\rangle $ \\ \hline
5.38 & $\langle a,b\;|\;pa,pb,\text{ class }3\rangle $ \\ \hline
5.39 & $\langle a,b\;|\;pa-bab,pb,\text{ class }3\rangle $ \\ \hline
5.40 & $\langle a,b\;|\;pa-\omega bab,pb,\text{ class }3\rangle $ \\ \hline
5.41 & $\langle a,b\;|\;pa-baa,pb,\text{ class }3\rangle $ \\ \hline
5.42 & $\langle a,b\;|\;pa-baa,pb+bab,\text{ class }3\rangle $ \\ \hline
5.45 & $\langle a,b\;|\;pa+bab,pb+\omega baa,\text{ class }3\rangle $ \\ 
\hline
5.47 & $\langle a,b\;|\;ba,p^{2}b,\text{ class }3\rangle $ \\ \hline
5.48 & $\langle a,b\;|\;ba-p^{2}a,p^{2}b,\text{ class }3\rangle $ \\ \hline
5.49 & $\langle a,b\;|\;baa,bab,pb,\text{ class }3\rangle $ \\ \hline
5.50 & $\langle a,b\;|\;bab,p^{2}a,pb,\text{ class }3\rangle $ \\ \hline
5.51 & $\langle a,b\;|\;bab,p^{2}a,pb-baa,\text{ class }3\rangle $ \\ \hline
5.52 & $\langle a,b\;|\;bab,p^{2}a,pb-\omega baa,\text{ class }3\rangle $ \\ 
\hline
5.54 & $\langle a,b\;|\;baa,p^{2}a,pb,\text{ class }3\rangle $ \\ \hline
5.58 & $\langle a,b\;|\;baa,pb-ba,\text{ class }3\rangle $ \\ \hline
5.60 & $\langle a,b\;|\;bab,pa,pb,\text{ class }4\rangle $ \\ \hline
5.65 & $\langle a,b\;|\;bab-baaa,pa,pb,\text{ class }4\rangle $ \\ \hline
5.73 & $\langle a,b\;|\;ba,pb,\text{ class }4\rangle $ \\ \hline
\end{tabular}%
\ 
\]%
}

We first give the cyclic and 6 generator Lie rings of order $p^{6}$, and
then list the remainder, grouped according to their immediate ancestors.

\begin{equation}
\langle a\,|\,\text{class }6\rangle  \tag{6.0}
\end{equation}

\begin{equation}
\langle a,b,c,d,e,f\,|\,\text{class }1\rangle  \tag{6.1}
\end{equation}

\subsection{Descendants of 3.1}

$3p+27$ algebras

\begin{equation}
\langle a,b,c\,|\,ba,\,ca,\,cb,\,\text{class }2\rangle  \tag{6.85}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pc,\,\text{class }2\rangle  \tag{6.86}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pc-ba,\,\text{class }2\rangle  \tag{6.87}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa,\,\text{class }2\rangle  \tag{6.88}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa-ba,\,\text{class }2\rangle  \tag{6.89}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,pb,\,pc,\,\text{class }2\rangle  \tag{6.90}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,pa,\,pc,\,\text{class }2\rangle  \tag{6.91}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,pb-ba,\,pc,\,\text{class }2\rangle  \tag{6.92}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,pb-ca,\,pc,\,\text{class }2\rangle  \tag{6.93}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,pa,\,pc-ba,\,\text{class }2\rangle  \tag{6.94}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,pa,\,pc-ca,\,\text{class }2\rangle  \tag{6.95}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,pa-ba,\,pc,\,\text{class }2\rangle  \tag{6.96}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,pa-ca,\,pc,\,\text{class }2\rangle  \tag{6.97}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,pb-ba,\,pc-xca,\,\text{class }2\rangle \;(x\neq
0,\,x\sim x^{-1})  \tag{6.98}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,pb-ba-ca,\,pc-ca,\,\text{class }2\rangle  \tag{6.99}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,pb-\omega ca,\,pc-ba,\,\text{class }2\rangle 
\tag{6.100}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,pb-xca,\,pc-ba-ca,\,\text{class }2\rangle \;(\text{%
all }x\text{ with }1+4x\text{ not a square }\func{mod}p)  \tag{6.101}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,pa-ca,\,pc-ba,\,\text{class }2\rangle  \tag{6.102}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,pa-ba,\,pc-ca,\,\text{class }2\rangle  \tag{6.103}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pa,\,pb,\,pc,\,\text{class }2\rangle  \tag{6.104}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pa-cb,\,pb,\,pc,\,\text{class }2\rangle  \tag{6.105}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pa-ba,\,pb,\,pc,\,\text{class }2\rangle  \tag{6.106}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pa-ca,\,pb-xcb,\,pc,\,\text{class }2\rangle \;(x\neq
0,\,x\sim x^{-1})  \tag{6.108}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pa-ca-cb,\,pb-cb,\,pc,\,\text{class }2\rangle  \tag{6.109}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pa-\omega cb,\,pb-ca,\,pc,\,\text{class }2\rangle 
\tag{6.110}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pa-xcb,\,pb-ca-cb,\,pc,\,\text{class }2\rangle \;(\text{%
all }x\text{ with }1+4x\text{ not a square }\func{mod}p)  \tag{6.111}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pa-ba,\,pb-ca,\,pc,\,\text{class }2\rangle  \tag{6.112}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pa-ba,\,pb-cb,\,pc,\,\text{class }2\rangle  \tag{6.113}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pa-ba,\,pb-cb,\,pc-xba-ca,\,\text{class }2\rangle \;(\text{%
all }x)  \tag{6.114}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pa-ba,\,pb-ca,\,pc-cb,\,\text{class }2\rangle  \tag{6.115}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pa-ba,\,pb-ca,\,pc+cb,\,\text{class }2\rangle  \tag{6.116}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pa-ba,\,pb-ca,\,pc-\omega ba+cb,\,\text{class }2\rangle 
\tag{6.117}
\end{equation}

\subsection{Descendants of 4.1}

$4p+48$ algebras

\begin{equation}
\langle a,b,c,d\,|\,ba,\,ca,\,da,\,cb,\,db,\,dc,\,pc,\,pd,\,\text{class }%
2\rangle  \tag{6.9}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,\,da,\,cb,\,db,\,dc,\,pa,\,pb,\,pc-ba,\,\text{class }%
2\rangle  \tag{6.10}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,\,da,\,cb,\,db,\,dc,\,pb,\,pc,\,pd,\,\text{class }%
2\rangle  \tag{6.11}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,\,da,\,cb,\,db,\,dc,\,pb-ba,\,pc,\,pd,\,\text{class }%
2\rangle  \tag{6.12}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,\,da,\,cb,\,db,\,dc,\,pb,\,pc-ba,\,pd,\,\text{class }%
2\rangle  \tag{6.13}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,\,da,\,cb,\,db,\,dc,\,pa,\,pb,\,pd,\,\text{class }%
2\rangle  \tag{6.14}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,\,da,\,cb,\,db,\,dc,\,pa-ba,\,pb,\,pd,\,\text{class }%
2\rangle  \tag{6.15}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,\,da,\,cb,\,db,\,dc-ba,\,pb,\,pc,\,pd,\,\text{class }%
2\rangle  \tag{6.16}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,\,da,\,cb,\,db,\,dc-ba,\,pb-ba,\,pc,\,pd,\,\text{
class }2\rangle  \tag{6.17}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,\,da,\,cb,\,db,\,dc-ba,\,pb,\,pc-ba,\,pd,\,\text{
class }2\rangle  \tag{6.18}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db,\,dc,\,pa,\,pb,\,pc,\,pd,\,\text{class }%
2\rangle  \tag{6.19}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db,\,dc,\,pa-ba,\,pb,\,pc,\,pd,\,\text{class }%
2\rangle  \tag{6.20}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db,\,dc,\,pa,\,pb-ba,\,pc,\,pd,\,\text{class }%
2\rangle  \tag{6.21}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db,\,dc,\,pa-ca,\,pb-ba,\,pc,\,pd,\,\text{
class }2\rangle  \tag{6.22}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db,\,dc,\,pa,\,pb-ca,\,pc,\,pd,\,\text{class }%
2\rangle  \tag{6.23}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db,\,dc,\,pa-ba,\,pb-ca,\,pc,\,pd,\,\text{
class }2\rangle  \tag{6.24}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db,\,dc,\,pa,\,pb-ba,\,pc-xca,\,pd,\,\text{
class }2\rangle \;(x\neq 0,\,x\sim x^{-1})  \tag{6.25}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db,\,dc,\,pa,\,pb-ba-ca,\,pc-ca,\,pd,\,\text{
class }2\rangle  \tag{6.26}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db,\,dc,\,pa,\,pb-\omega ca,\,pc-ba,\,pd,\,%
\text{class }2\rangle  \tag{6.27}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db,\,dc,\,pa,\,pb-xca,\,pc-ba-ca,\,pd,\,\text{
class }2\rangle \;(\text{all }x\text{ with }1+4x\text{ not a square }\func{%
mod}p)  \tag{6.28}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db,\,dc,\,pa,\,pb,\,pc,\,pd-ca,\,\text{class }%
2\rangle  \tag{6.29}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db,\,dc,\,pa-ba,\,pb,\,pc,\,pd-ca,\,\text{
class }2\rangle  \tag{6.30}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db,\,dc,\,pa,\,pb-ba,\,pc,\,pd-ca,\,\text{
class }2\rangle  \tag{6.31}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db,\,dc,\,pa,\,pb,\,pc-ba,\,pd-ca,\,\text{
class }2\rangle  \tag{6.32}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ba,\,dc,\,pa,\,pb,\,pc,\,pd,\,\text{class }%
2\rangle  \tag{6.33}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ba,\,dc,\,pa-ba,\,pb,\,pc,\,pd,\,\text{
class }2\rangle  \tag{6.34}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ba,\,dc,\,pa,\,pb-ba,\,pc,\,pd,\,\text{
class }2\rangle  \tag{6.35}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ba,\,dc,\,pa-ba,\,pb-ba,\,pc,\,pd,\,\text{
class }2\rangle  \tag{6.36}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ba,\,dc,\,pa-ba-ca,\,pb,\,pc,\,pd,\,\text{
class }2\rangle  \tag{6.37}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ba,\,dc,\,pa-ba-ca,\,pb-ba-ca,\,pc,\,pd,\,%
\text{class }2\rangle  \tag{6.38}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ba,\,dc,\,pa-ba,\,pb-ba,\,pc-ca,\,pd-ca,\,%
\text{class }2\rangle  \tag{6.39}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ba,\,dc,\,pa-ba,\,pb-ba,\,pc,\,pd-ca,\,%
\text{class }2\rangle  \tag{6.39B}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ba,\,dc,\,pa-ba+ca,\,pb-ba,\,pc,\,pd-ca,\,%
\text{class }2\rangle  \tag{6.40}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ba,\,dc,\,pa+ca,\,pb-ba,\,pc,\,pd-ca,\,%
\text{class }2\rangle  \tag{6.41}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ba,\,dc,\,pa,\,pb-ba,\,pc,\,pd-ca,\,\text{
class }2\rangle  \tag{6.42}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ba,\,dc,\,pa-ba-ca,\,pb-ba-xca,\,pc,\,pd,\,%
\text{class }2\rangle\;(x \neq 0,\,1)  \tag{6.43}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ba,\,dc,\,pa-ba-ca,\,pb-ba,\,pc,\,pd,\,%
\text{class }2\rangle  \tag{6.44}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ba,\,dc,\,pa-ba-ca,\,pb-ca,\,pc,\,pd,\,%
\text{class }2\rangle  \tag{6.45}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ba,\,dc,\,pa-ba,\,pb-ca,\,pc,\,pd,\,\text{
class }2\rangle  \tag{6.46}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ba,\,dc,\,pa-ca,\,pb-ba,\,pc,\,pd,\,\text{
class }2\rangle  \tag{6.47}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ca,\,dc,\,pa,\,pb,\,pc,\,pd,\,\text{class }%
2\rangle  \tag{6.48}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ca,\,dc,\,pa-ba,\,pb,\,pc,\,pd,\,\text{
class }2\rangle  \tag{6.49}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ca,\,dc,\,pa,\,pb,\,pc-ba,\,pd,\,\text{
class }2\rangle  \tag{6.50}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ca,\,dc,\,pa-ca,\,pb,\,pc,\,pd,\,\text{
class }2\rangle  \tag{6.51}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ca,\,dc,\,pa,\,pb,\,pc-ca,\,pd,\,\text{
class }2\rangle  \tag{6.52}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ca,\,dc,\,pa-ba,\,pb-ca,\,pc,\,pd,\,\text{
class }2\rangle  \tag{6.53}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ca,\,dc,\,pa-ca,\,pb,\,pc-ba,\,pd,\,\text{
class }2\rangle  \tag{6.54}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ca,\,dc,\,pa,\,pb-ca,\,pc-ba,\,pd,\,\text{
class }2\rangle  \tag{6.55}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ca,\,dc,\,pa,\,pb-\omega ca,\,pc-ba,\,pd,\,%
\text{class }2\rangle  \tag{6.56}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ca,\,dc,\,pa-ba,\,pb,\,pc-ca,\,pd,\,\text{
class }2\rangle  \tag{6.57}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ca,\,dc,\,pa,\,pb-xba,\,pc-ca,\,pd,\,\text{
class }2\rangle\;(x \neq 0)  \tag{6.58}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ca,\,dc,\,pa,\,pb,\,pc-ba,\,pd-ca,\,\text{
class }2\rangle  \tag{6.59}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ca,\,dc-\omega ba,\,pa,\,pb,\,pc,\,pd,\,%
\text{class }2\rangle  \tag{6.60}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ca,\,dc-\omega ba,\,pa-ba,\,pb,\,pc,\,pd,\,%
\text{class }2\rangle  \tag{6.60B}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ca,\,dc-\omega
ba,\,pa,\,pb,\,pc-ba,\,pd-ca,\,\text{class }2\rangle  \tag{6.61}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,\,da,\,db-ca,\,dc-\omega
ba,\,pa,\,pb-ba,\,pc-xba-yca,\,pd,\,\text{class }2\rangle  \tag{6.62}
\end{equation}%
Algebra 6.62 has two parameters $x,y$, with $y\neq 0$. Parameter pairs $%
(x,y) $ and $(z,t)$ give isomorphic algebras if and only if%
\[
\left( 
\begin{array}{ll}
1 & 0 \\ 
z & t%
\end{array}%
\right) =\left( 
\begin{array}{ll}
\mu & \nu \\ 
\omega \nu & \mu%
\end{array}%
\right) \left( 
\begin{array}{ll}
1 & 0 \\ 
x & y%
\end{array}%
\right) \left( 
\begin{array}{ll}
\mu +\nu x & \nu y \\ 
\omega \nu y & \mu +\nu x%
\end{array}%
\right) ^{-1} 
\]%
for some non-singular matrix $\left( 
\begin{array}{ll}
\mu & \nu \\ 
\omega \nu & \mu%
\end{array}%
\right) $. There are $p$ distinct algebras.

\subsection{Descendants of 4.3}

$3p^{2}+13p+37+\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa,\,pb,\,pc,\,\text{class }3\rangle  \tag{6.118}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa-bab,\,pb,\,pc,\,\text{class }3\rangle 
\tag{6.119}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa-\omega bab,\,pb,\,pc,\,\text{class }3\rangle 
\tag{6.120}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa-baa,\,pb,\,pc,\,\text{class }3\rangle 
\tag{6.121}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa-baa,\,pb-xbab,\,pc,\,\text{class }3\rangle
\;(x\neq 0,\,x\sim x^{-1})  \tag{6.122}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa-baa-bab,\,pb-bab,\,pc,\,\text{class }3\rangle
\tag{6.123}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa-baa-\omega bab,\,pb-bab,\,pc,\,\text{class }%
3\rangle  \tag{6.124}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa-\omega bab,\,pb-baa,\,pc,\,\text{class }%
3\rangle  \tag{6.125}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa-xbab,\,pb-baa-bab,\,pc,\,\text{class }%
3\rangle \;(\text{all }x\text{ with }1+4x\text{ not a square }\func{mod}p) 
\tag{6.126}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa,\,pb,\,pc-bab,\,\text{class }3\rangle 
\tag{6.127}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa-baa,\,pb,\,pc-bab,\,\text{class }3\rangle 
\tag{6.128}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa,\,pb-baa,\,pc-bab,\,\text{class }3\rangle 
\tag{6.129}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa,\,pb-\omega baa,\,pc-bab,\,\text{class }%
3\rangle  \tag{6.130}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa,\,pb,\,pc,\,\text{class }3\rangle 
\tag{6.131}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa-bab,\,pb,\,pc,\,\text{class }3\rangle 
\tag{6.132}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa-\omega bab,\,pb,\,pc,\,\text{class }%
3\rangle  \tag{6.133}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa-baa,\,pb,\,pc,\,\text{class }3\rangle 
\tag{6.134}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa-xbaa,\,pb-bab,\,pc,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{6.135}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa-baa-bab,\,pb-bab,\,pc,\,\text{class }%
3\rangle  \tag{6.136}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa-baa-\omega bab,\,pb-bab,\,pc,\,\text{
class }3\rangle  \tag{6.137}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa-baa-xbab,\,pb-baa,\,pc,\,\text{class }%
3\rangle\;(\text{all }x)  \tag{6.138}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa,\,pb-baa,\,pc,\,\text{class }3\rangle 
\tag{6.139}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa-bab,\,pb-baa,\,pc,\,\text{class }3\rangle
\tag{6.140}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa-\omega bab,\,pb-baa,\,pc,\,\text{class }%
3\rangle  \tag{6.141}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa-baa-xbab,\,pb-\omega baa,\,pc,\,\text{
class }3\rangle\;(\text{all }x)  \tag{6.142}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa,\,pb-\omega baa,\,pc,\,\text{class }%
3\rangle  \tag{6.143}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa-bab,\,pb-\omega baa,\,pc,\,\text{class }%
3\rangle  \tag{6.144}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa-\omega bab,\,pb-\omega baa,\,pc,\,\text{
class }3\rangle  \tag{6.145}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa,\,pb-xbaa,\,pc-bab,\,\text{class }%
3\rangle\;(\text{all }x)  \tag{6.146}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa-baa,\,pb,\,pc-bab,\,\text{class }3\rangle
\tag{6.147}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa-xbab,\,pb,\,pc-baa,\,\text{class }%
3\rangle\;(x=0,\,1,\,\omega ,\,\omega ^2,\,\omega ^3,\,\;p=1 \func{mod} 4) 
\tag{6.148}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa-xbab,\,pb,\,pc-baa,\,\text{class }%
3\rangle\;(x=0,\,1,\,\omega ,\,\;p=3 \func{mod} 4)  \tag{6.148A}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa-xbab,\,pb-ybab,\,pc-baa,\,\text{class }%
3\rangle  \tag{6.149}
\end{equation}%
\[
(y=1,\,\omega ,\,\omega ^{2},\,\text{all }x,\,x\sim x^{\prime }\text{ if }%
x^{3}=x^{\prime 3}\func{mod}p,\,\;p=1\func{mod}3) 
\]

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb,\,pa-xbab,\,pb-bab,\,pc-baa,\,\text{class }%
3\rangle\;(\text{all }x,\,\;p=2 \func{mod} 3)  \tag{6.149A}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa,\,pb,\,pc,\,\text{class }3\rangle 
\tag{6.150}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa,\,pb,\,pc,\,\text{class }3\rangle 
\tag{6.151}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-bab,\,pb,\,pc,\,\text{class }3\rangle 
\tag{6.152}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-bab,\,pb,\,pc,\,\text{class }3\rangle
\tag{6.153}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-\omega bab,\,pb,\,pc,\,\text{class }%
3\rangle  \tag{6.154}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-\omega bab,\,pb,\,pc,\,\text{class }%
3\rangle  \tag{6.155}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-bab,\,pb-baa,\,pc,\,\text{class }%
3\rangle\;(p=1 \func{mod} 4)  \tag{6.156}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-bab,\,pb-\omega baa,\,pc,\,\text{class }%
3\rangle\;(p=1 \func{mod} 4)  \tag{6.157}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-\omega bab,\,pb-\omega baa,\,pc,\,\text{
class }3\rangle\;(p=1 \func{mod} 4)  \tag{6.158}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-bab,\,pb-baa,\,pc,\,\text{class }%
3\rangle\;(p=3 \func{mod} 4)  \tag{6.156A}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-bab,\,pb-\omega baa,\,pc,\,\text{class }%
3\rangle\;(p=3 \func{mod} 4)  \tag{6.157A}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-\omega bab,\,pb-baa,\,pc,\,\text{class }%
3\rangle\;(p=3 \func{mod} 4)  \tag{6.158A}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-xbab,\,pb-baa,\,pc,\,\text{class }%
3\rangle\;(\text{all }x)  \tag{6.159}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-xbab,\,pb-\omega baa,\,pc,\,\text{
class }3\rangle\;(\text{all }x)  \tag{6.160}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa,\,pb-xbab,\,pc,\,\text{class }%
3\rangle\;(x \neq 0,\, x\sim x^{-1})  \tag{6.160A}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-bab,\,pb-xbab,\,pc,\,\text{class }%
3\rangle\;(x \neq 0)  \tag{6.161}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-\omega bab,\,pb-xbab,\,pc,\,\text{
class }3\rangle\;(x \neq 0)  \tag{6.162}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-bab,\,pb-xbaa-ybab,\,pc,\,\text{
class }3\rangle\;(x \neq 0,\, y \neq 0,\,1,\,-1,\, y\sim y^{-1})  \tag{6.163}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-\omega bab,\,pb-xbaa-ybab,\,pc,\,%
\text{class }3\rangle\;(x \neq 0,\, y \neq 0,\,1,\,-1,\, y\sim y^{-1}) 
\tag{6.164}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-bab,\,pb-x^2baa+bab,\,pc,\,\text{
class }3\rangle\;(x \neq 0,\, x\sim -x)  \tag{6.165}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-bab,\,pb-x^2baa-bab,\,pc,\,\text{
class }3\rangle\;(x \neq 0,\, x\sim -x)  \tag{6.165}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-bab,\,pb-x^{2}\omega baa+bab,\,pc,\,%
\text{class }3\rangle \;(x\neq 0,\,x\sim -x)  \tag{6.166}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-bab,\,pb-x^{2}\omega baa-bab,\,pc,\,%
\text{class }3\rangle \;(x\neq 0,\,x\sim -x)  \tag{6.166}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-\omega bab,\,pb-x^{2}\omega
baa+bab,\,pc,\,\text{class }3\rangle \;(x\neq 0,\,x\sim -x,\,\;p=1\func{mod}%
4)  \tag{6.167}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-\omega bab,\,pb-x^{2}\omega
baa-bab,\,pc,\,\text{class }3\rangle \;(x\neq 0,\,x\sim -x,\,\;p=1\func{mod}%
4)  \tag{6.167}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-\omega bab,\,pb-x^2baa+bab,\,pc,\,%
\text{class }3\rangle\;(x \neq 0,\, x\sim -x,\,\;p=3 \func{mod} 4) 
\tag{6.167A}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-\omega bab,\,pb-x^2baa-bab,\,pc,\,%
\text{class }3\rangle\;(x \neq 0,\, x\sim -x,\,\;p=3 \func{mod} 4) 
\tag{6.167A}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa,\,pb,\,pc-baa,\,\text{class }3\rangle 
\tag{6.168}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-bab,\,pb,\,pc-baa,\,\text{class }3\rangle
\tag{6.169}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-\omega bab,\,pb,\,pc-baa,\,\text{class }%
3\rangle  \tag{6.170}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-xbab,\,pb-bab,\,pc-baa,\,\text{class }%
3\rangle\;(\text{all }x)  \tag{6.171}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-xbab,\,pb-ybab,\,pc-baa-bab,\,\text{
class }3\rangle \;(\text{all }x,\,y,\,(x,\,y)\sim (y,\,x))  \tag{6.172}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,\,pa,\,pb,\,pc,\,\text{class }%
3\rangle  \tag{6.173}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,\,pa-xbaa-bab,\,pb,\,pc,\,\text{
class }3\rangle\;(\text{all }x,\, x\sim -x)  \tag{6.174}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,\,pa-xbaa-\omega bab,\,pb,\,pc,\,%
\text{class }3\rangle\;(\text{all }x,\, x\sim -x)  \tag{6.175}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,\,pa-baa,\,pb,\,pc,\,\text{class }%
3\rangle  \tag{6.176}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,\,pa-\omega baa,\,pb,\,pc,\,\text{
class }3\rangle\;(p=1 \func{mod} 4)  \tag{6.177}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb-\omega
baa,\,pa-tbaa-xbab,\,pb-ybaa-zbab,\,pc,\,\text{class }3\rangle  \tag{6.178}
\end{equation}%
Algebra 6.178 has four parameters $x,y,z,t$. If we write the parameters in a
matrix $A=\left( 
\begin{array}{cc}
t & x \\ 
y & z%
\end{array}%
\right) $, then two such parameter matrices $A$ and $B$ define isomorphic
algebras if and only if%
\[
B=\frac{1}{\det P}PAP^{-1}\func{mod}p 
\]%
for some matrix $P$ of the form%
\[
\left( 
\begin{array}{ll}
\alpha & \beta \\ 
\omega \beta & \alpha%
\end{array}%
\right) \text{ or }\left( 
\begin{array}{ll}
\alpha & \beta \\ 
-\omega \beta & -\alpha%
\end{array}%
\right) . 
\]

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,\,pa-xbaa,\,pb-ybaa,\,pc-bab,\,%
\text{class }3\rangle\;(\text{all }x,\,y,\, x\sim -x)  \tag{6.179}
\end{equation}

\subsection{Descendants of 4.6}

2 algebras

\begin{equation}
\langle a,b\,|\,ba,\,\text{class }3\rangle  \tag{6.366}
\end{equation}

\begin{equation}
\langle a,b\,|\,ba-p^2a,\,\text{class }3\rangle  \tag{6.367}
\end{equation}

\subsection{Descendants of 4.7}

$p+15$ algebras

\begin{equation}
\langle a,b\,|\,p^2a,\,pb,\,\text{class }3\rangle  \tag{6.368}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a,\,pb-bab,\,\text{class }3\rangle  \tag{6.369}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a,\,pb-baa,\,\text{class }3\rangle  \tag{6.370}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a,\,pb-\omega baa,\,\text{class }3\rangle  \tag{6.371}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-bab,\,pb,\,\text{class }3\rangle  \tag{6.372}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-bab,\,pb-baa,\,\text{class }3\rangle  \tag{6.373}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-bab,\,pb-\omega baa,\,\text{class }3\rangle  \tag{6.374}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-\omega bab,\,pb,\,\text{class }3\rangle  \tag{6.375}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-\omega bab,\,pb-baa,\,\text{class }3\rangle  \tag{6.376}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-\omega bab,\,pb-\omega baa,\,\text{class }3\rangle 
\tag{6.377}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-baa,\,pb-xbab,\,\text{class }3\rangle\;(\text{all }x) 
\tag{6.378}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,\,pb,\,\text{class }3\rangle  \tag{6.379}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,\,pb-bab,\,\text{class }3\rangle  \tag{6.380}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,pb,\,\text{class }3\rangle  \tag{6.381}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,pb-baa,\,\text{class }3\rangle  \tag{6.382}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,pb-\omega baa,\,\text{class }3\rangle  \tag{6.383}
\end{equation}

\subsection{Descendants of 4.8}

1 algebra

\begin{equation}
\langle a,b\,|\,pb-ba,\,\text{class }3\rangle  \tag{6.384}
\end{equation}

\subsection{Descendants of 5.1}

7 algebras

\begin{equation}
\langle
a,b,c,d,\,e\,|\,ba,\,ca,\,da,\,ea,\,cb,\,db,\,eb,\,dc,\,ec,\,ed,\,pb,\,pc,%
\,pd,\,pe,\,\text{class }2\rangle  \tag{6.2}
\end{equation}

\begin{equation}
\langle
a,b,c,d,\,e\,|\,ca,\,da,\,ea,\,cb,\,db,\,eb,\,dc,\,ec,\,ed,\,pa,\,pb,\,pc,%
\,pd,\,pe,\,\text{class }2\rangle  \tag{6.3}
\end{equation}

\begin{equation}
\langle
a,b,c,d,\,e\,|\,ca,\,da,\,ea,\,cb,\,db,\,eb,\,dc,\,ec,\,ed,\,pa-ba,\,pb,%
\,pc,\,pd,\,pe,\,\text{class }2\rangle  \tag{6.4}
\end{equation}

\begin{equation}
\langle
a,b,c,d,\,e\,|\,ca,\,da,\,ea,\,cb,\,db,\,eb,\,dc,\,ec,\,ed,\,pa,\,pb,%
\,pc-ba,\,pd,\,pe,\,\text{class }2\rangle  \tag{6.5}
\end{equation}

\begin{equation}
\langle
a,b,c,d,\,e\,|\,ca,\,da,\,ea,\,cb,\,db,\,eb,\,dc-ba,\,ec,\,ed,\,pa,\,pb,%
\,pc,\,pd,\,pe,\,\text{class }2\rangle  \tag{6.6}
\end{equation}

\begin{equation}
\langle
a,b,c,d,\,e\,|\,ca,\,da,\,ea,\,cb,\,db,\,eb,\,dc-ba,\,ec,\,ed,\,pa-ba,\,pb,%
\,pc,\,pd,\,pe,\,\text{class }2\rangle  \tag{6.7}
\end{equation}

\begin{equation}
\langle
a,b,c,d,\,e\,|\,ca,\,da,\,ea,\,cb,\,db,\,eb,\,dc-ba,\,ec,\,ed,\,pa,\,pb,%
\,pc,\,pd,\,pe-ba,\,\text{class }2\rangle  \tag{6.8}
\end{equation}

\subsection{Descendants of 5.2}

4 algebras

\begin{equation}
\langle a,b,c,d\,|\,ba,\,ca,\,da,\,cb,\,db,\,dc,\,pb,\,pc,\,pd,\,\text{
class }3\rangle  \tag{6.63}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ba-p^2a,\,ca,\,da,\,cb,\,db,\,dc,\,pb,\,pc,\,pd,\,\text{
class }3\rangle  \tag{6.64}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ba,\,ca,\,da,\,cb-p^2a,\,db,\,dc,\,pb,\,pc,\,pd,\,\text{
class }3\rangle  \tag{6.65}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ba,\,ca,\,da-p^2a,\,cb-p^2a,\,db,\,dc,\,pb,\,pc,\,pd,\,%
\text{class }3\rangle  \tag{6.66}
\end{equation}

\subsection{Descendants of 5.3}

18 algebras

\begin{equation}
\langle a,b,c,d\,|\,bab,\,ca,\,cb,\,da,\,db,\,dc,\,pa,\,pb,\,pc,\,pd,\,\text{
class }3\rangle  \tag{6.67}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,\,ca,\,cb,\,da,\,db,\,dc,\,pa-baa,\,pb,\,pc,\,pd,\,%
\text{class }3\rangle  \tag{6.68}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,\,ca,\,cb,\,da,\,db,\,dc,\,pa,\,pb-baa,\,pc,\,pd,\,%
\text{class }3\rangle  \tag{6.69}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,\,ca,\,cb,\,da,\,db,\,dc,\,pa,\,pb-\omega
baa,\,pc,\,pd,\,\text{class }3\rangle  \tag{6.70}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,\,ca,\,cb,\,da,\,db,\,dc,\,pa,\,pb,\,pc-baa,\,pd,\,%
\text{class }3\rangle  \tag{6.71}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,\,cb-baa,\,bab,\,da,\,db,\,dc,\,pa,\,pb,\,pc,\,pd,\,%
\text{class }3\rangle  \tag{6.72}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,ca,\,cb-baa,\,bab,\,da,\,db,\,dc,\,pa-baa,\,pb,\,pc,\,pd,\,\text{
class }3\rangle  \tag{6.73}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,ca,\,cb-baa,\,bab,\,da,\,db,\,dc,\,pa,\,pb-baa,\,pc,\,pd,\,\text{
class }3\rangle  \tag{6.74}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,\,cb-baa,\,bab,\,da,\,db,\,dc,\,pa,\,pb-\omega
baa,\,pc,\,pd,\,\text{class }3\rangle  \tag{6.75}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,ca,\,cb-baa,\,bab,\,da,\,db,\,dc,\,pa,\,pb,\,pc-baa,\,pd,\,\text{
class }3\rangle  \tag{6.76}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,ca,\,cb-baa,\,bab,\,da,\,db,\,dc,\,pa,\,pb,\,pc,\,pd-baa,\,\text{
class }3\rangle  \tag{6.77}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,\,ca,\,cb,\,da,\,db,\,dc-baa,\,pa,\,pb,\,pc,\,pd,\,%
\text{class }3\rangle  \tag{6.78}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,bab,\,ca,\,cb,\,da,\,db,\,dc-baa,\,pa-baa,\,pb,\,pc,\,pd,\,\text{
class }3\rangle  \tag{6.79}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,bab,\,ca,\,cb,\,da,\,db,\,dc-baa,\,pa,\,pb-baa,\,pc,\,pd,\,\text{
class }3\rangle  \tag{6.80}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,\,ca,\,cb,\,da,\,db,\,dc-baa,\,pa,\,pb-\omega
baa,\,pc,\,pd,\,\text{class }3\rangle  \tag{6.81}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,bab,\,ca,\,cb,\,da,\,db,\,dc-baa,\,pa,\,pb,\,pc-baa,\,pd,\,\text{
class }3\rangle  \tag{6.82}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,bab,\,ca,\,cb,\,da,\,db,\,dc-baa,\,pa,\,pb-baa,\,pc-baa,\,pd,\,%
\text{class }3\rangle  \tag{6.83}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,\,ca,\,cb,\,da,\,db,\,dc-baa,\,pa,\,pb-\omega
baa,\,pc-baa,\,pd,\,\text{class }3\rangle  \tag{6.84}
\end{equation}

\subsection{Descendants of 5.8}

4 algebras

\begin{equation}
\langle a,b,c\,|\,ba,\,ca,\,cb-p^2a,\,p^2b,\,pc,\,\text{class }3\rangle 
\tag{6.180}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba,\,ca-p^2a,\,cb,\,p^2b,\,pc,\,\text{class }3\rangle 
\tag{6.181}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba,\,ca,\,cb,\,p^2b,\,pc,\,\text{class }3\rangle 
\tag{6.182}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba-p^2a,\,ca,\,cb,\,p^2b,\,pc,\,\text{class }3\rangle 
\tag{6.183}
\end{equation}

\subsection{Descendants of 5.9}

23 algebras

\begin{equation}
\langle a,b,c\,|\,baa,\,bab,\,ca,\,cb,\,pb,\,pc,\,\text{class }3\rangle 
\tag{6.184}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,\,bab,\,ca-p^2a,\,cb,\,pb,\,pc,\,\text{class }3\rangle
\tag{6.185}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,\,bab,\,ca,\,cb-p^2a,\,pb,\,pc,\,\text{class }3\rangle
\tag{6.186}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb,\,p^2a,\,pb,\,pc,\,\text{class }3\rangle 
\tag{6.187}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,p^2a,\,pb,\,pc,\,\text{class }3\rangle
\tag{6.188}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb,\,p^2a,\,pb-baa,\,pc,\,\text{class }3\rangle
\tag{6.189}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,p^2a,\,pb-baa,\,pc,\,\text{class }%
3\rangle  \tag{6.190}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb,\,p^2a,\,pb-\omega baa,\,pc,\,\text{class }%
3\rangle  \tag{6.191}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,p^2a,\,pb-\omega baa,\,pc,\,\text{
class }3\rangle  \tag{6.192}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb,\,p^2a,\,pb,\,pc-baa,\,\text{class }3\rangle
\tag{6.193}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,p^2a,\,pb,\,pc-baa,\,\text{class }%
3\rangle  \tag{6.194}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb,\,p^2a-baa,\,pb,\,pc,\,\text{class }3\rangle
\tag{6.195}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,p^2a-baa,\,pb,\,pc,\,\text{class }%
3\rangle  \tag{6.196}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,\,ca,\,cb,\,p^2a,\,pb,\,pc,\,\text{class }3\rangle 
\tag{6.197}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,\,ca-bab,\,cb,\,p^2a,\,pb,\,pc,\,\text{class }3\rangle
\tag{6.198}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,\,ca,\,cb,\,p^2a,\,pb-bab,\,pc,\,\text{class }3\rangle
\tag{6.199}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,\,ca-bab,\,cb,\,p^2a,\,pb-bab,\,pc,\,\text{class }%
3\rangle  \tag{6.200}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,\,ca,\,cb,\,p^2a,\,pb,\,pc-bab,\,\text{class }3\rangle
\tag{6.201}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,\,ca-bab,\,cb,\,p^2a,\,pb,\,pc-bab,\,\text{class }%
3\rangle  \tag{6.202}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,\,ca,\,cb,\,p^2a-bab,\,pb,\,pc,\,\text{class }3\rangle
\tag{6.203}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,\,ca-bab,\,cb,\,p^2a-bab,\,pb,\,pc,\,\text{class }%
3\rangle  \tag{6.204}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,\,ca,\,cb,\,p^2a-\omega bab,\,pb,\,pc,\,\text{class }%
3\rangle  \tag{6.205}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,\,ca-bab,\,cb,\,p^2a-\omega bab,\,pb,\,pc,\,\text{
class }3\rangle  \tag{6.206}
\end{equation}

\subsection{Descendants of 5.10}

5 algebras

\begin{equation}
\langle a,b,c\,|\,baa,\,ca,\,cb,\,pb-ba,\,pc,\,\text{class }3\rangle 
\tag{6.207}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,\,ca-p^2a,\,cb,\,pb-ba,\,pc,\,\text{class }3\rangle 
\tag{6.208}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,\,ca,\,cb-p^2a,\,pb-ba,\,pc,\,\text{class }3\rangle 
\tag{6.209}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,p^2a,\,pb-ba,\,pc,\,\text{class }3\rangle 
\tag{6.210}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,p^2a,\,pb-ba,\,pc-baa,\,\text{class }3\rangle 
\tag{6.211}
\end{equation}

\subsection{Descendants of 5.11}

4 algebras

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pb,\,pc-ba,\,\text{class }3\rangle  \tag{6.212}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb-p^2a,\,pb,\,pc-ba,\,\text{class }3\rangle 
\tag{6.213}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb-\omega p^2a,\,pb,\,pc-ba,\,\text{class }3\rangle 
\tag{6.214}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-p^2a,\,cb,\,pb,\,pc-ba,\,\text{class }3\rangle 
\tag{6.215}
\end{equation}

\subsection{Descendants of 5.12}

12 algebras

\begin{equation}
\langle a,b,c\,|\,baa,\,bab,\,ca,\,cb,\,pa,\,pb,\,\text{class }3\rangle 
\tag{6.216}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,\,bab,\,ca-p^2c,\,cb,\,pa,\,pb,\,\text{class }3\rangle
\tag{6.217}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb,\,pa,\,pb,\,p^2c,\,\text{class }3\rangle 
\tag{6.218}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb,\,pa-baa,\,pb,\,p^2c,\,\text{class }3\rangle
\tag{6.219}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb,\,pa,\,pb-baa,\,p^2c,\,\text{class }3\rangle
\tag{6.220}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb,\,pa,\,pb-\omega baa,\,p^2c,\,\text{class }%
3\rangle  \tag{6.221}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,pa,\,pb,\,p^2c,\,\text{class }3\rangle
\tag{6.222}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,pa-baa,\,pb,\,p^2c,\,\text{class }%
3\rangle  \tag{6.223}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,pa,\,pb-baa,\,p^2c,\,\text{class }%
3\rangle  \tag{6.224}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,pa,\,pb-\omega baa,\,p^2c,\,\text{
class }3\rangle  \tag{6.225}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb,\,pa,\,pb,\,p^2c-baa,\,\text{class }3\rangle
\tag{6.226}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,pa,\,pb,\,p^2c-baa,\,\text{class }%
3\rangle  \tag{6.227}
\end{equation}

\subsection{Descendants of 5.13}

$p+1$ algebras

\begin{equation}
\langle a,b,c\,|\,ca,\,cb,\,pa-ba,\,pb,\,\text{class }3\rangle  \tag{6.228}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-p^2c,\,cb,\,pa-ba,\,pb,\,\text{class }3\rangle 
\tag{6.229}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,\,cb-xp^2c,\,pa-ba,\,pb,\,\text{class }3\rangle\;(x
\neq 0)  \tag{6.230}
\end{equation}

\subsection{Descendants of 5.14}

35 algebras

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac,\,pa,\,pb,\,pc,\,\text{class }%
3\rangle  \tag{6.231}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac,\,pa,\,pb-bab,\,pc,\,\text{
class }3\rangle  \tag{6.232}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac,\,pa-bab,\,pb,\,pc,\,\text{
class }3\rangle  \tag{6.233}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac,\,pa-bab,\,pb-bab,\,pc,\,\text{
class }3\rangle  \tag{6.234}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac,\,pa-\omega bab,\,pb,\,pc,\,%
\text{class }3\rangle  \tag{6.235}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac,\,pa-\omega bab,\,pb-bab,\,pc,\,%
\text{class }3\rangle  \tag{6.236}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac,\,pa,\,pb,\,pc-bab,\,\text{
class }3\rangle  \tag{6.237}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa-bab,\,cac,\,pa,\,pb,\,pc,\,\text{
class }3\rangle  \tag{6.238}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa-bab,\,cac,\,pa-bab,\,pb,\,pc,\,\text{
class }3\rangle  \tag{6.239}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa-bab,\,cac,\,pa-\omega bab,\,pb,\,pc,\,%
\text{class }3\rangle  \tag{6.240}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa-bab,\,cac,\,pa,\,pb-bab,\,pc,\,\text{
class }3\rangle  \tag{6.241}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa-bab,\,cac,\,pa,\,pb,\,pc-bab,\,\text{
class }3\rangle  \tag{6.242}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa-bab,\,cac,\,pa,\,pb,\,pc-\omega bab,\,%
\text{class }3\rangle  \tag{6.243}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac+bab,\,pa,\,pb,\,pc,\,\text{
class }3\rangle  \tag{6.244}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac+bab,\,pa-bab,\,pb,\,pc,\,\text{
class }3\rangle  \tag{6.245}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac+bab,\,pa,\,pb-bab,\,pc,\,\text{
class }3\rangle  \tag{6.246}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac+bab,\,pa-bab,\,pb-bab,\,pc,\,%
\text{class }3\rangle  \tag{6.247}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac+bab,\,pa-\omega
bab,\,pb-bab,\,pc,\,\text{class }3\rangle  \tag{6.248}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac+bab,\,pa,\,pb-bab,\,pc-bab,\,%
\text{class }3\rangle  \tag{6.249}
\end{equation}

\begin{equation}
\langle
a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac+bab,\,pa-bab,\,pb-bab,\,pc-bab,\,\text{
class }3\rangle  \tag{6.250}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac+\omega bab,\,pa,\,pb,\,pc,\,%
\text{class }3\rangle  \tag{6.251}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac+\omega bab,\,pa-bab,\,pb,\,pc,\,%
\text{class }3\rangle  \tag{6.252}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac+\omega bab,\,pa,\,pb-bab,\,pc,\,%
\text{class }3\rangle  \tag{6.253}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac+\omega
bab,\,pa-bab,\,pb-bab,\,pc,\,\text{class }3\rangle  \tag{6.254}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,bac,\,caa,\,cac+\omega bab,\,pa-\omega
bab,\,pb-bab,\,pc,\,\text{class }3\rangle  \tag{6.255}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab,\,bac,\,caa,\,cac,\,pa,\,pb,\,pc,\,\text{class }%
3\rangle  \tag{6.256}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab,\,bac,\,caa,\,cac,\,pa-baa,\,pb,\,pc,\,\text{
class }3\rangle  \tag{6.257}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab,\,bac,\,caa,\,cac,\,pa,\,pb-baa,\,pc,\,\text{
class }3\rangle  \tag{6.258}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab,\,bac,\,caa,\,cac,\,pa,\,pb-\omega baa,\,pc,\,%
\text{class }3\rangle  \tag{6.259}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab,\,bac,\,caa,\,cac,\,pa,\,pb,\,pc-baa,\,\text{
class }3\rangle  \tag{6.260}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,\,bab,\,bac,\,caa,\,cac,\,pa,\,pb,\,pc,\,\text{
class }3\rangle  \tag{6.261}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,\,bab,\,bac,\,caa,\,cac,\,pa-baa,\,pb,\,pc,\,\text{
class }3\rangle  \tag{6.262}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,\,bab,\,bac,\,caa,\,cac,\,pa,\,pb-baa,\,pc,\,\text{
class }3\rangle  \tag{6.263}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,\,bab,\,bac,\,caa,\,cac,\,pa,\,pb-\omega baa,\,pc,\,%
\text{class }3\rangle  \tag{6.264}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,\,bab,\,bac,\,caa,\,cac,\,pa,\,pb,\,pc-baa,\,\text{
class }3\rangle  \tag{6.265}
\end{equation}

\subsection{Descendants of 5.15}

$2p+13$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,\,caa,\,pa-ba,\,pb-cac,\,pc,\,\text{class }3\rangle 
\tag{6.266}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,caa,\,pa-ba,\,pb,\,pc,\,\text{class }3\rangle 
\tag{6.267}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,caa,\,pa-ba,\,pb,\,pc-cac,\,\text{class }3\rangle 
\tag{6.268}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,caa,\,pa-ba-cac,\,pb,\,pc,\,\text{class }3\rangle 
\tag{6.269}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,caa,\,pa-ba-cac,\,pb,\,pc-cac,\,\text{class }3\rangle
\tag{6.270}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,caa,\,pa-ba-\omega cac,\,pb,\,pc,\,\text{class }%
3\rangle  \tag{6.271}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,caa,\,pa-ba-\omega cac,\,pb,\,pc-cac,\,\text{class }%
3\rangle  \tag{6.272}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,cac,\,pa-ba,\,pb-caa,\,pc,\,\text{class }3\rangle 
\tag{6.273}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,cac,\,pa-ba,\,pb,\,pc,\,\text{class }3\rangle 
\tag{6.274}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,cac,\,pa-ba,\,pb,\,pc-caa,\,\text{class }3\rangle 
\tag{6.275}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,cac,\,pa-ba,\,pb,\,pc-\omega caa,\,\text{class }%
3\rangle  \tag{6.276}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,\,cac,\,pa-ba,\,pb-caa,\,pc,\,\text{class }3\rangle
\tag{6.277}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,\,cac,\,pa-ba,\,pb,\,pc-xcaa,\,\text{class }%
3\rangle\;(\text{all }x)  \tag{6.278}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,\,cac,\,pa-ba,\,pb-caa,\,pc,\,\text{class }%
3\rangle  \tag{6.279}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,\,cac,\,pa-ba,\,pb,\,pc-xcaa,\,\text{class }%
3\rangle\;(\text{all }x)  \tag{6.280}
\end{equation}

\subsection{Descendants of 5.16}

$4p+8$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,\,caa,\,pa,\,pb-ba,\,pc,\,\text{class }3\rangle 
\tag{6.281}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,caa,\,pa,\,pb-ba-cac,\,pc,\,\text{class }3\rangle 
\tag{6.282}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,caa,\,pa-cac,\,pb-ba,\,pc,\,\text{class }3\rangle 
\tag{6.283}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,caa,\,pa-cac,\,pb-ba-cac,\,pc,\,\text{class }3\rangle
\tag{6.284}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,caa,\,pa-\omega cac,\,pb-ba,\,pc,\,\text{class }%
3\rangle  \tag{6.285}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,caa,\,pa-\omega cac,\,pb-ba-cac,\,pc,\,\text{class }%
3\rangle  \tag{6.286}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,caa,\,pa-xcac,\,pb-ba,\,pc-cac,\,\text{class }%
3\rangle\;(\text{all }x)  \tag{6.287}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,caa,\,pa-xcac,\,pb-ba-cac,\,pc-cac,\,\text{class }%
3\rangle\;(\text{all }x)  \tag{6.288}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,cac,\,pa,\,pb-ba,\,pc-xcaa,\,\text{class }3\rangle\;(%
\text{all }x)  \tag{6.289}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,\,cac,\,pa,\,pb-ba,\,pc-xcaa,\,\text{class }%
3\rangle\;(\text{all }x)  \tag{6.290}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,cac,\,pa-caa,\,pb-ba,\,pc,\,\text{class }3\rangle 
\tag{6.291}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,\,cac,\,pa-caa,\,pb-ba,\,pc,\,\text{class }3\rangle
\tag{6.292}
\end{equation}

\subsection{Descendants of 5.18}

$2p+13+3\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,pa,\,pb-ca,\,pc-bab,\,\text{class }3\rangle 
\tag{6.293}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,pa,\,pb-ca,\,pc,\,\text{class }3\rangle 
\tag{6.294}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,pa-bab,\,pb-ca,\,pc,\,\text{class }3\rangle 
\tag{6.295}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,pa-\omega bab,\,pb-ca,\,pc,\,\text{class }%
3\rangle  \tag{6.296}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,pa,\,pb-ca-bab,\,pc,\,\text{class }3\rangle 
\tag{6.297}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,pa-bab,\,pb-ca-bab,\,pc,\,\text{class }3\rangle
\tag{6.298}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,baa,\,pa-\omega bab,\,pb-ca-bab,\,pc,\,\text{class }%
3\rangle  \tag{6.299}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab-baa,\,pa,\,pb-ca,\,pc-baa,\,\text{class }3\rangle
\tag{6.300}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab-baa,\,pa,\,pb-ca,\,pc-\omega baa,\,\text{class }%
3\rangle\;(p=1 \func{mod} 3)  \tag{6.301}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab-baa,\,pa,\,pb-ca,\,pc-\omega ^2baa,\,\text{ class 
}3\rangle\;(p=1 \func{mod} 3)  \tag{6.302}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab-baa,\,pa,\,pb-ca,\,pc,\,\text{class }3\rangle 
\tag{6.303}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab-baa,\,pa,\,pb-ca-baa,\,pc,\,\text{class }3\rangle
\tag{6.304}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab-baa,\,pa,\,pb-ca-\omega baa,\,pc,\,\text{class }%
3\rangle  \tag{6.305}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab-baa,\,pa-baa,\,pb-ca-xbaa,\,pc,\,\text{class }%
3\rangle\;(\text{all }x)  \tag{6.305B}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab-baa,\,pa-\omega baa,\,pb-ca-xbaa,\,pc,\,\text{
class }3\rangle\;(\text{all }x)  \tag{6.305C}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab,\,pa,\,pb-ca,\,pc-baa,\,\text{class }3\rangle 
\tag{6.306}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab,\,pa,\,pb-ca,\,pc-\omega baa,\,\text{class }%
3\rangle\;(p=1 \func{mod} 3)  \tag{6.307}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab,\,pa,\,pb-ca,\,pc-\omega ^2baa,\,\text{class }%
3\rangle\;(p=1 \func{mod} 3)  \tag{6.308}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,\,bab,\,pa,\,pb-ca,\,pc-baa,\,\text{class }3\rangle
\tag{6.309}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,\,bab,\,pa,\,pb-ca,\,pc-\omega baa,\,\text{class }%
3\rangle\;(p=1 \func{mod} 3)  \tag{6.310}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,\,bab,\,pa,\,pb-ca,\,pc-\omega ^2baa,\,\text{ class 
}3\rangle\;(p=1 \func{mod} 3)  \tag{6.311}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab,\,pa,\,pb-ca,\,pc,\,\text{class }3\rangle 
\tag{6.312}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,\,bab,\,pa,\,pb-ca,\,pc,\,\text{class }3\rangle 
\tag{6.313}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,\,bab,\,pa-baa,\,pb-ca,\,pc,\,\text{class }3\rangle 
\tag{6.314}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,\,bab,\,pa-baa,\,pb-ca,\,pc,\,\text{class }3\rangle
\tag{6.315}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega baa,\,bab,\,pa-baa,\,pb-ca,\,pc,\,\text{class }%
3\rangle  \tag{6.316}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega ^2baa,\,bab,\,pa-baa,\,pb-ca,\,pc,\,\text{ class 
}3\rangle\;(p=1 \func{mod} 4)  \tag{6.317}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega ^3baa,\,bab,\,pa-baa,\,pb-ca,\,pc,\,\text{ class 
}3\rangle\;(p=1 \func{mod} 4)  \tag{6.318}
\end{equation}

\subsection{Descendants of 5.19}

3 algebras

\begin{equation}
\langle a,b,c\,|\,cb,\,pa-ba,\,pb-ca,\,pc,\,\text{class }3\rangle 
\tag{6.319}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-bab,\,pa-ba,\,pb-ca,\,pc,\,\text{class }3\rangle 
\tag{6.320}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega bab,\,pa-ba,\,pb-ca,\,pc,\,\text{class }3\rangle
\tag{6.321}
\end{equation}

\subsection{Descendants of 5.24}

3 algebras

\begin{equation}
\langle a,b,c\,|\,ba,\,ca,\,cb,\,pb,\,pc,\,\text{class }4\rangle  \tag{6.322}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba-p^3a,\,ca,\,cb,\,pb,\,pc,\,\text{class }4\rangle 
\tag{6.323}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba,\,ca,\,cb-p^3a,\,pb,\,pc,\,\text{class }4\rangle 
\tag{6.324}
\end{equation}

\subsection{Descendants of 5.27}

$11+4\gcd (p-1,3)+2\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb,\,pa,\,pb,\,pc,\,\text{class }4\rangle 
\tag{6.325}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,\,ca,\,cb,\,pa,\,pb,\,pc,\,\text{class }4\rangle 
\tag{6.326}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baaa,\,pa,\,pb,\,pc,\,\text{class }4\rangle 
\tag{6.327}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,\,ca,\,cb-baaa,\,pa,\,pb,\,pc,\,\text{class }%
4\rangle  \tag{6.328}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb,\,pa-baaa,\,pb,\,pc,\,\text{class }4\rangle 
\tag{6.329}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,\,ca,\,cb,\,pa-baaa,\,pb,\,pc,\,\text{class }%
4\rangle  \tag{6.330}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-\omega baaa,\,ca,\,cb,\,pa-baaa,\,pb,\,pc,\,\text{
class }4\rangle  \tag{6.331}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-\omega ^2baaa,\,ca,\,cb,\,pa-baaa,\,pb,\,pc,\,\text{
class }4\rangle\;(p=1 \func{mod} 4)  \tag{6.332}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-\omega ^3baaa,\,ca,\,cb,\,pa-baaa,\,pb,\,pc,\,\text{
class }4\rangle\;(p=1 \func{mod} 4)  \tag{6.333}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baaa,\,pa-baaa,\,pb,\,pc,\,\text{class }%
4\rangle  \tag{6.334}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,\,ca,\,cb-baaa,\,pa-baaa,\,pb,\,pc,\,\text{ class 
}4\rangle  \tag{6.335}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-\omega baaa,\,ca,\,cb-baaa,\,pa-baaa,\,pb,\,pc,\,\text{
class }4\rangle  \tag{6.336}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-\omega ^2baaa,\,ca,\,cb-baaa,\,pa-baaa,\,pb,\,pc,\,%
\text{class }4\rangle\;(p=1 \func{mod} 4)  \tag{6.337}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-\omega ^3baaa,\,ca,\,cb-baaa,\,pa-baaa,\,pb,\,pc,\,%
\text{class }4\rangle\;(p=1 \func{mod} 4)  \tag{6.338}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb,\,pa,\,pb-baaa,\,pc,\,\text{class }4\rangle 
\tag{6.339}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,\,ca,\,cb,\,pa,\,pb-baaa,\,pc,\,\text{class }%
4\rangle  \tag{6.340}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baaa,\,pa,\,pb-baaa,\,pc,\,\text{class }%
4\rangle  \tag{6.341}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,\,ca,\,cb-baaa,\,pa,\,pb-baaa,\,pc,\,\text{ class 
}4\rangle  \tag{6.342}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb,\,pa,\,pb-\omega baaa,\,pc,\,\text{class }%
4\rangle\;(p=1 \func{mod} 3)  \tag{6.343}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,\,ca,\,cb,\,pa,\,pb-\omega baaa,\,pc,\,\text{
class }4\rangle\;(p=1 \func{mod} 3)  \tag{6.344}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baaa,\,pa,\,pb-\omega baaa,\,pc,\,\text{
class }4\rangle\;(p=1 \func{mod} 3)  \tag{6.345}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,\,ca,\,cb-baaa,\,pa,\,pb-\omega baaa,\,pc,\,\text{
class }4\rangle\;(p=1 \func{mod} 3)  \tag{6.346}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb,\,pa,\,pb-\omega ^2baaa,\,pc,\,\text{class }%
4\rangle\;(p=1 \func{mod} 3)  \tag{6.347}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,\,ca,\,cb,\,pa,\,pb-\omega ^2baaa,\,pc,\,\text{
class }4\rangle\;(p=1 \func{mod} 3)  \tag{6.348}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baaa,\,pa,\,pb-\omega ^2baaa,\,pc,\,\text{
class }4\rangle\;(p=1 \func{mod} 3)  \tag{6.349}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,\,ca,\,cb-baaa,\,pa,\,pb-\omega ^2baaa,\,pc,\,%
\text{class }4\rangle\;(p=1 \func{mod} 3)  \tag{6.350}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb,\,pa,\,pb,\,pc-baaa,\,\text{class }4\rangle 
\tag{6.351}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baaa,\,pa,\,pb,\,pc-baaa,\,\text{class }%
4\rangle  \tag{6.352}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,\,ca,\,cb,\,pa,\,pb,\,pc-baaa,\,\text{class }%
4\rangle  \tag{6.353}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,\,ca,\,cb-baaa,\,pa,\,pb,\,pc-baaa,\,\text{ class 
}4\rangle  \tag{6.354}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,\,ca,\,cb-\omega baaa,\,pa,\,pb,\,pc-baaa,\,\text{
class }4\rangle  \tag{6.355}
\end{equation}

\subsection{Descendants of 5.32}

$4+2\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,pa,\,pb-baaa,\,pc,\,\text{class }%
4\rangle  \tag{6.356}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,pa,\,pb-\omega baaa,\,pc,\,\text{
class }4\rangle\;(p=1 \func{mod} 3)  \tag{6.357}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,pa,\,pb-\omega ^2baaa,\,pc,\,\text{
class }4\rangle\;(p=1 \func{mod} 3)  \tag{6.358}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,pa,\,pb-baaa,\,pc-baaa,\,\text{class }%
4\rangle  \tag{6.359}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,pa,\,pb-\omega baaa,\,pc-baaa,\,\text{
class }4\rangle\;(p=1 \func{mod} 3)  \tag{6.360}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,pa,\,pb-\omega ^2baaa,\,pc-baaa,\,%
\text{class }4\rangle\;(p=1 \func{mod} 3)  \tag{6.361}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,pa,\,pb,\,pc,\,\text{class }4\rangle 
\tag{6.362}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,pa-baaa,\,pb,\,pc,\,\text{class }%
4\rangle  \tag{6.363}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,pa,\,pb,\,pc-baaa,\,\text{class }%
4\rangle  \tag{6.364}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,\,ca,\,cb-baa,\,pa-baaa,\,pb,\,pc-baaa,\,\text{class }%
4\rangle  \tag{6.365}
\end{equation}

\subsection{Descendants of 5.37}

$p+8$ algebras

\begin{equation}
\langle a,b\,|\,baa,\,bab,\,p^2a,\,p^2b,\,\text{class }3\rangle  \tag{6.385}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,\,bab,\,pba,\,p^2b,\,\text{class }3\rangle  \tag{6.386}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,\,bab,\,p^2a-pba,\,p^2b,\,\text{class }3\rangle 
\tag{6.387}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,pba,\,p^2a,\,p^2b,\,\text{class }3\rangle  \tag{6.388}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,pba,\,p^2a-baa,\,p^2b,\,\text{class }3\rangle 
\tag{6.389}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,pba,\,p^2a,\,p^2b-baa,\,\text{class }3\rangle 
\tag{6.390}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,pba,\,p^2a,\,p^2b-\omega baa,\,\text{class }3\rangle 
\tag{6.391}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,pba-baa,\,p^2a,\,p^2b-xbaa,\,\text{class }3\rangle\;(%
\text{all }x)  \tag{6.392}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,pba-baa,\,p^2a-baa,\,p^2b,\,\text{class }3\rangle 
\tag{6.393}
\end{equation}

\subsection{Descendants of 5.38}

$5+3\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b\,|\,baab,\,babb,\,pa,\,pb,\,\text{class }4\rangle  \tag{6.394}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,\,babb,\,pa-baaa,\,pb,\,\text{class }4\rangle 
\tag{6.395}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,\,babb,\,pa,\,pb-baaa,\,\text{class }4\rangle 
\tag{6.396}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,\,babb,\,pa,\,pb-\omega baaa,\,\text{class }%
4\rangle\;(p=1 \func{mod} 3)  \tag{6.397}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,\,babb,\,pa,\,pb-\omega ^2baaa,\,\text{class }%
4\rangle\;(p=1 \func{mod} 3)  \tag{6.398}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,\,babb+baaa,\,pa,\,pb,\,\text{class }4\rangle 
\tag{6.399}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,\,babb+baaa,\,pa-baaa,\,pb,\,\text{class }4\rangle 
\tag{6.400}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,\,babb+baaa,\,pa-\omega baaa,\,pb,\,\text{class }%
4\rangle\;(p=1 \func{mod} 3)  \tag{6.401}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,\,babb+baaa,\,pa-\omega ^2baaa,\,pb,\,\text{class }%
4\rangle\;(p=1 \func{mod} 3)  \tag{6.402}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,\,babb+baaa,\,pa-baaa,\,pb-baaa,\,\text{class }4\rangle
\tag{6.403}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,\,babb+\omega baaa,\,pa,\,pb,\,\text{class }4\rangle 
\tag{6.404}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,\,babb+\omega baaa,\,pa-baaa,\,pb,\,\text{class }%
4\rangle  \tag{6.405}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,\,babb+\omega baaa,\,pa-\omega baaa,\,pb,\,\text{ class 
}4\rangle\;(p=1 \func{mod} 3)  \tag{6.406}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,\,babb+\omega baaa,\,pa-\omega ^2baaa,\,pb,\,\text{
class }4\rangle\;(p=1 \func{mod} 3)  \tag{6.407}
\end{equation}

\subsection{Descendants of 5.39}

$1+\gcd (p-1,3)+(1/2)\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b\,|\,pa-bab,\,pb-baaa,\,\text{class }4\rangle  \tag{6.408}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-bab,\,pb-\omega baaa,\,\text{class }4\rangle\;(p=1 \func{%
mod} 3)  \tag{6.409}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-bab,\,pb-\omega ^2baaa,\,\text{class }4\rangle\;(p=1 
\func{mod} 3)  \tag{6.410}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-bab,\,pb,\,\text{class }4\rangle  \tag{6.411}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-bab-baaa,\,pb,\,\text{class }4\rangle  \tag{6.412}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-bab-\omega baaa,\,pb,\,\text{class }4\rangle\;(p=1 \func{%
mod} 4)  \tag{6.413}
\end{equation}

\subsection{Descendants of 5.40}

$1+\gcd (p-1,3)+(1/2)\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b\,|\,pa-\omega bab,\,pb-baaa,\,\text{class }4\rangle  \tag{6.414}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-\omega bab,\,pb-\omega baaa,\,\text{class }4\rangle\;(p=1 
\func{mod} 3)  \tag{6.415}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-\omega bab,\,pb-\omega ^2baaa,\,\text{class }%
4\rangle\;(p=1 \func{mod} 3)  \tag{6.416}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-\omega bab,\,pb,\,\text{class }4\rangle  \tag{6.417}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-\omega bab-baaa,\,pb,\,\text{class }4\rangle  \tag{6.418}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-\omega bab-\omega baaa,\,pb,\,\text{class }4\rangle\;(p=1 
\func{mod} 4)  \tag{6.419}
\end{equation}

\subsection{Descendants of 5.41}

$p+1+\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b\,|\,pa-baa,\,pb,\,\text{class }4\rangle  \tag{6.420}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-baa-babb,\,pb,\,\text{class }4\rangle  \tag{6.421}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-baa-\omega babb,\,pb,\,\text{class }4\rangle\;(p=1 \func{%
mod} 3)  \tag{6.422}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-baa-\omega ^2babb,\,pb,\,\text{class }4\rangle\;(p=1 
\func{mod} 3)  \tag{6.423}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-baa-xbabb,\,pb-babb,\,\text{class }4\rangle \;(\text{all }%
x)  \tag{6.424}
\end{equation}

\subsection{Descendants of 5.42}

$p+1$ algebras

\begin{equation}
\langle a,b\,|\,pa-baa,\,pb+bab,\,\text{class }4\rangle  \tag{6.425}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-baa-baab,\,pb+bab-xbaab,\,\text{class }4\rangle\;(\text{%
all }x)  \tag{6.426}
\end{equation}

\subsection{Descendants of 5.45}

Here we have a two parameter family of $p$ algebras, with the isomorphism
type depending on the value of $y^{2}-\omega x^{2}\func{mod}p$.

\begin{equation}
\langle a,b\,|\,pa+bab-xbaaa,\,pb+\omega baa-ybaaa,\,\text{class }4\rangle 
\tag{6.427}
\end{equation}

\subsection{Descendants of 5.47}

2 algebras

\begin{equation}
\langle a,b\,|\,ba,\,p^2b,\,\text{class }4\rangle  \tag{6.428}
\end{equation}

\begin{equation}
\langle a,b\,|\,ba-p^3a,\,p^2b,\,\text{class }4\rangle  \tag{6.429}
\end{equation}

\subsection{Descendants of 5.48}

1 algebra

\begin{equation}
\langle a,b\,|\,ba-p^2a,\,p^2b,\,\text{class }4\rangle  \tag{6.430}
\end{equation}

\subsection{Descendants of 5.49}

4 algebras

\begin{equation}
\langle a,b\,|\,baa,\,bab,\,pb,\,\text{class }4\rangle  \tag{6.431}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-p^3a,\,bab,\,pb,\,\text{class }4\rangle  \tag{6.432}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,\,bab-p^3a,\,pb,\,\text{class }4\rangle  \tag{6.433}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,\,bab-\omega p^3a,\,pb,\,\text{class }4\rangle 
\tag{6.434}
\end{equation}

\subsection{Descendants of 5.50}

$3+2\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b\,|\,bab,\,p^2a,\,pb,\,\text{class }4\rangle  \tag{6.435}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,p^2a,\,pb,\,\text{class }4\rangle  \tag{6.436}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,p^2a-baaa,\,pb,\,\text{class }4\rangle  \tag{6.437}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,p^2a-baaa,\,pb,\,\text{class }4\rangle 
\tag{6.438}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-\omega baaa,\,p^2a-baaa,\,pb,\,\text{class }4\rangle 
\tag{6.439}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-\omega ^2baaa,\,p^2a-baaa,\,pb,\,\text{class }%
4\rangle\;(p=1 \func{mod} 4)  \tag{6.440}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-\omega ^3baaa,\,p^2a-baaa,\,pb,\,\text{class }%
4\rangle\;(p=1 \func{mod} 4)  \tag{6.441}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,p^2a,\,pb-baaa,\,\text{class }4\rangle  \tag{6.442}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,p^2a,\,pb-\omega baaa,\,\text{class }4\rangle\;(p=1 
\func{mod} 3)  \tag{6.443}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,p^2a,\,pb-\omega ^2baaa,\,\text{class }4\rangle\;(p=1 
\func{mod} 3)  \tag{6.444}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,p^2a,\,pb-baaa,\,\text{class }4\rangle 
\tag{6.445}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,p^2a,\,pb-\omega baaa,\,\text{class }%
4\rangle\;(p=1 \func{mod} 3)  \tag{6.446}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,p^2a,\,pb-\omega ^2baaa,\,\text{class }%
4\rangle\;(p=1 \func{mod} 3)  \tag{6.447}
\end{equation}

\subsection{Descendants of 5.51}

$3(p+1)/2$ algebras

\begin{equation}
\langle a,b\,|\,bab,\,p^2a,\,pb-baa-xbaaa,\,\text{class }4\rangle\;(\text{%
all }x,\, x\sim -x)  \tag{6.448}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,p^2a-baaa,\,pb-baa,\,\text{class }4\rangle  \tag{6.449}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,p^2a-xbaaa,\,pb-baa,\,\text{class }4\rangle\;(%
\text{all }x)  \tag{6.450}
\end{equation}

\subsection{Descendants of 5.52}

$3(p+1)/2$ algebras

\begin{equation}
\langle a,b\,|\,bab,\,p^2a,\,pb-\omega baa-xbaaa,\,\text{class }4\rangle\;(%
\text{all }x,\, x\sim -x)  \tag{6.451}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,p^2a-baaa,\,pb-\omega baa,\,\text{class }4\rangle 
\tag{6.452}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,p^2a-xbaaa,\,pb-\omega baa,\,\text{class }%
4\rangle\;(\text{all }x)  \tag{6.453}
\end{equation}

\subsection{Descendants of 5.54}

$3+2\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b\,|\,baa,\,p^2a,\,pb,\,\text{class }4\rangle  \tag{6.454}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,\,p^2a,\,pb-babb,\,\text{class }4\rangle  \tag{6.455}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,\,p^2a-babb,\,pb,\,\text{class }4\rangle  \tag{6.456}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,\,p^2a-\omega babb,\,pb,\,\text{class }4\rangle\;(p=1 
\func{mod} 3)  \tag{6.457}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,\,p^2a-\omega ^2babb,\,pb,\,\text{class }4\rangle\;(p=1 
\func{mod} 3)  \tag{6.458}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,\,p^2a,\,pb,\,\text{class }4\rangle  \tag{6.459}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,\,p^2a,\,pb-babb,\,\text{class }4\rangle 
\tag{6.460}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,\,p^2a,\,pb-\omega babb,\,\text{class }4\rangle 
\tag{6.461}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,\,p^2a,\,pb-\omega ^2babb,\,\text{class }%
4\rangle\;(p=1 \func{mod} 4)  \tag{6.462}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,\,p^2a,\,pb-\omega ^3babb,\,\text{class }%
4\rangle\;(p=1 \func{mod} 4)  \tag{6.463}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,\,p^2a-babb,\,pb,\,\text{class }4\rangle 
\tag{6.464}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,\,p^2a-\omega babb,\,pb,\,\text{class }%
4\rangle\;(p=1 \func{mod} 3)  \tag{6.465}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,\,p^2a-\omega ^2babb,\,pb,\,\text{class }%
4\rangle\;(p=1 \func{mod} 3)  \tag{6.466}
\end{equation}

\subsection{Descendants of 5.58}

2 algebras

\begin{equation}
\langle a,b\,|\,baa,\,pb-ba,\,\text{class }4\rangle  \tag{6.467}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-p^3a,\,pb-ba,\,\text{class }4\rangle  \tag{6.468}
\end{equation}

\subsection{Descendants of 5.60}

$7+2\gcd (p-1,3)+3\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b\,|\,bab,\,baaab,\,pa,\,pb,\,\text{class }5\rangle  \tag{6.469}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,baaab,\,pa-baaaa,\,pb,\,\text{class }5\rangle 
\tag{6.470}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,baaab,\,pa,\,pb-baaaa,\,\text{class }5\rangle 
\tag{6.471}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,baaab,\,pa,\,pb-\omega baaaa,\,\text{class }5\rangle 
\tag{6.472}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,baaab,\,pa,\,pb-\omega ^2baaaa,\,\text{class }%
5\rangle\;(p=1 \func{mod} 4)  \tag{6.473}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,baaab,\,pa,\,pb-\omega ^3baaaa,\,\text{class }%
5\rangle\;(p=1 \func{mod} 4)  \tag{6.474}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,\,baaab,\,pa,\,pb,\,\text{class }5\rangle 
\tag{6.475}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,\,baaab,\,pa-baaaa,\,pb,\,\text{class }5\rangle 
\tag{6.476}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,\,baaab,\,pa-\omega baaaa,\,pb,\,\text{class }%
5\rangle  \tag{6.477}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,\,baaab,\,pa-\omega ^2baaaa,\,pb,\,\text{class }%
5\rangle\;(p=1 \func{mod} 3)  \tag{6.478}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,\,baaab,\,pa-\omega ^3baaaa,\,pb,\,\text{class }%
5\rangle\;(p=1 \func{mod} 3)  \tag{6.479}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,\,baaab,\,pa-\omega ^4baaaa,\,pb,\,\text{class }%
5\rangle\;(p=1 \func{mod} 3)  \tag{6.480}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,\,baaab,\,pa-\omega ^5baaaa,\,pb,\,\text{class }%
5\rangle\;(p=1 \func{mod} 3)  \tag{6.481}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,\,baaab,\,pa,\,pb-baaaa,\,\text{class }5\rangle 
\tag{6.482}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,\,baaab,\,pa,\,pb-\omega baaaa,\,\text{class }%
5\rangle  \tag{6.483}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,\,baaab,\,pa,\,pb-\omega ^2baaaa,\,\text{class }%
5\rangle\;(p=1 \func{mod} 4)  \tag{6.484}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,\,baaab,\,pa,\,pb-\omega ^3baaaa,\,\text{class }%
5\rangle\;(p=1 \func{mod} 4)  \tag{6.485}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,baaaa,\,pa,\,pb,\,\text{class }5\rangle  \tag{6.486}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,baaaa,\,pa-baaab,\,pb,\,\text{class }5\rangle 
\tag{6.487}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,baaaa,\,pa-\omega baaab,\,pb,\,\text{class }5\rangle 
\tag{6.488}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,baaaa,\,pa,\,pb-baaab,\,\text{class }5\rangle 
\tag{6.489}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,baaaa,\,pa-baaab,\,pb-baaab,\,\text{class }5\rangle 
\tag{6.490}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,baaaa,\,pa-\omega baaab,\,pb-baaab,\,\text{class }%
5\rangle  \tag{6.491}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,baaaa,\,pa-\omega ^2baaab,\,pb-baaab,\,\text{class }%
5\rangle\;(p=1 \func{mod} 4)  \tag{6.492}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,\,baaaa,\,pa-\omega ^3baaab,\,pb-baaab,\,\text{class }%
5\rangle\;(p=1 \func{mod} 4)  \tag{6.493}
\end{equation}

\subsection{Descendants of 5.65}

$2\gcd (p-1,3)+\gcd (p-1,4)+2\gcd (p-1,5)$ algebras

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaaa,\,pa,\,pb,\,\text{class }5\rangle 
\tag{6.494}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaaa,\,pa,\,pb-baaab,\,\text{class }5\rangle 
\tag{6.495}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaaa,\,pa,\,pb-\omega baaab,\,\text{class }%
5\rangle\;(p=1 \func{mod} 5)  \tag{6.496}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaaa,\,pa,\,pb-\omega ^2baaab,\,\text{class }%
5\rangle\;(p=1 \func{mod} 5)  \tag{6.497}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaaa,\,pa,\,pb-\omega ^3baaab,\,\text{class }%
5\rangle\;(p=1 \func{mod} 5)  \tag{6.498}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaaa,\,pa,\,pb-\omega ^4baaab,\,\text{class }%
5\rangle\;(p=1 \func{mod} 5)  \tag{6.499}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaaa,\,pa-baaab,\,pb,\,\text{class }5\rangle 
\tag{6.500}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaaa,\,pa-\omega baaab,\,pb,\,\text{class }%
5\rangle  \tag{6.501}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaaa,\,pa-\omega ^2baaab,\,pb,\,\text{class }%
5\rangle\;(p=1 \func{mod} 3)  \tag{6.502}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaaa,\,pa-\omega ^3baaab,\,pb,\,\text{class }%
5\rangle\;(p=1 \func{mod} 3)  \tag{6.503}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaaa,\,pa-\omega ^4baaab,\,pb,\,\text{class }%
5\rangle\;(p=1 \func{mod} 3)  \tag{6.504}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaaa,\,pa-\omega ^{5}baaab,\,pb,\,\text{class }%
5\rangle \;(p=1\func{mod}3)  \tag{6.505}
\end{equation}

For the next 5 algebras, we have $x\neq 0,\,x\sim x^{\prime }$ if $%
x^{5}=x^{\prime 5}\func{mod}p$:

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaaa,\,pa-xbaaab,\,pb-baaab,\,\text{class }%
5\rangle  \tag{6.506}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaaa,\,pa-xbaaab,\,pb-\omega baaab,\,\text{
class }5\rangle \;(p=1\func{mod}5)  \tag{6.506A}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaaa,\,pa-xbaaab,\,pb-\omega ^{2}baaab,\,\text{
class }5\rangle \;(p=1\func{mod}5)  \tag{6.506A}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaaa,\,pa-xbaaab,\,pb-\omega ^{3}baaab,\,\text{
class }5\rangle \;(p=1\func{mod}5)  \tag{6.506A}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaaa,\,pa-xbaaab,\,pb-\omega ^{4}baaab,\,\text{
class }5\rangle \;(p=1\func{mod}5)  \tag{6.506A}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaab,\,pa,\,pb,\,\text{class }5\rangle 
\tag{6.507}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaab,\,pa,\,pb-baaaa,\,\text{class }5\rangle 
\tag{6.508}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaab,\,pa,\,pb-\omega baaaa,\,\text{class }%
5\rangle  \tag{6.509}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaab,\,pa,\,pb-\omega ^2baaaa,\,\text{class }%
5\rangle\;(p=1 \func{mod} 4)  \tag{6.510}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaab,\,pa,\,pb-\omega ^3baaaa,\,\text{class }%
5\rangle\;(p=1 \func{mod} 4)  \tag{6.511}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaab,\,pa-baaaa,\,pb,\,\text{class }5\rangle 
\tag{6.512}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaab,\,pa-\omega baaaa,\,pb,\,\text{class }%
5\rangle\;(p=1 \func{mod} 5)  \tag{6.513}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaab,\,pa-\omega ^2baaaa,\,pb,\,\text{class }%
5\rangle\;(p=1 \func{mod} 5)  \tag{6.514}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaab,\,pa-\omega ^3baaaa,\,pb,\,\text{class }%
5\rangle\;(p=1 \func{mod} 5)  \tag{6.515}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaab,\,pa-\omega ^{4}baaaa,\,pb,\,\text{class }%
5\rangle \;(p=1\func{mod}5)  \tag{6.516}
\end{equation}

For the next 5 algebras, we have $x\neq 0,\,x\sim x^{\prime }$ if $%
x^{5}=x^{\prime 5}\func{mod}p$:

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaab,\,pa-baaaa,\,pb-xbaaaa,\,\text{class }%
5\rangle  \tag{6.517}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaab,\,pa-\omega baaaa,\,pb-xbaaaa,\,\text{
class }5\rangle \;(p=1\func{mod}5)  \tag{6.517A}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaab,\,pa-\omega ^{2}baaaa,\,pb-xbaaaa,\,\text{
class }5\rangle \;(p=1\func{mod}5)  \tag{6.517A}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaab,\,pa-\omega ^{3}baaaa,\,pb-xbaaaa,\,\text{
class }5\rangle \;(p=1\func{mod}5)  \tag{6.517A}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,\,baaab,\,pa-\omega ^{4}baaaa,\,pb-xbaaaa,\,\text{
class }5\rangle \;(p=1\func{mod}5)  \tag{6.517A}
\end{equation}

\subsection{Descendants of 5.73}

2 algebras

\begin{equation}
\langle a,b\,|\,ba,\,pb,\,\text{class }5\rangle  \tag{6.518}
\end{equation}

\begin{equation}
\langle a,b\,|\,ba-p^{4}a,\,pb,\,\text{class }5\rangle  \tag{6.519}
\end{equation}

\section{Order $p^{7}$}

For $p\geq 5$ the number of nilpotent Lie rings of order $p^{7}$ is

\begin{eqnarray*}
&&\allowbreak 3p^{5}+12p^{4}+44p^{3}+\allowbreak 170p^{2}+707p+2455 \\
&&+(4p^{2}+44p+291)\gcd (p-1,3)+\allowbreak (p^{2}+19p+135)\gcd (p-1,4) \\
&&+(3p+31)\gcd (p-1,5)+4\gcd (p-1,7)+5\gcd (p-1,8)+\gcd (p-1,9).
\end{eqnarray*}

The following table lists the nilpotent Lie rings of order at most $p^{5}$
which have immediate descendants of order $p^{7}$, and gives the number of
descendants.

{\ }

{\small 
\[
\begin{tabular}{|l|l|}
\hline
3.1 & $p+14$ \\ \hline
4.1 & $1361$ if $p=3$, otherwise $p^{5}+2p^{4}+7p^{3}+25p^{2}+88p+270+(p+4)%
\gcd (p-1,3)+\gcd (p-1,4)$ \\ \hline
4.7 & $4$ \\ \hline
5.1 & $178$ if $p=3$, otherwise $p^{2}+15p+125$ \\ \hline
5.3 & $p^{4}+5p^{3}+19p^{2}+64p+140+(p+6)\gcd (p-1,3)+(p+7)\gcd (p-1,4)+\gcd
(p-1,5)$ \\ \hline
5.8 & $p+8$ \\ \hline
5.9 & $4p^{2}+26p+107+5\gcd (p-1,3)+(p+4)\gcd (p-1,4)$ \\ \hline
5.10 & $2p+7$ \\ \hline
5.12 & $3p^{2}+17p+53+\gcd (p-1,3)+\gcd (p-1,4)$ \\ \hline
5.14 & $2p^{5}+7p^{4}+19p^{3}+49p^{2}+128p+256+(p^{2}+7p+29)\gcd (p-1,3)$ \\ 
\hline
& $+(p^{2}+7p+24)\gcd (p-1,4)+(p+3)\gcd (p-1,5)$ \\ \hline
5.15 & $3p^{2}+12p+14+(p+2)\gcd (p-1,4)$ \\ \hline
5.16 & $p^{4}+2p^{3}+5p^{2}+14p$ \\ \hline
5.18 & $3p^{3}+6p^{2}+6p+11+(p+7)\gcd (p-1,3)+(p+1)\gcd (p-1,4)+\gcd (p-1,5)$
\\ \hline
5.37 & $p^{2}+8p+25$ \\ \hline
5.38 & $p+6+(p^{2}+3p+10)\gcd (p-1,3)$ \\ \hline
5.60 & $2p^{2}+p+3+2(p+1)\gcd (p-1,3)+(2p+4)\gcd (p-1,4)+\gcd (p-1,8)$ \\ 
\hline
5.65 & $p^{3}+p^{2}+p-2+2\gcd (p-1,3)+\gcd (p-1,4)+(p+1)\gcd (p-1,5)$ \\ 
\hline
\end{tabular}%
\ 
\]%
}

In the two tables below we give the list of 42 nilpotent Lie rings of order
at most $p^{5}$ which have immediate descendants of order $p^{6}$. For each
of these Lie rings we give the number of immediate descendants of order $%
p^{6}$ which are capable, and we give the number of \textquotedblleft
grandchildren\textquotedblright\ of order $p^{7}$. Note that by
\textquotedblleft grandchildren\textquotedblright\ we mean Lie rings whose
immediate ancestors have order $p^{6}$.

{\ }

{\small 
\[
\begin{tabular}{|l|l|}
\hline
& capable descendants of order $p^{6}$ \\ \hline
3.1 & $3p+27$ \\ \hline
4.1 & $24$ \\ \hline
4.3 & $5p+37+\gcd (p-1,4)$ \\ \hline
4.6 & $2$ \\ \hline
4.7 & $16$ \\ \hline
4.8 & $1$ \\ \hline
5.0 & $1$ \\ \hline
5.1 & $2$ \\ \hline
5.2 & $1$ \\ \hline
5.3 & $2$ \\ \hline
5.8 & $2$ \\ \hline
5.9 & $9$ \\ \hline
5.10 & $1$ \\ \hline
5.11 & $2$ \\ \hline
5.12 & $3$ \\ \hline
5.13 & $1$ \\ \hline
5.14 & $3$ \\ \hline
5.15 & $2p+9$ \\ \hline
5.16 & $4$ \\ \hline
5.18 & $11$ \\ \hline
5.19 & $0$ \\ \hline
5.24 & $1$ \\ \hline
5.27 & $4$ \\ \hline
5.32 & $1$ \\ \hline
5.37 & $2$ \\ \hline
5.38 & $3$ \\ \hline
5.39 & $1+\gcd (p-1,3)+\gcd (p-1,4)/2$ \\ \hline
5.40 & $1+\gcd (p-1,3)+\gcd (p-1,4)/2$ \\ \hline
5.41 & $p+1+\gcd (p-1,3)$ \\ \hline
5.42 & $p+1$ \\ \hline
5.45 & $p$ \\ \hline
5.47 & $2$ \\ \hline
5.48 & $0$ \\ \hline
5.49 & $1$ \\ \hline
5.50 & $2+2\gcd (p-1,3)$ \\ \hline
5.51 & $(p+1)/2$ \\ \hline
5.52 & $(p+1)/2$ \\ \hline
5.54 & $3+\gcd (p-1,4)$ \\ \hline
5.58 & $1$ \\ \hline
5.60 & $2$ \\ \hline
5.65 & $1$ \\ \hline
5.73 & $1$ \\ \hline
\end{tabular}%
\ 
\]%
}

{\small 
\[
\begin{tabular}{|l|l|}
\hline
& grandchildren of order $p^{7}$ \\ \hline
3.1 & $2p^{2}+63p+362+(p+19)\gcd (p-1,3)+5\gcd (p-1,4)+\gcd (p-1,5)$ \\ 
\hline
4.1 & $p^{3}+13p^{2}+96p+595+(3p+21)\gcd (p-1,3)+(p+11)\gcd (p-1,4)+\gcd
(p-1,5)$ \\ \hline
4.3 & $p^{4}+\allowbreak 4p^{3}+\allowbreak 17p^{2}+39p+72+(p^{2}+9p+47)\gcd
(p-1,3)$ \\ \hline
& $+(2p+8)\gcd (p-1,4)+2\gcd (p-1,5)+\gcd (p-1,7)$ \\ \hline
4.6 & $4$ \\ \hline
4.7 & $15p+41+16\gcd (p-1,3)+4\gcd (p-1,4)$ \\ \hline
4.8 & $2$ \\ \hline
5.0 & $1$ \\ \hline
5.1 & $30$ \\ \hline
5.2 & $4$ \\ \hline
5.3 & $35+(p+15)\gcd (p-1,3)+4\gcd (p-1,4)$ \\ \hline
5.8 & $6$ \\ \hline
5.9 & $5p+49+11\gcd (p-1,3)+4\gcd (p-1,4)$ \\ \hline
5.10 & $5$ \\ \hline
5.11 & $7$ \\ \hline
5.12 & $2p+20+7\gcd (p-1,3)+3\gcd (p-1,4)$ \\ \hline
5.13 & $p+1$ \\ \hline
5.14 & $p^{2}+9p+36+(p^{2}+5p+29)\gcd (p-1,3)+(p+7)\gcd (p-1,4)$ \\ \hline
& $+\gcd (p-1,7)+\gcd (p-1,8)$ \\ \hline
5.15 & $10p+16+(2p+7)\gcd (p-1,3)+2\gcd (p-1,4)+2\gcd (p-1,5)$ \\ \hline
5.16 & $p^{3}+5p^{2}+13p+6+3\gcd (p-1,3)$ \\ \hline
5.18 & $2p^{2}+14p+10+(2p+8)\gcd (p-1,3)+7\gcd (p-1,4)+\gcd (p-1,5)$ \\ 
\hline
5.19 & $0$ \\ \hline
5.24 & $3$ \\ \hline
5.27 & $p^{2}+10p+34+(p+14)\gcd (p-1,3)+13\gcd (p-1,4)+6\gcd (p-1,5)+\gcd
(p-1,7)$ \\ \hline
5.32 & $p^{2}+7p+3+2\gcd (p-1,3)+3\gcd (p-1,4)+\gcd (p-1,5)$ \\ \hline
5.37 & $5p+10+2\gcd (p-1,3)+\gcd (p-1,4)$ \\ \hline
5.38 & $p^{3}+\allowbreak 3p^{2}+8p+18+5\gcd (p-1,3)+(p+5)\gcd (p-1,4)$ \\ 
\hline
& $+3\gcd (p-1,5)+2\gcd (p-1,8)+\gcd (p-1,9)$ \\ \hline
5.39 & $\frac{1}{2}\allowbreak (p^{2}+2p+1+(p+5)\gcd (p-1,3)+(p+3)\gcd
(p-1,4))$ \\ \hline
5.40 & $\frac{1}{2}\allowbreak (p^{2}+2p+1+(p+5)\gcd (p-1,3)+(p+3)\gcd
(p-1,4))$ \\ \hline
5.41 & $3p^{2}+4p+(p+1)\gcd (p-1,3)+\gcd (p-1,4)$ \\ \hline
5.42 & $p+3$ \\ \hline
5.45 & $p+1$ \\ \hline
5.47 & $3$ \\ \hline
5.48 & $0$ \\ \hline
5.49 & $4$ \\ \hline
5.50 & $4p+5+(p+7)\gcd (p-1,3)+3\gcd (p-1,4)+2\gcd (p-1,5)$ \\ \hline
5.51 & $\frac{1}{2}(p+1)$ \\ \hline
5.52 & $\frac{1}{2}(p+1)$ \\ \hline
5.54 & $7p+9+4\gcd (p-1,3)+6\gcd (p-1,4)+2\gcd (p-1,5)$ \\ \hline
5.58 & $2$ \\ \hline
5.60 & $4p+3+2\gcd (p-1,3)+4\gcd (p-1,5)+\gcd (p-1,7)+\gcd (p-1,8)$ \\ \hline
5.65 & $2p^{2}+p+2p\gcd (p-1,3)+p\gcd (p-1,5)$ \\ \hline
5.73 & $2$ \\ \hline
\end{tabular}%
\ 
\]%
}

We now give the cyclic and seven generator nilpotent Lie rings of order $%
p^{7}$, and then list the rest, grouped according to their immediate
ancestors.

\begin{equation}
\langle a\,|\,\text{class }7\rangle  \tag{7.0}
\end{equation}

\begin{equation}
\langle a,b,c,d,e,f,g\,|\,\text{class }1\rangle .  \tag{7.1}
\end{equation}

\subsection{Descendants of 4.7}

4 algebras

\begin{equation}
\langle a,b\,|\,pb,\,\text{class }3\rangle  \tag{7.2}
\end{equation}

\begin{equation}
\langle a,b\,|\,pb-baa,\,\text{class }3\rangle  \tag{7.3}
\end{equation}

\begin{equation}
\langle a,b\,|\,pb-\omega baa,\,\text{class }3\rangle  \tag{7.4}
\end{equation}

\begin{equation}
\langle a,b\,|\,pb-bab,\,\text{class }3\rangle  \tag{7.5}
\end{equation}

\subsection{Descendants of 5.37}

$p^{2}+8p+25$ algebras%
\begin{equation}
\langle a,b\,|\,baa,bab,pba,\,\text{class }3\rangle  \tag{7.6}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,bab,p^2b,\,\text{class }3\rangle  \tag{7.7}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,bab,p^2b-pba,\,\text{class }3\rangle  \tag{7.8}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^2a,p^2b-xbaa-pba,\,\text{class }3\rangle \;(\text{all }%
x)  \tag{7.9}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^2a-baa,p^2b-xbaa-pba,\,\text{class }3\rangle \;(\text{%
all }x)  \tag{7.10}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^2a,p^2b-baa,\,\text{class }3\rangle  \tag{7.11}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^2a-pba,p^2b-baa,\,\text{class }3\rangle  \tag{7.12}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^{2}a,p^{2}b-\omega baa,\,\text{class }3\rangle 
\tag{7.13}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^{2}a-pba,p^{2}b-\omega baa,\,\text{class }3\rangle 
\tag{7.14}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^2a,p^2b,\,\text{class }3\rangle  \tag{7.15}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^2a-baa,p^2b,\,\text{class }3\rangle  \tag{7.16}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^2a-pba,p^2b,\,\text{class }3\rangle  \tag{7.17}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^2a-baa-pba,p^2b,\,\text{class }3\rangle  \tag{7.18}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pba,p^2a,\,\text{class }3\rangle  \tag{7.19}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pba,p^2a-baa,\,\text{class }3\rangle  \tag{7.20}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pba-baa,p^2a,\,\text{class }3\rangle  \tag{7.21}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pba-baa,p^2a-baa,\,\text{class }3\rangle  \tag{7.22}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pba,p^2b,\,\text{class }3\rangle  \tag{7.23}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pba,p^2b-baa,\,\text{class }3\rangle  \tag{7.24}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pba,p^{2}b-\omega baa,\,\text{class }3\rangle  \tag{7.25}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pba-baa,p^2b-xbaa,\,\text{class }3\rangle \;(\text{all }%
x)  \tag{7.26}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-baa,p^2b,pba,\,\text{class }3\rangle  \tag{7.27}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-baa,p^2b-xbab,pba,\,\text{class }3\rangle \;(x \neq 0,
x \sim x^{-1})  \tag{7.28}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-baa-bab,p^2b-bab,pba,\,\text{class }3\rangle  \tag{7.29}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-baa-\omega bab,p^{2}b-bab,pba,\,\text{class }3\rangle
\tag{7.30}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a,p^2b,pba,\,\text{class }3\rangle  \tag{7.31}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-bab,p^2b,pba,\,\text{class }3\rangle  \tag{7.32}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-\omega bab,p^{2}b,pba,\,\text{class }3\rangle 
\tag{7.33}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-\omega bab,p^{2}b-baa,pba,\,\text{class }3\rangle 
\tag{7.34}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-xbab,p^2b-baa-bab,pba,\,\text{class }3\rangle \;(\text{%
all }x\text{ with }1+4x\text{ not a square }\func{\func{mod}}p)  \tag{7.35}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-bab,p^2b-xbaa,pba-baa,\,\text{class }3\rangle \;(\text{%
all }x)  \tag{7.36}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-\omega bab,p^{2}b-xbaa,pba-baa,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.37}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-ybab,p^{2}b-xbaa-bab,pba-baa,\,\text{class }3\rangle
\;(\text{all }x,\,y\neq 0)  \tag{7.38}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a,p^2b-xbaa,pba-baa,\,\text{class }3\rangle \;(\text{all }%
x)  \tag{7.39}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-baa,p^2b-xbaa-bab,pba-baa,\,\text{class }3\rangle \;(x
\neq 0)  \tag{7.40}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a,p^2b-bab,pba-baa,\,\text{class }3\rangle  \tag{7.41}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-baa,p^2b-xbab,pba-baa,\,\text{class }3\rangle \;(\text{%
all }x)  \tag{7.42}
\end{equation}

\subsection{Descendants of 5.38}

$p+6+(p^{2}+3p+10)\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b\,|\,babb,pa,pb,\,\text{class }4\rangle  \tag{7.43}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa,pb-baab,\,\text{class }4\rangle  \tag{7.44}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa-baaa,pb-xbaab,\,\text{class }4\rangle \;(\text{all }%
x)  \tag{7.45}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa-baab,pb,\,\text{class }4\rangle  \tag{7.46}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa-baaa-baab,pb-2baab,\,\text{class }4\rangle 
\tag{7.47}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa-baaa-\omega baab,pb-2baab,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.48}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa-baaa-\omega ^{2}baab,pb-2baab,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.49}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa,pb-baaa,\,\text{class }4\rangle  \tag{7.50}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa-baab,pb-baaa,\,\text{class }4\rangle  \tag{7.51}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa-\omega baab,pb-baaa,\,\text{class }4\rangle 
\tag{7.52}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa-xbaab,pb-baaa-baab,\,\text{class }4\rangle \;(\text{%
all }x)  \tag{7.53}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa,pb-\omega baaa,\,\text{class }4\rangle \;(p=1\func{%
mod}3)  \tag{7.54}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa-baab,pb-\omega baaa,\,\text{class }4\rangle \;(p=1%
\func{mod}3)  \tag{7.55}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa-\omega baab,pb-\omega baaa,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.56}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa-xbaab,pb-\omega baaa-baab,\,\text{class }4\rangle \;(%
\text{all }x,\;p=1\func{mod}3)  \tag{7.57}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa,pb-\omega ^{2}baaa,\,\text{class }4\rangle \;(p=1%
\func{mod}3)  \tag{7.58}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa-baab,pb-\omega ^{2}baaa,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.59}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa-\omega baab,pb-\omega ^{2}baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.60}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,pa-xbaab,pb-\omega ^{2}baaa-baab,\,\text{class }%
4\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.61}
\end{equation}

Algebra 5.38 has$\allowbreak $ $\gcd (p-1,3)(p^{2}+3p+10)+p+6$ immediate
descendants of order $p^{7}$. Of these $\frac{1}{2}((p^{2}+3p+11)\gcd
(p-1,3)+1)$ come from one 4 parameter family of algebras, and $\frac{1}{2}%
(\gcd (p-1,3)(p^{2}+p+1)+5)$ come from another four parameter family. These
two families have presentations:

\begin{equation}
\langle a,b\,|\,babb-baaa,pa-xbaaa-ybaab,pb-zbaaa-tbaab,\,\text{class }%
4\rangle  \tag{7.62}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb-\omega baaa,pa-xbaaa-ybaab,pb-zbaaa-tbaab,\,\text{class 
}4\rangle  \tag{7.63}
\end{equation}%
In both 7.62 and 7.63 we take the four parameters as entries in a $2\times 2$
matrix%
\[
A=\left( 
\begin{array}{cc}
x & y \\ 
z & t%
\end{array}%
\right) , 
\]%
and in both cases we consider the orbits of matrices $A$ of this form over GF%
$(p)$ under an action of the subgroup of GL$(2,p)$ consisting of
non-singular matrices of the form%
\[
\left( 
\begin{array}{ll}
\alpha & \beta \\ 
\beta & \alpha%
\end{array}%
\right) \text{ or }\left( 
\begin{array}{ll}
\alpha & \beta \\ 
-\beta & -\alpha%
\end{array}%
\right) . 
\]

In 7.62 two matrices $A$ and $B$ give isomorphic Lie rings if and only if%
\[
B=\left( 
\begin{array}{ll}
\alpha & \beta \\ 
\beta & \alpha%
\end{array}%
\right) A\left( 
\begin{array}{ll}
(\alpha ^{4}-\beta ^{4}) & 2\alpha \beta (\alpha ^{2}-\beta ^{2}) \\ 
2\alpha \beta (\alpha ^{2}-\beta ^{2}) & \alpha ^{4}-\beta ^{4}%
\end{array}%
\right) ^{-1} 
\]%
or 
\[
B=\left( 
\begin{array}{ll}
\alpha & \beta \\ 
-\beta & -\alpha%
\end{array}%
\right) A\left( 
\begin{array}{ll}
-(\alpha ^{4}-\beta ^{4}) & -2\alpha \beta (\alpha ^{2}-\beta ^{2}) \\ 
2\alpha \beta (\alpha ^{2}-\beta ^{2}) & \alpha ^{4}-\beta ^{4}%
\end{array}%
\right) ^{-1} 
\]%
for some $\alpha ,\beta $.

In 7.63, two matrices $A$ and $B$ give isomorphic Lie rings if and only if%
\[
B=\left( 
\begin{array}{ll}
\alpha & \beta \\ 
\beta & \alpha%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha ^{4}-\omega ^{2}\beta ^{4} & 2\alpha \beta (\alpha ^{2}-\omega \beta
^{2}) \\ 
2\omega \alpha \beta (\alpha ^{2}-\omega \beta ^{2}) & \alpha ^{4}-\omega
^{2}\beta ^{4}%
\end{array}%
\right) ^{-1} 
\]%
or 
\[
B=\left( 
\begin{array}{ll}
\alpha & \beta \\ 
-\beta & -\alpha%
\end{array}%
\right) A\left( 
\begin{array}{ll}
-(\alpha ^{4}-\omega ^{2}\beta ^{4}) & -2\alpha \beta (\alpha ^{2}-\omega
\beta ^{2}) \\ 
2\omega \alpha \beta (\alpha ^{2}-\omega \beta ^{2}) & \alpha ^{4}-\omega
^{2}\beta ^{4}%
\end{array}%
\right) ^{-1} 
\]%
for some $\alpha ,\beta $.

\subsection{Descendants of 5.60}

$2p^{2}+p+3+2(p+1)\gcd (p-1,3)+(2p+4)\gcd (p-1,4)+\gcd (p-1,8)$ algebras

\begin{equation}
\langle a,b\,|\,bab,pa,pb,\,\text{class }5\rangle  \tag{7.64}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa,pb-baaab,\,\text{class }5\rangle  \tag{7.65}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-baaaa,pb-xbaaab,\,\text{class }5\rangle \;(\text{all }%
x)  \tag{7.66}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-baaab,pb,\,\text{class }5\rangle  \tag{7.67}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-\omega baaab,pb,\,\text{class }5\rangle  \tag{7.68}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-baaaa-baaab,pb-baaab,\,\text{class }5\rangle 
\tag{7.69}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-baaaa-\omega baaab,pb-baaab,\,\text{class }5\rangle 
\tag{7.70}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-baaaa-\omega ^{2}baaab,pb-baaab,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.71}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-baaaa-\omega ^{3}baaab,pb-baaab,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.72}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa,pb-baaaa,\,\text{class }5\rangle  \tag{7.73}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-baaab,pb-baaaa,\,\text{class }5\rangle  \tag{7.74}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-\omega baaab,pb-baaaa,\,\text{class }5\rangle 
\tag{7.75}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-xbaaab,pb-baaaa-baaab,\,\text{class }5\rangle \;(%
\text{all }x)  \tag{7.76}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa,pb-\omega baaaa,\,\text{class }5\rangle  \tag{7.77}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-baaab,pb-\omega baaaa,\,\text{class }5\rangle 
\tag{7.78}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-\omega baaab,pb-\omega baaaa,\,\text{class }5\rangle 
\tag{7.79}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-xbaaab,pb-\omega baaaa-baaab,\,\text{class }5\rangle
\;(\text{all }x)  \tag{7.80}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa,pb-\omega ^{2}baaaa,\,\text{class }5\rangle \;(p=1%
\func{mod}4)  \tag{7.81}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-baaab,pb-\omega ^{2}baaaa,\,\text{class }5\rangle
\;(p=1\func{mod}4)  \tag{7.82}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-\omega baaab,pb-\omega ^{2}baaaa,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.83}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-xbaaab,pb-\omega ^{2}baaaa-baaab,\,\text{class }%
5\rangle \;(\text{all }x,\;p=1\func{mod}4)  \tag{7.84}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa,pb-\omega ^{3}baaaa,\,\text{class }5\rangle \;(p=1%
\func{mod}4)  \tag{7.85}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-baaab,pb-\omega ^{3}baaaa,\,\text{class }5\rangle
\;(p=1\func{mod}4)  \tag{7.86}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-\omega baaab,pb-\omega ^{3}baaaa,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.87}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pa-xbaaab,pb-\omega ^{3}baaaa-baaab,\,\text{class }%
5\rangle \;(\text{all }x,\;p=1\func{mod}4)  \tag{7.88}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa,pb,\,\text{class }5\rangle  \tag{7.89}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa,pb-baaab,\,\text{class }5\rangle  \tag{7.90}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa,pb-\omega baaab,\,\text{class }5\rangle 
\tag{7.91}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa,pb-\omega ^{2}baaab,\,\text{class }5\rangle
\;(p=1\func{mod}3)  \tag{7.92}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa,pb-\omega ^{3}baaab,\,\text{class }5\rangle
\;(p=1\func{mod}3)  \tag{7.93}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa,pb-\omega ^{4}baaab,\,\text{class }5\rangle
\;(p=1\func{mod}3)  \tag{7.94}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa,pb-\omega ^{5}baaab,\,\text{class }5\rangle
\;(p=1\func{mod}3)  \tag{7.95}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-baaaa,pb-xbaaab,\,\text{class }5\rangle \;(%
\text{all }x)  \tag{7.96}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-\omega baaaa,pb-xbaaab,\,\text{class }5\rangle
\;(\text{all }x)  \tag{7.97}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-\omega ^{2}baaaa,pb-xbaaab,\,\text{class }%
5\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.98}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-\omega ^{3}baaaa,pb-xbaaab,\,\text{class }%
5\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.99}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-\omega ^{4}baaaa,pb-xbaaab,\,\text{class }%
5\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.100}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-\omega ^{5}baaaa,pb-xbaaab,\,\text{class }%
5\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.101}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-baaab,pb,\,\text{class }5\rangle  \tag{7.102}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-\omega baaab,pb,\,\text{class }5\rangle 
\tag{7.103}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-\omega ^{2}baaab,pb,\,\text{class }5\rangle
\;(p=1\func{mod}4)  \tag{7.104}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-\omega ^{3}baaab,pb,\,\text{class }5\rangle
\;(p=1\func{mod}4)  \tag{7.105}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-\omega ^{4}baaab,pb,\,\text{class }5\rangle
\;(p=1\func{mod}8)  \tag{7.106}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-\omega ^{5}baaab,pb,\,\text{class }5\rangle
\;(p=1\func{mod}8)  \tag{7.107}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-\omega ^{6}baaab,pb,\,\text{class }5\rangle
\;(p=1\func{mod}8)  \tag{7.108}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-\omega ^{7}baaab,pb,\,\text{class }5\rangle
\;(p=1\func{mod}8)  \tag{7.109}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-baaaa-xbaaab,pb-baaab,\,\text{class }5\rangle
\;(x\neq 0,x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}\func{mod}p) 
\tag{7.110}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-\omega baaaa-xbaaab,pb-\omega baaab,\,\text{%
class }5\rangle \;(x\neq 0,x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}%
\func{mod}p)  \tag{7.111}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-\omega ^{2}baaaa-xbaaab,pb-\omega ^{2}baaab,\,%
\text{class }5\rangle \;(x\neq 0,x\sim x^{\prime }\text{ if }x^{3}=x^{\prime
3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.112}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-\omega ^{3}baaaa-xbaaab,pb-\omega ^{3}baaab,\,%
\text{class }5\rangle \;(x\neq 0,x\sim x^{\prime }\text{ if }x^{3}=x^{\prime
3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.113}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-\omega ^{4}baaaa-xbaaab,pb-\omega ^{4}baaab,\,%
\text{class }5\rangle \;(x\neq 0,x\sim x^{\prime }\text{ if }x^{3}=x^{\prime
3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.114}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-\omega ^{5}baaaa-xbaaab,pb-\omega ^{5}baaab,\,%
\text{class }5\rangle \;(x\neq 0,x\sim x^{\prime }\text{ if }x^{3}=x^{\prime
3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.115}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-xbaaab,pb-baaaa-ybaaab,\,\text{class }5\rangle
\;(\text{all }x,\,y,\;p=3\func{mod}4)  \tag{7.116}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-xbaaab,pb-\omega baaaa-ybaaab,\,\text{class }%
5\rangle \;(\text{all }x,\,y,\;p=3\func{mod}4)  \tag{7.117}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-xbaaab,pb-baaaa-ybaaab,\,\text{class }5\rangle
\;(\text{all }x,\,y,\;y\sim -y,\;p=1\func{mod}4)  \tag{7.118}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-xbaaab,pb-\omega baaaa-ybaaab,\,\text{class }%
5\rangle \;(\text{all }x,\,y,\;y\sim -y,\;p=1\func{mod}4)  \tag{7.119}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-xbaaab,pb-\omega ^{2}baaaa-ybaaab,\,\text{class 
}5\rangle \;(\text{all }x,\,y,\;y\sim -y,\;p=1\func{mod}4)  \tag{7.120}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,pa-xbaaab,pb-\omega ^{3}baaaa-ybaaab,\,\text{class 
}5\rangle \;(\text{all }x,\,y,\;y\sim -y,\;p=1\func{mod}4)  \tag{7.121}
\end{equation}

\subsection{Descendants of 5.65}

$p^{3}+p^{2}+p-2+2\gcd (p-1,3)+\gcd (p-1,4)+(p+1)\gcd (p-1,5)$ algebras

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaaa-ybaaab,pb-baaaa-zbaaab,\,\text{class }%
5\rangle \;(x\neq 0,\text{all }y,z,\;x\sim x^{\prime }\text{ if }%
x^{4}=x^{\prime 4}\func{mod}p)  \tag{7.122}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaaa-ybaaab,pb-\omega baaaa-zbaaab,\,\text{%
class }5\rangle \;(x\neq 0,\text{all }y,z,\;x\sim x^{\prime }\text{ if }%
x^{4}=x^{\prime 4}\func{mod}p)  \tag{7.123}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaaa-ybaaab,pb-\omega ^{2}baaaa-zbaaab,\,\text{%
class }5\rangle \;(x\neq 0,\text{all }y,z,\;x\sim x^{\prime }\text{ if }%
x^{4}=x^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.124}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaaa-ybaaab,pb-\omega ^{3}baaaa-zbaaab,\,\text{%
class }5\rangle \;(x\neq 0,\text{all }y,z,\;x\sim x^{\prime }\text{ if }%
x^{4}=x^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.125}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaab,pb-baaaa-ybaaab,\,\text{class }5\rangle
\;(\text{all }x,\,y\neq 0,\;y\sim y^{\prime }\text{ if }y^{4}=y^{\prime 4}%
\func{mod}p)  \tag{7.126}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaab,pb-\omega baaaa-ybaaab,\,\text{class }%
5\rangle \;(\text{all }x,\,y\neq 0,\;y\sim y^{\prime }\text{ if }%
y^{4}=y^{\prime 4}\func{mod}p)  \tag{7.127}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaab,pb-\omega ^{2}baaaa-ybaaab,\,\text{class }%
5\rangle \;(\text{all }x,\,y\neq 0,\;y\sim y^{\prime }\text{ if }%
y^{4}=y^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.128}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaab,pb-\omega ^{3}baaaa-ybaaab,\,\text{class }%
5\rangle \;(\text{all }x,\,y\neq 0,\;y\sim y^{\prime }\text{ if }%
y^{4}=y^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.129}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaab,pb-baaaa,\,\text{class }5\rangle \;(x
\neq 0,\;p=3 \func{mod} 4)  \tag{7.130}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaab,pb-\omega baaaa,\,\text{class }5\rangle
\;(x\neq 0,\;p=3\func{mod}4)  \tag{7.131}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaab,pb-baaaa,\,\text{class }5\rangle \;(x\neq
0,\;x\sim -x,\;p=1\func{mod}4)  \tag{7.132}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaab,pb-\omega baaaa,\,\text{class }5\rangle
\;(x\neq 0,\;x\sim -x,\;p=1\func{mod}4)  \tag{7.133}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaab,pb-\omega ^{2}baaaa,\,\text{class }%
5\rangle \;(x\neq 0,\;x\sim -x,\;p=1\func{mod}4)  \tag{7.134}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaab,pb-\omega ^{3}baaaa,\,\text{class }%
5\rangle \;(x\neq 0,\;x\sim -x,\;p=1\func{mod}4)  \tag{7.135}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa,pb-baaaa,\,\text{class }5\rangle  \tag{7.136}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa,pb-\omega baaaa,\,\text{class }5\rangle 
\tag{7.137}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa,pb-\omega ^{2}baaaa,\,\text{class }5\rangle
\;(p=1\func{mod}4)  \tag{7.138}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa,pb-\omega ^{3}baaaa,\,\text{class }5\rangle
\;(p=1\func{mod}4)  \tag{7.139}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-baaaa,pb-xbaaab,\,\text{class }5\rangle \;(\text{%
all }x)  \tag{7.140}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-\omega baaaa,pb-xbaaab,\,\text{class }5\rangle
\;(\text{all }x,\;p=1\func{mod}5)  \tag{7.141}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-\omega ^{2}baaaa,pb-xbaaab,\,\text{class }%
5\rangle \;(\text{all }x,\;p=1\func{mod}5)  \tag{7.142}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-\omega ^{3}baaaa,pb-xbaaab,\,\text{class }%
5\rangle \;(\text{all }x,\;p=1\func{mod}5)  \tag{7.143}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-\omega ^{4}baaaa,pb-xbaaab,\,\text{class }%
5\rangle \;(\text{all }x,\;p=1\func{mod}5)  \tag{7.144}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-baaaa-ybaaab,pb-xbaaab,\,\text{class }5\rangle
\;(\text{all }x,\,\;y\neq 0,\;y\sim y^{\prime }\text{ if }y^{5}=y^{\prime 5}%
\func{mod}p)  \tag{7.145}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-\omega baaaa-ybaaab,pb-xbaaab,\,\text{class }%
5\rangle \;(\text{all }x,\;\,y\neq 0,\;y\sim y^{\prime }\text{ if }%
y^{5}=y^{\prime 5}\func{mod}p,\;p=1\func{mod}5)  \tag{7.146}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-\omega ^{2}baaaa-ybaaab,pb-xbaaab,\,\text{class }%
5\rangle \;(\text{all }x,\,y\neq 0,\;y\sim y^{\prime }\text{ if }%
y^{5}=y^{\prime 5}\func{mod}p,\;p=1\func{mod}5)  \tag{7.147}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-\omega ^{3}baaaa-ybaaab,pb-xbaaab,\,\text{class }%
5\rangle \;(\text{all }x,\,y\neq 0,\;y\sim y^{\prime }\text{ if }%
y^{5}=y^{\prime 5}\func{mod}p,\;p=1\func{mod}5)  \tag{7.148}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-\omega ^{4}baaaa-ybaaab,pb-xbaaab,\,\text{class }%
5\rangle \;(\text{all }x,\,y\neq 0,\;y\sim y^{\prime }\text{ if }%
y^{5}=y^{\prime 5}\func{mod}p,\;p=1\func{mod}5)  \tag{7.149}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa,pb-baaab,\,\text{class }5\rangle  \tag{7.150}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa,pb-\omega baaab,\,\text{class }5\rangle \;(p=1%
\func{mod}5)  \tag{7.151}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa,pb-\omega ^{2}baaab,\,\text{class }5\rangle
\;(p=1\func{mod}5)  \tag{7.152}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa,pb-\omega ^{3}baaab,\,\text{class }5\rangle
\;(p=1\func{mod}5)  \tag{7.153}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa,pb-\omega ^{4}baaab,\,\text{class }5\rangle
\;(p=1\func{mod}5)  \tag{7.154}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaab,pb-baaab,\,\text{class }5\rangle \;(x\neq
0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod}p)  \tag{7.155}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaab,pb-\omega baaab,\,\text{class }5\rangle
\;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod}p,\;p=1%
\func{mod}5)  \tag{7.156}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaab,pb-\omega ^{2}baaab,\,\text{class }%
5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod%
}p,\;p=1\func{mod}5)  \tag{7.157}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaab,pb-\omega ^{3}baaab,\,\text{class }%
5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod%
}p,\;p=1\func{mod}5)  \tag{7.158}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-xbaaab,pb-\omega ^{4}baaab,\,\text{class }%
5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod%
}p,\;p=1\func{mod}5)  \tag{7.159}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa,pb,\,\text{class }5\rangle  \tag{7.160}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-baaab,pb,\,\text{class }5\rangle  \tag{7.161}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-\omega baaab,pb,\,\text{class }5\rangle 
\tag{7.162}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-\omega ^{2}baaab,pb,\,\text{class }5\rangle
\;(p=1\func{mod}3)  \tag{7.163}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-\omega ^{3}baaab,pb,\,\text{class }5\rangle
\;(p=1\func{mod}3)  \tag{7.164}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-\omega ^{4}baaab,pb,\,\text{class }5\rangle
\;(p=1\func{mod}3)  \tag{7.165}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pa-\omega ^{5}baaab,pb,\,\text{class }5\rangle
\;(p=1\func{mod}3)  \tag{7.166}
\end{equation}

\subsection{Descendants of 6.366}

2 algebras

\begin{equation}
\langle a,b\,|\,ba,p^3b,\,\text{class }4\rangle  \tag{7.167}
\end{equation}

\begin{equation}
\langle a,b\,|\,ba-p^3a,p^3b,\,\text{class }4\rangle  \tag{7.168}
\end{equation}

\subsection{Descendants of 6.367}

2 algebras

\begin{equation}
\langle a,b\,|\,ba-p^2a,p^3a,\,\text{class }4\rangle  \tag{7.169}
\end{equation}

\begin{equation}
\langle a,b\,|\,ba-p^2a,p^3b,\,\text{class }4\rangle  \tag{7.170}
\end{equation}

\subsection{Descendants of 6.368}

$9+6\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b\,|\,baaa,baab,p^2a,pb,\,\text{class }4\rangle  \tag{7.171}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,baab,p^2a,pb-babb,\,\text{class }4\rangle  \tag{7.172}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,baab,p^2a-babb,pb,\,\text{class }4\rangle  \tag{7.173}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,baab,p^{2}a-\omega babb,pb,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.174}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,baab,p^{2}a-\omega ^{2}babb,pb,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.175}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb-baaa,p^2a,pb,\,\text{class }4\rangle  \tag{7.176}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb-baaa,p^2a,pb-baaa,\,\text{class }4\rangle 
\tag{7.177}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb-baaa,p^{2}a,pb-\omega baaa,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.178}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb-baaa,p^{2}a,pb-\omega ^{2}baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.179}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb-baaa,p^2a-baaa,pb,\,\text{class }4\rangle 
\tag{7.180}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb-baaa,p^{2}a-\omega baaa,pb,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.181}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb-baaa,p^{2}a-\omega ^{2}baaa,pb,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.182}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb-\omega baaa,p^{2}a,pb,\,\text{class }4\rangle 
\tag{7.183}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb-\omega baaa,p^{2}a,pb-baaa,\,\text{class }4\rangle
\tag{7.184}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb-\omega baaa,p^{2}a,pb-\omega baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.185}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb-\omega baaa,p^{2}a,pb-\omega ^{2}baaa,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.186}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb-\omega baaa,p^{2}a-baaa,pb,\,\text{class }4\rangle
\tag{7.187}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb-\omega baaa,p^{2}a-\omega baaa,pb,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.188}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb-\omega baaa,p^{2}a-\omega ^{2}baaa,pb,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.189}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,babb,p^2a,pb,\,\text{class }4\rangle  \tag{7.190}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,babb,p^2a,pb-baab,\,\text{class }4\rangle  \tag{7.191}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,babb,p^2a-baab,pb,\,\text{class }4\rangle  \tag{7.192}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb,p^2a,pb,\,\text{class }4\rangle  \tag{7.193}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb,p^2a,pb-baaa,\,\text{class }4\rangle  \tag{7.194}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb,p^{2}a,pb-\omega baaa,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.195}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb,p^{2}a,pb-\omega ^{2}baaa,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.196}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,babb,p^2a-baaa,pb,\,\text{class }4\rangle  \tag{7.197}
\end{equation}

\subsection{Descendants of 6.369}

$5+\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b\,|\,baaa,p^2a,pb-bab,\,\text{class }4\rangle  \tag{7.198}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,p^2a,pb-bab-baab,\,\text{class }4\rangle  \tag{7.199}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,p^2a-baab,pb-bab,\,\text{class }4\rangle  \tag{7.200}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,p^2a,pb-bab,\,\text{class }4\rangle  \tag{7.201}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,p^2a,pb-bab-baaa,\,\text{class }4\rangle  \tag{7.202}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,p^{2}a,pb-bab-\omega baaa,\,\text{class }4\rangle \;(p=1%
\func{mod}3)  \tag{7.203}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,p^{2}a,pb-bab-\omega ^{2}baaa,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.204}
\end{equation}

\begin{equation}
\langle a,b\,|\,baab,p^2a-baaa,pb-bab,\,\text{class }4\rangle  \tag{7.205}
\end{equation}

\subsection{Descendants of 6.370}

$2p+2+(p+1)/2+\gcd (p-1,3)+\gcd (p-1,4)/2$ algebras

\begin{equation}
\langle a,b\,|\,baaa,p^2a,pb-baa,\,\text{class }4\rangle  \tag{7.206}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,p^2a,pb-baa-babb,\,\text{class }4\rangle  \tag{7.207}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,p^{2}a,pb-baa-\omega babb,\,\text{class }4\rangle \;(p=1%
\func{mod}4)  \tag{7.208}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,p^2a-babb,pb-baa,\,\text{class }4\rangle  \tag{7.209}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,p^{2}a-\omega babb,pb-baa,\,\text{class }4\rangle \;(p=1%
\func{mod}3)  \tag{7.210}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,p^{2}a-\omega ^{2}babb,pb-baa,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.211}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,p^{2}a,pb-baa-xbaaa,\,\text{class }4\rangle \;(\text{%
all }x,\,x\sim -x)  \tag{7.212}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,p^2a-baaa,pb-baa,\,\text{class }4\rangle  \tag{7.213}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb-baaa,p^{2}a,pb-baa-xbaaa,\,\text{class }4\rangle \;(%
\text{all }x,\,x\sim -x)  \tag{7.214}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb-baaa,p^2a-xbaaa,pb-baa,\,\text{class }4\rangle \;(x
\neq 0, x \sim -x)  \tag{7.215}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb-\omega baaa,p^{2}a,pb-baa-xbaaa,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.216}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb-\omega baaa,p^{2}a-xbaaa,pb-baa,\,\text{class }4\rangle
\;(x\neq 0,\;x\sim -x)  \tag{7.217}
\end{equation}

\subsection{Descendants of 6.371}

$2p+2+(p+1)/2+\gcd (p-1,3)+\gcd (p-1,4)/2$ algebras

\begin{equation}
\langle a,b\,|\,baaa,p^{2}a,pb-\omega baa,\,\text{class }4\rangle 
\tag{7.218}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,p^{2}a,pb-\omega baa-babb,\,\text{class }4\rangle 
\tag{7.219}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,p^{2}a,pb-\omega baa-\omega babb,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.220}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,p^{2}a-babb,pb-\omega baa,\,\text{class }4\rangle 
\tag{7.221}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,p^{2}a-\omega babb,pb-\omega baa,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.222}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,p^{2}a-\omega ^{2}babb,pb-\omega baa,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.223}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,p^{2}a,pb-\omega baa-xbaaa,\,\text{class }4\rangle \;(%
\text{all }x,\,x\sim -x)  \tag{7.224}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,p^{2}a-baaa,pb-\omega baa,\,\text{class }4\rangle 
\tag{7.225}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb-baaa,p^{2}a,pb-\omega baa-xbaaa,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.226}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb-baaa,p^{2}a-xbaaa,pb-\omega baa,\,\text{class }4\rangle
\;(x\neq 0,x\sim -x)  \tag{7.227}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb-\omega baaa,p^{2}a,pb-\omega baa-xbaaa,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.228}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb-\omega baaa,p^{2}a-xbaaa,pb-\omega baa,\,\text{class }%
4\rangle \;(x\neq 0,x\sim -x)  \tag{7.229}
\end{equation}

\subsection{Descendants of 6.372}

$(p+1)/2+\gcd (p-1,3)+\gcd (p-1,4)/2$ algebras

\begin{equation}
\langle a,b\,|\,p^2a-bab,pb,\,\text{class }4\rangle  \tag{7.230}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-bab,pb-baaa,\,\text{class }4\rangle  \tag{7.231}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-bab,pb-\omega baaa,\,\text{class }4\rangle \;(p=1%
\func{mod}3)  \tag{7.232}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-bab,pb-\omega ^{2}baaa,\,\text{class }4\rangle \;(p=1%
\func{mod}3)  \tag{7.233}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-bab-baaa,pb-xbaaa,\,\text{class }4\rangle \;(\text{%
all }x,\,x\sim x^{\prime }\text{ if }x^{4}=x^{\prime 4}\func{mod}p,\;p=1%
\func{mod}4)  \tag{7.234}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-bab-\omega baaa,pb-xbaaa,\,\text{class }4\rangle \;(%
\text{all }x,\,x\sim x^{\prime }\text{ if }x^{4}=x^{\prime 4}\func{mod}%
p,\;p=1\func{mod}4)  \tag{7.235}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-bab-baaa,pb-xbaaa,\,\text{class }4\rangle \;(\text{%
all }x,\,x\sim -x,\;p=3\func{mod}4)  \tag{7.236}
\end{equation}

\subsection{Descendants of 6.373}

$p$ algebras

\begin{equation}
\langle a,b\,|\,p^{2}a-bab,pb-baa-xbaaa,\,\text{class }4\rangle \;(\text{all 
}x,\,x\sim -x)  \tag{7.237}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-bab-xbaaa,pb-baa,\,\text{class }4\rangle \;(x \neq 0, x
\sim -x)  \tag{7.238}
\end{equation}

\subsection{Descendants of 6.374}

$p$ algebras

\begin{equation}
\langle a,b\,|\,p^{2}a-bab,pb-\omega baa-xbaaa,\,\text{class }4\rangle \;(%
\text{all }x,\,x\sim -x)  \tag{7.239}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-bab-xbaaa,pb-\omega baa,\,\text{class }4\rangle
\;(x\neq 0,\,x\sim -x)  \tag{7.240}
\end{equation}

\subsection{Descendants of 6.375}

$(p+1)/2+\gcd (p-1,3)+\gcd (p-1,4)/2$ algebras

\begin{equation}
\langle a,b\,|\,p^{2}a-\omega bab,pb,\,\text{class }4\rangle  \tag{7.241}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-\omega bab,pb-baaa,\,\text{class }4\rangle 
\tag{7.242}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-\omega bab,pb-\omega baaa,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.243}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-\omega bab,pb-\omega ^{2}baaa,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.244}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-\omega bab-baaa,pb-xbaaa,\,\text{class }4\rangle \;(%
\text{all }x,\,x\sim x^{\prime }\text{ if }x^{4}=x^{\prime 4}\func{mod}%
p,\;p=1\func{mod}4)  \tag{7.245}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-\omega bab-\omega baaa,pb-xbaaa,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim x^{\prime }\ if\ x^{4}=x^{\prime 4}\func{mod%
}p,\;p=1\func{mod}4)  \tag{7.246}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-\omega bab-baaa,pb-xbaaa,\,\text{class }4\rangle \;(%
\text{all }x,\,x\sim -x,\;p=3\func{mod}4)  \tag{7.247}
\end{equation}

\subsection{Descendants of 6.376}

$p$ algebras

\begin{equation}
\langle a,b\,|\,p^{2}a-\omega bab,pb-baa-xbaaa,\,\text{class }4\rangle \;(%
\text{all }x,\,x\sim -x)  \tag{7.248}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-\omega bab-xbaaa,pb-baa,\,\text{class }4\rangle
\;(x\neq 0,x\sim -x)  \tag{7.249}
\end{equation}

\subsection{Descendants of 6.377}

$p$ algebras

\begin{equation}
\langle a,b\,|\,p^{2}a-\omega bab,pb-\omega baa-xbaaa,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.250}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-\omega bab-xbaaa,pb-\omega baa,\,\text{class }%
4\rangle \;(x\neq 0,x\sim -x)  \tag{7.251}
\end{equation}

\subsection{Descendants of 6.378}

$p+1+\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b\,|\,p^2a-baa,pb,\,\text{class }4\rangle  \tag{7.252}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-baa-babb,pb,\,\text{class }4\rangle  \tag{7.253}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-baa-\omega babb,pb,\,\text{class }4\rangle \;(p=1%
\func{mod}3)  \tag{7.254}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{2}a-baa-\omega ^{2}babb,pb,\,\text{class }4\rangle \;(p=1%
\func{mod}3)  \tag{7.255}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^2a-baa-xbabb,pb-babb,\,\text{class }4\rangle \;(\text{all }%
x)  \tag{7.256}
\end{equation}

\subsection{Descendants of 6.379}

$5+2\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b\,|\,p^3a,baa,pb,\,\text{class }4\rangle  \tag{7.257}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^3a,baa,pb-babb,\,\text{class }4\rangle  \tag{7.258}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^3a,baa-babb,pb,\,\text{class }4\rangle  \tag{7.259}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^3a,baa-babb,pb-babb,\,\text{class }4\rangle  \tag{7.260}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a,baa-babb,pb-\omega babb,\,\text{class }4\rangle 
\tag{7.261}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a,baa-babb,pb-\omega ^{2}babb,\,\text{class }4\rangle
\;(p=1\func{mod}4)  \tag{7.262}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a,baa-babb,pb-\omega ^{3}babb,\,\text{class }4\rangle
\;(p=1\func{mod}4)  \tag{7.263}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,baa,pb,\,\text{class }4\rangle  \tag{7.264}
\end{equation}

\begin{equation}
\langle a,b\,|\,babb,baa-p^3a,pb,\,\text{class }4\rangle  \tag{7.265}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^3a-babb,baa,pb,\,\text{class }4\rangle  \tag{7.266}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^3a-babb,baa-babb,pb,\,\text{class }4\rangle  \tag{7.267}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a-\omega babb,baa,pb,\,\text{class }4\rangle \;(p=1%
\func{mod}3)  \tag{7.268}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a-\omega babb,baa-babb,pb,\,\text{class }4\rangle \;(p=1%
\func{mod}3)  \tag{7.269}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a-\omega ^{2}babb,baa,pb,\,\text{class }4\rangle \;(p=1%
\func{mod}3)  \tag{7.270}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a-\omega ^{2}babb,baa-babb,pb,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.271}
\end{equation}

\subsection{Descendants of 6.380}

$p$ algebras

\begin{equation}
\langle a,b\,|\,baa-xp^3a,pb-bab,\,\text{class }4\rangle \;(\text{all }x) 
\tag{7.272}
\end{equation}

\subsection{Descendants of 6.381}

$6+2\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b\,|\,baaa,bab,pb,\,\text{class }4\rangle  \tag{7.273}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,bab-p^3a,pb,\,\text{class }4\rangle  \tag{7.274}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,bab-\omega p^{3}a,pb,\,\text{class }4\rangle 
\tag{7.275}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^3a,bab,pb,\,\text{class }4\rangle  \tag{7.276}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^3a,bab-baaa,pb,\,\text{class }4\rangle  \tag{7.277}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^3a,bab,pb-baaa,\,\text{class }4\rangle  \tag{7.278}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^3a,bab-baaa,pb-baaa,\,\text{class }4\rangle  \tag{7.279}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a,bab,pb-\omega baaa,\,\text{class }4\rangle \;(p=1%
\func{mod}3)  \tag{7.280}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a,bab-baaa,pb-\omega baaa,\,\text{class }4\rangle \;(p=1%
\func{mod}3)  \tag{7.281}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a,bab,pb-\omega ^{2}baaa,\,\text{class }4\rangle \;(p=1%
\func{mod}3)  \tag{7.282}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a,bab-baaa,pb-\omega ^{2}baaa,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.283}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^3a-baaa,bab,pb,\,\text{class }4\rangle  \tag{7.284}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^3a-baaa,bab-baaa,pb,\,\text{class }4\rangle  \tag{7.285}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a-baaa,bab-\omega baaa,pb,\,\text{class }4\rangle 
\tag{7.286}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a-baaa,bab-\omega ^{2}baaa,pb,\,\text{class }4\rangle
\;(p=1\func{mod}4)  \tag{7.287}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a-baaa,bab-\omega ^{3}baaa,pb,\,\text{class }4\rangle
\;(p=1\func{mod}4)  \tag{7.288}
\end{equation}

\subsection{Descendants of 6.382}

$p+5+(p-1)/2$ algebras

\begin{equation}
\langle a,b\,|\,baaa,bab,pb-baa,\,\text{class }4\rangle  \tag{7.289}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,bab-p^3a,pb-baa,\,\text{class }4\rangle  \tag{7.290}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,bab-\omega p^{3}a,pb-baa,\,\text{class }4\rangle 
\tag{7.291}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a,bab,pb-baa-xbaaa,\,\text{class }4\rangle \;(\text{all 
}x,\,x\sim -x)  \tag{7.292}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^3a,bab-baaa,pb-baa,\,\text{class }4\rangle  \tag{7.293}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^3a-baaa,bab-xbaaa,pb-baa,\,\text{class }4\rangle \;(\text{%
all }x)  \tag{7.294}
\end{equation}

\subsection{Descendants of 6.383}

$p+5+(p-1)/2$ algebras

\begin{equation}
\langle a,b\,|\,baaa,bab,pb-\omega baa,\,\text{class }4\rangle  \tag{7.295}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,bab-p^{3}a,pb-\omega baa,\,\text{class }4\rangle 
\tag{7.296}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaa,bab-\omega p^{3}a,pb-\omega baa,\,\text{class }4\rangle
\tag{7.297}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a,bab,pb-\omega baa-xbaaa,\,\text{class }4\rangle \;(%
\text{all }x,\,x\sim -x)  \tag{7.298}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a,bab-baaa,pb-\omega baa,\,\text{class }4\rangle 
\tag{7.299}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^{3}a-baaa,bab-xbaaa,pb-\omega baa,\,\text{class }4\rangle
\;(\text{all }x)  \tag{7.300}
\end{equation}

\subsection{Descendants of 6.384}

2 algebras

\begin{equation}
\langle a,b\,|\,baaa,pb-ba,\,\text{class }4\rangle  \tag{7.301}
\end{equation}

\begin{equation}
\langle a,b\,|\,p^3a,pb-ba,\,\text{class }4\rangle  \tag{7.302}
\end{equation}

\subsection{Descendants of 6.386}

$p+5$ algebras

\begin{equation}
\langle a,b\,|\,baa,bab,pba,p^2b,\,\text{class }4\rangle  \tag{7.303}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-p^3a,bab,pba,p^2b,\,\text{class }4\rangle  \tag{7.304}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,bab-p^3a,pba,p^2b,\,\text{class }4\rangle  \tag{7.305}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,bab-\omega p^{3}a,pba,p^{2}b,\,\text{class }4\rangle 
\tag{7.306}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,bab-xp^3a,pba-p^3a,p^2b,\,\text{class }4\rangle \;(\text{%
all }x)  \tag{7.307}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-p^3a,bab,pba-p^3a,p^2b,\,\text{class }4\rangle 
\tag{7.308}
\end{equation}

\subsection{Descendants of 6.388}

$4p+5+2\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b\,|\,bab,pba,p^2a,p^2b,\,\text{class }4\rangle  \tag{7.309}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pba,p^2a,p^2b,\,\text{class }4\rangle  \tag{7.310}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pba-baaa,p^2a,p^2b,\,\text{class }4\rangle  \tag{7.311}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pba-baaa,p^2a,p^2b,\,\text{class }4\rangle 
\tag{7.312}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pba-\omega baaa,p^{2}a,p^{2}b,\,\text{class }4\rangle 
\tag{7.313}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pba-\omega baaa,p^{2}a,p^{2}b,\,\text{class }%
4\rangle  \tag{7.314}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pba,p^2a-baaa,p^2b,\,\text{class }4\rangle  \tag{7.315}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pba,p^2a-baaa,p^2b,\,\text{class }4\rangle 
\tag{7.316}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-\omega baaa,pba,p^{2}a-baaa,p^{2}b,\,\text{class }%
4\rangle  \tag{7.317}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-\omega ^{2}baaa,pba,p^{2}a-baaa,p^{2}b,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.318}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-\omega ^{3}baaa,pba,p^{2}a-baaa,p^{2}b,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.319}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-xbaaa,pba-baaa,p^2a-baaa,p^2b,\,\text{class }4\rangle \;(%
\text{all }x)  \tag{7.320}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-xbaaa,pba-\omega baaa,p^{2}a-baaa,p^{2}b,\,\text{class }%
4\rangle \;(\text{all }x)  \tag{7.321}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pba,p^2a,p^2b-baaa,\,\text{class }4\rangle  \tag{7.322}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pba,p^2a,p^2b-baaa,\,\text{class }4\rangle 
\tag{7.323}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pba-xbaaa,p^2a,p^2b-baaa,\,\text{class }4\rangle \;(x
\neq 0, x \sim x^{\prime }\text{ if }x^3=x^{\prime 3 }\func{mod} p) 
\tag{7.324}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pba-xbaaa,p^2a,p^2b-baaa,\,\text{class }4\rangle
\;(x \neq 0, x \sim x^{\prime }\text{ if }x^3=x^{\prime 3 }\func{mod} p) 
\tag{7.325}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pba,p^{2}a,p^{2}b-\omega baaa,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.326}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pba,p^{2}a,p^{2}b-\omega baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.327}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pba-xbaaa,p^{2}a,p^{2}b-\omega baaa,\,\text{class }%
4\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}%
p,\;p=1\func{mod}3)  \tag{7.328}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pba-xbaaa,p^{2}a,p^{2}b-\omega baaa,\,\text{class }%
4\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}%
p,\;p=1\func{mod}3)  \tag{7.329}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pba,p^{2}a,p^{2}b-\omega ^{2}baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.330}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pba,p^{2}a,p^{2}b-\omega ^{2}baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.331}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,pba-xbaaa,p^{2}a,p^{2}b-\omega ^{2}baaa,\,\text{class }%
4\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}%
p,\;p=1\func{mod}3)  \tag{7.332}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,pba-xbaaa,p^{2}a,p^{2}b-\omega ^{2}baaa,\,\text{%
class }4\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}%
\func{mod}p,\;p=1\func{mod}3)  \tag{7.333}
\end{equation}

\subsection{Descendants of 6.394}

$p^{2}+3p+10+2\gcd (p-1,3)+3\gcd (p-1,4)+2\gcd (p-1,5)+\gcd (p-1,8)+\gcd
(p-1,9)$ algebras

\begin{equation}
\langle a,b\,|\,baaaa,baab,babb,pa,pb,\,\text{class }5\rangle  \tag{7.334}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,baab,babb,pa-baaab,pb,\,\text{class }5\rangle 
\tag{7.335}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,baab,babb,pa-\omega baaab,pb,\,\text{class }5\rangle 
\tag{7.336}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,baab,babb,pa,pb-baaab,\,\text{class }5\rangle 
\tag{7.337}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,baab,babb,pa-baaab,pb-baaab,\,\text{class }5\rangle 
\tag{7.338}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,baab,babb,pa-\omega baaab,pb-baaab,\,\text{class }%
5\rangle  \tag{7.339}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,baab,babb,pa-\omega ^{2}baaab,pb-baaab,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.340}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,baab,babb,pa-\omega ^{3}baaab,pb-baaab,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.341}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,baab,babb-baaab,pa,pb,\,\text{class }5\rangle 
\tag{7.342}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,baab,babb-baaab,pa,pb-baaab,\,\text{class }5\rangle 
\tag{7.343}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,baab,babb-baaab,pa,pb-xbaaab,\,\text{class }5\rangle
\;(x=\omega ,\omega ^{2},\omega ^{3},\omega ^{4},\;p=1\func{mod}5) 
\tag{7.344}
\end{equation}%
In the following presentation we have$\;y=1,\omega ,\omega ^{2},\omega
^{3},\omega ^{4},\omega ^{5}$, all $x,\,x\sim x^{\prime }$ if $%
x^{6}=x^{\prime 6}\func{mod}p,\;p=1\func{mod}3$:%
\begin{equation}
\langle a,b\,|\,baaaa,baab,babb-baaab,pa-ybaaab,pb-xbaaab,\,\text{class }%
5\rangle  \tag{7.345}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,baab,babb-baaab,pa-ybaaab,pb-xbaaab,\,\text{class }%
5\rangle \;(y=1,\omega ,\text{all }x,\,x\sim -x,\;p=2\func{mod}3) 
\tag{7.346}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,baab-baaab,babb,pa,pb,\,\text{class }5\rangle 
\tag{7.347}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,baab-baaab,babb,pa-baaab,pb,\,\text{class }5\rangle 
\tag{7.348}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,baab-baaab,babb,pa-\omega baaab,pb,\,\text{class }%
5\rangle  \tag{7.349}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,baab-baaab,babb,pa-xbaaab,pb-baaab,\,\text{class }%
5\rangle \;(\text{all }x)  \tag{7.350}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,baab-baaab,babb-baaab,pa-xbaaab,pb-ybaaab,\,\text{%
class }5\rangle \;(\text{all }x,\,y)  \tag{7.351}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb,pa,pb,\,\text{class }5\rangle  \tag{7.352}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb,pa,pb-baaaa,\,\text{class }5\rangle 
\tag{7.353}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb,pa,pb-\omega baaaa,\,\text{class }5\rangle 
\tag{7.354}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb,pa,pb-\omega ^{2}baaaa,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.355}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb,pa,pb-\omega ^{3}baaaa,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.356}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb,pa-baaaa,pb,\,\text{class }5\rangle 
\tag{7.357}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab-baaaa,babb,pa,pb,\,\text{class }5\rangle 
\tag{7.358}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab-baaaa,babb,pa-baaaa,pb,\,\text{class }5\rangle 
\tag{7.359}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab-baaaa,babb,pa-xbaaaa,pb,\,\text{class }5\rangle
\;(x=\omega ,\omega ^{2},\omega ^{3},\omega ^{4},\;p=1\func{mod}5) 
\tag{7.360}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab-baaaa,babb,pa,pb-baaaa,\,\text{class }5\rangle 
\tag{7.361}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab-baaaa,babb,pa,pb-\omega baaaa,\,\text{class }%
5\rangle  \tag{7.362}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab-baaaa,babb,pa,pb-\omega ^{2}baaaa,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.363}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab-baaaa,babb,pa,pb-\omega ^{3}baaaa,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.364}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb-baaaa,pa,pb,\,\text{class }5\rangle 
\tag{7.365}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb-baaaa,pa-baaaa,pb,\,\text{class }5\rangle 
\tag{7.366}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb-baaaa,pa-ybaaaa,pb,\,\text{class }5\rangle
\;(y=\omega ,\omega ^{2},\;p=1\func{mod}3)  \tag{7.367}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb-baaaa,pa-ybaaaa,pb,\,\text{class }5\rangle
\;(y=\omega ^{3},\omega ^{4},\omega ^{5},\omega ^{6},\omega ^{7},\omega
^{8},\;p=1\func{mod}9)  \tag{7.368}
\end{equation}
\begin{equation}
\langle a,b\,|\,baaab,baab,babb-baaaa,pa-xbaaaa,pb-ybaaaa,\,\text{class }%
5\rangle \;(y=1,\omega ,\text{all }x,\,x \sim x^{\prime }\text{ if }%
x^8=x^{\prime 8}\func{mod}p)  \tag{7.369}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb-baaaa,pa-xbaaaa,pb-ybaaaa,\,\text{class }%
5\rangle \;(y=\omega ^{2},\omega ^{3},\text{all }x,\,x\sim x^{\prime}\text{
if }x^8=x^{\prime 8}\func{mod}p,\;p=1\func{mod}4)  \tag{7.370}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb-baaaa,pa-xbaaaa,pb-ybaaaa,\,\text{class }%
5\rangle \;(y=\omega ^{4},\omega ^{5},\omega ^{6},\omega ^{7},\text{all }%
x,\,x \sim x^{\prime }\text{ if }x^8=x^{\prime 8}\func{mod}p,\;p=1\func{mod}%
8)  \tag{7.371}
\end{equation}

\subsection{Descendants of 6.399}

$(1/2)p^{3}+(3/2)p^{2}+4p+7+3\gcd (p-1,3)+(p+3)\gcd (p-1,4)/2+\gcd
(p-1,5)+\gcd (p-1,8)/2$ algebras

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+baaa,pa,pb,\,\text{class }5\rangle 
\tag{7.372}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+baaa,pa-baaaa,pb,\,\text{class }5\rangle 
\tag{7.373}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+baaa,pa-\omega baaaa,pb,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.374}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+baaa,pa-\omega ^{2}baaaa,pb,\,\text{class }%
5\rangle \;(p=1\func{mod}8)  \tag{7.375}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+baaa,pa-\omega ^{3}baaaa,pb,\,\text{class }%
5\rangle \;(p=1\func{mod}8)  \tag{7.376}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+baaa,pa-xbaaaa,pb-baaaa,\,\text{class }%
5\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.377}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+baaa,pa-xbaaaa,pb-\omega baaaa,\,\text{class 
}5\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.378}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+baaa,pa-xbaaaa,pb-\omega ^{2}baaaa,\,\text{%
class }5\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.379}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+baaa,pa-xbaaaa,pb-\omega ^{3}baaaa,\,\text{%
class }5\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.380}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab-baaaa,babb+baaa,pa-xbaaaa,pb-ybaaaa,\,\text{class 
}5\rangle \;(\text{all }x,\,y,x\sim -x)  \tag{7.381}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+baaa-baaaa,pa-xbaaaa,pb-ybaaaa,\,\text{class 
}5\rangle \;(\text{all }x,\,y,x\sim -x)  \tag{7.382}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab-xbaaaa,babb+baaa-baaaa,pa-ybaaaa,pb-zbaaaa,\,%
\text{class }5\rangle \;(x\neq 0,\text{ all }y,z,x\sim -x)  \tag{7.383}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab,babb+baaa,pa,pb,\,\text{class }5\rangle 
\tag{7.384}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab,babb+baaa,pa,pb-baaaa,\,\text{class }%
5\rangle  \tag{7.385}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab,babb+baaa,pa,pb-\omega baaaa,\,\text{class }%
5\rangle  \tag{7.386}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab,babb+baaa,pa,pb-\omega ^{2}baaaa,\,\text{%
class }5\rangle \;(p=1\func{mod}4)  \tag{7.387}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab,babb+baaa,pa,pb-\omega ^{3}baaaa,\,\text{%
class }5\rangle \;(p=1\func{mod}4)  \tag{7.388}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab,babb+baaa,pa-baaaa,pb-baaaa,\,\text{class }%
5\rangle  \tag{7.389}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab,babb+baaa,pa-\omega baaaa,pb-\omega baaaa,\,%
\text{class }5\rangle  \tag{7.390}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab,babb+baaa,pa-baaaa,pb+baaaa,\,\text{class }%
5\rangle  \tag{7.391}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab,babb+baaa-baaaa,pa-xbaaaa,pb-ybaaaa,\,\text{%
class }5\rangle \;(\text{all }x,\,y,(x,\,y)\sim (y,x))  \tag{7.392}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab-baaaa,babb+baaa-2baaaa,pa,pb-xbaaaa,\,\text{%
class }5\rangle \;(\text{all }x)  \tag{7.393}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab-baaaa,babb+baaa-2baaaa,pa-baaaa,pb-baaaa,\,%
\text{class }5\rangle  \tag{7.394}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab-baaaa,babb+baaa-2baaaa,pa-\omega
baaaa,pb-\omega baaaa,\,\text{class }5\rangle  \tag{7.395}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab-baaaa,babb+baaa-2baaaa,pa-baaaa,pb+baaaa,\,%
\text{class }5\rangle  \tag{7.396}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab-baaaa,babb+baaa-2baaaa,pa-\omega
baaaa,pb+\omega baaaa,\,\text{class }5\rangle \;(p=1\func{mod}3)  \tag{7.397}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab-baaaa,babb+baaa-2baaaa,pa-\omega
^{2}baaaa,pb+\omega ^{2}baaaa,\,\text{class }5\rangle \;(p=1\func{mod}3) 
\tag{7.398}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab-baaaa,babb+baaa+2baaaa,pa,pb-xbaaaa,\,\text{%
class }5\rangle \;(\text{all }x)  \tag{7.399}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab-baaaa,babb+baaa+2baaaa,pa-baaaa,pb-baaaa,\,%
\text{class }5\rangle  \tag{7.400}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab-baaaa,babb+baaa+2baaaa,pa-\omega
baaaa,pb-\omega baaaa,\,\text{class }5\rangle  \tag{7.401}
\end{equation}

\begin{equation}
\langle
a,b\,|\,baaab-baaaa,baab-baaaa,babb+baaa+2baaaa,pa-xbaaaa,pb-xbaaaa,\,\text{%
class }5\rangle \;(x=\omega ^{2},\omega ^{3},\omega ^{4},\omega ^{5},\;p=1%
\func{mod}3)  \tag{7.402}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab-baaaa,baab-baaaa,babb+baaa+2baaaa,pa-baaaa,pb+baaaa,\,%
\text{class }5\rangle  \tag{7.403}
\end{equation}

\begin{equation}
\langle
a,b\,|\,baaab-baaaa,baab-baaaa,babb+baaa+2baaaa,pa-xbaaaa,pb+xbaaaa,\,\text{%
class }5\rangle \;(x=\omega ,\omega ^{2},\omega ^{3},\omega ^{4},\;p=1\func{%
mod}5)  \tag{7.404}
\end{equation}

In the following presentation, $z$ is chosen so as to be any \emph{one}
integer such that $z^{2}-4$ is not a square $\func{mod}p$:

\begin{equation}
\langle
a,b\,|\,baaab-baaaa,baab-baaaa,babb+baaa-zbaaaa,pa-xbaaaa,pb-ybaaaa,\,\text{%
class }5\rangle \;(\text{all }x,\,y,(x,\,y)\sim (y,x))  \tag{7.405}
\end{equation}

\subsection{Descendants of 6.404}

$(p^{3}+p^{2}+2p+2+(p+1)\gcd (p-1,4)+\gcd (p-1,8))/2$ algebras

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+\omega baaa,pa,pb,\,\text{class }5\rangle 
\tag{7.406}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+\omega baaa,pa-baaaa,pb,\,\text{class }%
5\rangle  \tag{7.407}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+\omega baaa,pa-\omega baaaa,pb,\,\text{class 
}5\rangle \;(p=1\func{mod}4)  \tag{7.408}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+\omega baaa,pa-\omega ^{2}baaaa,pb,\,\text{%
class }5\rangle \;(p=1\func{mod}8)  \tag{7.409}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+\omega baaa,pa-\omega ^{3}baaaa,pb,\,\text{%
class }5\rangle \;(p=1\func{mod}8)  \tag{7.410}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+\omega baaa,pa-xbaaaa,pb-baaaa,\,\text{class 
}5\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.411}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+\omega baaa,pa-xbaaaa,pb-\omega baaaa,\,%
\text{class }5\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.412}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+\omega baaa,pa-xbaaaa,pb-\omega ^{2}baaaa,\,%
\text{class }5\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4) 
\tag{7.413}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+\omega baaa,pa-xbaaaa,pb-\omega ^{3}baaaa,\,%
\text{class }5\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4) 
\tag{7.414}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab-baaaa,babb+\omega baaa,pa-xbaaaa,pb-ybaaaa,\,%
\text{class }5\rangle \;(\text{all }x,\,y,x\sim -x)  \tag{7.415}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab,babb+\omega baaa-baaaa,pa-xbaaaa,pb-ybaaaa,\,%
\text{class }5\rangle \;(\text{all }x,\,y,x\sim -x)  \tag{7.416}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,baab-xbaaaa,babb+\omega
baaa-baaaa,pa-ybaaaa,pb-zbaaaa,\,\text{class }5\rangle \;(x\neq 0,\text{ all 
}y,z,x\sim -x)  \tag{7.417}
\end{equation}

\subsection{Descendants of 6.408}

$(1/2)(p+1)(p-1+\gcd (p-1,3))$ algebras 
\begin{equation}
\langle a,b\,|\,pa-bab-xbaaaa,pb-baaa-ybaaaa,\,\text{class }5\rangle \;(%
\text{all }x,\,y,x\sim -x,\,y\sim y^{\prime }\text{ if }y^{3}=y^{\prime 3}%
\func{mod}p,\;p=1\func{mod}3)  \tag{7.418}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-bab-xbaaaa,pb-\omega baaa-ybaaaa,\,\text{class }5\rangle
\;(\text{all }x,\,y,x\sim -x,\,y \sim y^{\prime }\text{ if }y^3=y^{\prime 3}%
\func{mod}p,\;p=1\func{mod}3)  \tag{7.419}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-bab-xbaaaa,pb-\omega ^{2}baaa-ybaaaa,\,\text{class }%
5\rangle \;(\text{all }x,\,y,x\sim -x,\,y \sim y^{\prime }\text{ if }%
y^3=y^{\prime 3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.420}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-bab-xbaaaa,pb-baaa-ybaaaa,\,\text{class }5\rangle \;(%
\text{all }x,\,y,x\sim -x,\;p=2\func{mod}3)  \tag{7.421}
\end{equation}

\subsection{Descendants of 6.411}

$2+2\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b\,|\,baaaa,pa-bab,pb,\,\text{class }5\rangle  \tag{7.422}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,pa-bab,pb-baaab,\,\text{class }5\rangle  \tag{7.423}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,pa-bab,pb-\omega baaab,\,\text{class }5\rangle \;(p=1%
\func{mod}3)  \tag{7.424}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,pa-bab,pb-\omega ^{2}baaab,\,\text{class }5\rangle
\;(p=1\func{mod}3)  \tag{7.425}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-bab,pb,\,\text{class }5\rangle  \tag{7.426}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-bab-baaaa,pb,\,\text{class }5\rangle  \tag{7.427}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-bab-\omega baaaa,pb,\,\text{class }5\rangle \;(p=1%
\func{mod}3)  \tag{7.428}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-bab-\omega ^{2}baaaa,pb,\,\text{class }5\rangle
\;(p=1\func{mod}3)  \tag{7.429}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-bab,pb-baaaa,\,\text{class }5\rangle  \tag{7.430}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-bab,pb-\omega baaaa,\,\text{class }5\rangle 
\tag{7.431}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-bab,pb-\omega ^{2}baaaa,\,\text{class }5\rangle
\;(p=1\func{mod}4)  \tag{7.432}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-bab,pb-\omega ^{3}baaaa,\,\text{class }5\rangle
\;(p=1\func{mod}4)  \tag{7.433}
\end{equation}

\subsection{Descendants of 6.412}

$p-1+(1/2)(p+1)\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b\,|\,baaaa,pa-bab-baaa,pb-xbaaab,\,\text{class }5\rangle \;(\text{%
all }x,\,x\sim x^{\prime }\ \text{if}\ x^{4}=x^{\prime 4}\func{mod}p,\;p=1%
\func{mod}4)  \tag{7.434}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,pa-bab-\omega baaa,pb-xbaaab,\,\text{class }5\rangle
\;(\text{all }x,\,x\sim x^{\prime }\ \text{if}\ x^{4}=x^{\prime 4}\func{mod}%
p,\;p=1\func{mod}4)  \tag{7.435}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,pa-bab-baaa,pb-xbaaab,\,\text{class }5\rangle \;(\text{%
all }x,\,x\sim -x,\;p=3\func{mod}4)  \tag{7.436}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-bab-baaa,pb-xbaaaa,\,\text{class }5\rangle \;(\text{%
all }x)  \tag{7.437}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-bab-\omega baaa,pb-xbaaaa,\,\text{class }5\rangle
\;(\text{all }x,\;p=1\func{mod}4)  \tag{7.438}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-bab-baaa-xbaaaa,pb+2baaaa,\,\text{class }5\rangle
\;(x\neq 0,x\sim x^{\prime }\ \text{if}\ x^{4}=x^{\prime 4}\func{mod}p,\;p=1%
\func{mod}4)  \tag{7.439}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-bab-\omega baaa-xbaaaa,pb+(2\ast \omega
^{2})baaaa,\,\text{class }5\rangle \;(x\neq 0,x\sim x^{\prime }\ \text{if}\
x^{4}=x^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.440}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-bab-baaa-xbaaaa,pb+2baaaa,\,\text{class }5\rangle
\;(x \neq 0, x \sim -x,\;p=3 \func{mod} 4)  \tag{7.441}
\end{equation}

\subsection{Descendants of 6.414}

$(1/2)(p+1)(p-1+\gcd (p-1,3))$ algebras

\begin{equation}
\langle a,b\,|\,pa-\omega bab-xbaaaa,pb-baaa-ybaaaa,\,\text{class }5\rangle
\;(\text{all }x,\,y,x\sim -x,\,y \sim y^{\prime }\text{ if }y^3=y^{\prime 3}%
\func{mod}p,\;p=1\func{mod}3)  \tag{7.442}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-\omega bab-xbaaaa,pb-\omega baaa-ybaaaa,babb+baaaa,\,%
\text{class }5\rangle \;(\text{all }x,\,y,x\sim -x,\,y\sim y^{\prime}\text{
if }y^3=y^{\prime 3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.443}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-\omega bab-xbaaaa,pb-\omega ^{2}baaa-ybaaaa,babb+\omega
baaaa,\,\text{class }5\rangle \;(\text{all }x,\,y,x\sim -x,\,y\sim y^{\prime}%
\text{ if }y^3=y^{\prime 3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.444}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa-\omega bab-xbaaaa,pb-baaa-ybaaaa,\,\text{class }5\rangle
\;(\text{all }x,\,y,x\sim -x,\;p=2\func{mod}3)  \tag{7.445}
\end{equation}

\subsection{Descendants of 6.417}

$2+2\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b\,|\,baaaa,pa-\omega bab,pb,\,\text{class }5\rangle  \tag{7.446}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,pa-\omega bab,pb-baaab,\,\text{class }5\rangle 
\tag{7.447}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,pa-\omega bab,pb-\omega baaab,\,\text{class }5\rangle
\;(p=1\func{mod}3)  \tag{7.448}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,pa-\omega bab,pb-\omega ^{2}baaab,\,\text{class }%
5\rangle \;(p=1\func{mod}3)  \tag{7.449}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-\omega bab,pb,\,\text{class }5\rangle  \tag{7.450}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-\omega bab-baaaa,pb,\,\text{class }5\rangle 
\tag{7.451}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-\omega bab-\omega baaaa,pb,\,\text{class }5\rangle
\;(p=1\func{mod}3)  \tag{7.452}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-\omega bab-\omega ^{2}baaaa,pb,\,\text{class }%
5\rangle \;(p=1\func{mod}3)  \tag{7.453}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-\omega bab,pb-baaaa,\,\text{class }5\rangle 
\tag{7.454}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-\omega bab,pb-\omega baaaa,\,\text{class }5\rangle 
\tag{7.455}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-\omega bab,pb-\omega ^{2}baaaa,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.456}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-\omega bab,pb-\omega ^{3}baaaa,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.457}
\end{equation}

\subsection{Descendants of 6.418}

$p-1+(1/2)(p+1)\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b\,|\,baaaa,pa-\omega bab-baaa,pb-xbaaab,\,\text{class }5\rangle
\;(\text{all }x,\,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}\func{mod}%
p,\;p=1\func{mod}4)  \tag{7.458}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,pa-\omega bab-\omega baaa,pb-xbaaab,babb+baaab,\,\text{%
class }5\rangle \;(\text{all }x,\,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}%
\func{mod}p,\;p=1\func{mod}4)  \tag{7.459}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,pa-\omega bab-baaa,pb-xbaaab,\,\text{class }5\rangle
\;(\text{all }x,\,x\sim -x,\;p=3\func{mod}4)  \tag{7.460}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-\omega bab-baaa,pb-xbaaaa,\,\text{class }5\rangle
\;(\text{all }x)  \tag{7.461}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-\omega bab-\omega baaa,pb-xbaaaa,baab+baaaa,\,\text{%
class }5\rangle \;(\text{all }x,\;p=1\func{mod}4)  \tag{7.462}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-\omega bab-baaa-xbaaaa,pb+2\omega ^{-1}baaaa,\,%
\text{class }5\rangle \;(x\neq 0,x\sim x^{\prime }\text{ if }x^{4}=x^{\prime
4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.463}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-\omega bab-\omega baaa-xbaaaa,pb+2\omega
baaaa,baab+baaaa,\,\text{class }5\rangle \;(x\neq 0,x\sim x^{\prime }\text{
if }x^{4}=x^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.464}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,pa-\omega bab-baaa-xbaaaa,pb+2\omega ^{-1}baaaa,\,%
\text{class }5\rangle \;(x\neq 0,x\sim -x,\;p=3\func{mod}4)  \tag{7.465}
\end{equation}

\subsection{Descendants of 6.420}

$5p+1+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b\,|\,babba,pa-baa-xbabbb,pb-babbb,\,\text{class }5\rangle \;(%
\text{all }x)  \tag{7.466}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,pa-baa-xbabbb,pb-\omega babbb,\,\text{class }5\rangle
\;(\text{all }x)  \tag{7.467}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,pa-baa,pb,\,\text{class }5\rangle  \tag{7.468}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,pa-baa-babbb,pb,\,\text{class }5\rangle  \tag{7.469}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,pa-baa-\omega babbb,pb,\,\text{class }5\rangle 
\tag{7.470}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,pa-baa-\omega ^{2}babbb,pb,\,\text{class }5\rangle
\;(p=1\func{mod}4)  \tag{7.471}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,pa-baa-\omega ^{3}babbb,pb,\,\text{class }5\rangle
\;(p=1\func{mod}4)  \tag{7.472}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb,pa-baa,pb-xbabba,\,\text{class }5\rangle \;(\text{all }%
x)  \tag{7.473}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb-babba,pa-baa,pb-xbabba,\,\text{class }5\rangle \;(%
\text{all }x)  \tag{7.474}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb-\omega babba,pa-baa,pb-xbabba,\,\text{class }5\rangle
\;(\text{all }x)  \tag{7.475}
\end{equation}

\subsection{Descendants of 6.421}

$2p^{2}-p-1+(p+1)\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b\,|\,babba,pa-baa-babb-xbabbb,pb-ybabbb,\,\text{class }5\rangle
\;(x\neq 0,\text{ all }y,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{%
mod}p)  \tag{7.476}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,pa-baa-\omega babb-xbabbb,pb-ybabbb,\,\text{class }%
5\rangle \;(x\neq 0,\text{ all }y,x \sim x^{\prime }\text{ if }x^3=x^{\prime
3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.477}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,pa-baa-\omega ^{2}babb-xbabbb,pb-ybabbb,\,\text{class }%
5\rangle \;(x\neq 0,\text{ all }y,x \sim x^{\prime }\text{ if }x^3=x^{\prime
3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.478}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,pa-baa-babb,pb-xbabbb,\,\text{class }5\rangle \;(\text{%
all }x,\,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}p) 
\tag{7.479}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,pa-baa-\omega babb,pb-xbabbb,\,\text{class }5\rangle
\;(\text{all }x,\,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}%
p,\;p=1\func{mod}3)  \tag{7.480}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,pa-baa-\omega ^{2}babb,pb-xbabbb,\,\text{class }%
5\rangle \;(\text{all }x,\,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}%
\func{mod}p,\;p=1\func{mod}3)  \tag{7.481}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb-xbabba,pa-baa-babb,pb-ybabba,\,\text{class }5\rangle
\;(\text{all }x,\,y,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}p)
\tag{7.482}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb-xbabba,pa-baa-\omega babb,pb-ybabba,\,\text{class }%
5\rangle \;(\text{all }x,\,y,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}%
\func{mod}p,\;p=1\func{mod}3)  \tag{7.483}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb-xbabba,pa-baa-\omega ^{2}babb,pb-ybabba,\,\text{class }%
5\rangle \;(\text{all }x,\,y,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}%
\func{mod}p,\;p=1\func{mod}3)  \tag{7.484}
\end{equation}

\subsection{Descendants of 6.424}

$p^{2}$ algebras

\begin{equation}
\langle a,b\,|\,pa-baa-xbabb,pb-babb-ybabba,\,\text{class }5\rangle \;(\text{%
all }x,\,y)  \tag{7.485}
\end{equation}

\subsection{Descendants of 6.425}

3 algebras

\begin{equation}
\langle a,b\,|\,baabb,pa-baa,pb+bab,\,\text{class }5\rangle  \tag{7.486}
\end{equation}

\begin{equation}
\langle a,b\,|\,baabb-baaba,pa-baa,pb+bab,\,\text{class }5\rangle 
\tag{7.487}
\end{equation}

\begin{equation}
\langle a,b\,|\,baabb-\omega baaba,pa-baa,pb+bab,\,\text{class }5\rangle 
\tag{7.488}
\end{equation}

\subsection{Descendants of 6.426}

$p$ algebras

\begin{equation}
\langle a,b\,|\,pa-baa-baab,pb+bab-xbaab,\,\text{class }5\rangle \;(\text{%
all }x)  \tag{7.489}
\end{equation}

\subsection{Descendants of 6.427}

$p+1$ algebras

\begin{equation}
\langle a,b\,|\,pa+bab,pb+\omega baa-xbaaa,baaab,\,\text{class }5\rangle \;(%
\text{all }x,\,x\sim -x)  \tag{7.490}
\end{equation}

In the following two presentation, $y$ is chosen so as to be any \emph{one}
integer such that $1-\omega y^{2}$ is not a square $\func{mod}p$:

\begin{equation}
\langle a,b\,|\,pa+bab-xybaaa,pb+\omega baa-xbaaa,baaab-\omega ybaaaa,\,%
\text{class }5\rangle \;(x\neq 0,x\sim -x)  \tag{7.491}
\end{equation}

\begin{equation}
\langle a,b\,|\,pa+bab,pb+\omega baa,baaab+\omega ybaaaa,\,\text{class }%
5\rangle  \tag{7.492}
\end{equation}

\subsection{Descendants of 6.428}

2 algebras

\begin{equation}
\langle a,b\,|\,ba,p^2b,\,\text{class }5\rangle  \tag{7.493}
\end{equation}

\begin{equation}
\langle a,b\,|\,ba-p^4a,p^2b,\,\text{class }5\rangle  \tag{7.494}
\end{equation}

\subsection{Descendants of 6.429}

1 algebra

\begin{equation}
\langle a,b\,|\,ba-p^3a,p^2b,\,\text{class }5\rangle  \tag{7.495}
\end{equation}

\subsection{Descendants of 6.431}

4 algebras

\begin{equation}
\langle a,b\,|\,baa,bab,pb,\,\text{class }5\rangle  \tag{7.496}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,bab-p^4a,pb,\,\text{class }5\rangle  \tag{7.497}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,bab-\omega p^{4}a,pb,\,\text{class }5\rangle  \tag{7.498}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-p^4a,bab,pb,\,\text{class }5\rangle  \tag{7.499}
\end{equation}

\subsection{Descendants of 6.435}

$7+2\gcd (p-1,3)+2\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b\,|\,baaaa,bab,p^2a,pb,\,\text{class }5\rangle  \tag{7.500}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,bab,p^2a-baaab,pb,\,\text{class }5\rangle  \tag{7.501}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,bab,p^{2}a-\omega baaab,pb,\,\text{class }5\rangle 
\tag{7.502}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,bab,p^2a,pb-baaab,\,\text{class }5\rangle  \tag{7.503}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab,p^2a-baaaa,pb,\,\text{class }5\rangle  \tag{7.504}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-baaaa,p^2a-baaaa,pb,\,\text{class }5\rangle 
\tag{7.505}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-\omega baaaa,p^{2}a-baaaa,pb,\,\text{class }%
5\rangle  \tag{7.506}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-\omega ^{2}baaaa,p^{2}a-baaaa,pb,\,\text{class }%
5\rangle \;(p=1\func{mod}3)  \tag{7.507}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-\omega ^{3}baaaa,p^{2}a-baaaa,pb,\,\text{class }%
5\rangle \;(p=1\func{mod}3)  \tag{7.508}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-\omega ^{4}baaaa,p^{2}a-baaaa,pb,\,\text{class }%
5\rangle \;(p=1\func{mod}3)  \tag{7.509}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-\omega ^{5}baaaa,p^{2}a-baaaa,pb,\,\text{class }%
5\rangle \;(p=1\func{mod}3)  \tag{7.510}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab,p^2a,pb,\,\text{class }5\rangle  \tag{7.511}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-baaaa,p^2a,pb,\,\text{class }5\rangle  \tag{7.512}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab,p^2a,pb-baaaa,\,\text{class }5\rangle  \tag{7.513}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-baaaa,p^2a,pb-baaaa,\,\text{class }5\rangle 
\tag{7.514}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab,p^{2}a,pb-\omega baaaa,\,\text{class }5\rangle 
\tag{7.515}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-baaaa,p^{2}a,pb-\omega baaaa,\,\text{class }%
5\rangle  \tag{7.516}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab,p^{2}a,pb-\omega ^{2}baaaa,\,\text{class }5\rangle
\;(p=1\func{mod}4)  \tag{7.517}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-baaaa,p^{2}a,pb-\omega ^{2}baaaa,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.518}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab,p^{2}a,pb-\omega ^{3}baaaa,\,\text{class }5\rangle
\;(p=1\func{mod}4)  \tag{7.519}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-baaaa,p^{2}a,pb-\omega ^{3}baaaa,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.520}
\end{equation}

\subsection{Descendants of 6.436}

$2+2\gcd (p-1,3)+\gcd (p-1,4)+2\gcd (p-1,5)$ algebras

\begin{equation}
\langle a,b\,|\,baaaa,bab-baaa,p^2a,pb,\,\text{class }5\rangle  \tag{7.521}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,bab-baaa,p^2a-baaab,pb,\,\text{class }5\rangle 
\tag{7.522}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,bab-baaa,p^{2}a-\omega baaab,pb,\,\text{class }5\rangle
\tag{7.523}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,bab-baaa,p^{2}a-xbaaab,pb,\,\text{class }5\rangle
\;(x=\omega ^{2},\omega ^{3},\omega ^{4},\omega ^{5},\;p=1\func{mod}3) 
\tag{7.524}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,bab-baaa,p^2a,pb-baaab,\,\text{class }5\rangle 
\tag{7.525}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaaa,bab-baaa,p^{2}a,pb-xbaaab,\,\text{class }5\rangle
\;(x=\omega ,\omega ^{2},\omega ^{3},\omega ^{4},\;p=1\func{mod}5) 
\tag{7.526}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-baaa,p^2a,pb,\,\text{class }5\rangle  \tag{7.527}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-baaa,p^2a-baaaa,pb,\,\text{class }5\rangle 
\tag{7.528}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-baaa,p^{2}a-xbaaaa,pb,\,\text{class }5\rangle
\;(x=\omega ,\omega ^{2},\omega ^{3},\omega ^{4},\;p=1\func{mod}5) 
\tag{7.529}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-baaa,p^2a,pb-baaaa,\,\text{class }5\rangle 
\tag{7.530}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-baaa,p^{2}a,pb-\omega baaaa,\,\text{class }5\rangle
\tag{7.531}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-baaa,p^{2}a,pb-\omega ^{2}baaaa,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.532}
\end{equation}

\begin{equation}
\langle a,b\,|\,baaab,bab-baaa,p^{2}a,pb-\omega ^{3}baaaa,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.533}
\end{equation}

\subsection{Descendants of 6.442}

$2p-2+(p+2)\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b\,|\,bab,p^2a,pb-baaa,\,\text{class }5\rangle  \tag{7.534}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^2a-baaaa,pb-baaa,\,\text{class }5\rangle  \tag{7.535}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,p^2a-xbaaaa,pb-baaa,\,\text{class }5\rangle \;(%
\text{all }x)  \tag{7.536}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^{2}a,pb-\omega baaa,\,\text{class }5\rangle \;(p=1%
\func{mod}3)  \tag{7.537}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^{2}a-baaaa,pb-\omega baaa,\,\text{class }5\rangle
\;(p=1\func{mod}3)  \tag{7.538}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,p^{2}a-xbaaaa,pb-\omega baaa,\,\text{class }%
5\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.539}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^{2}a,pb-\omega ^{2}baaa,\,\text{class }5\rangle \;(p=1%
\func{mod}3)  \tag{7.540}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^{2}a-baaaa,pb-\omega ^{2}baaa,\,\text{class }5\rangle
\;(p=1\func{mod}3)  \tag{7.541}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,p^{2}a-xbaaaa,pb-\omega ^{2}baaa,\,\text{class }%
5\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.542}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^2a,pb-baaa-xbaaaa,\,\text{class }5\rangle \;(x \neq
0,\;p=2 \func{mod} 3)  \tag{7.543}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,p^2a,pb-baaa-xbaaaa,\,\text{class }5\rangle \;(x
\neq 0,\;p=2 \func{mod} 3)  \tag{7.544}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^2a,pb-baaa-xbaaaa,\,\text{class }5\rangle \;(x \neq 0,
x \sim x^{\prime }\text{ if }x^3=x^{\prime 3 }\func{mod} p,\;p=1 \func{mod}
3)  \tag{7.545}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,p^2a,pb-baaa-xbaaaa,\,\text{class }5\rangle \;(x
\neq 0, x \sim x^{\prime }\text{ if }x^3=x^{\prime 3 }\func{mod} p,\;p=1 
\func{mod} 3)  \tag{7.546}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^{2}a,pb-\omega baaa-xbaaaa,\,\text{class }5\rangle
\;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}p,\;p=1%
\func{mod}3)  \tag{7.547}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,p^{2}a,pb-\omega baaa-xbaaaa,\,\text{class }%
5\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}%
p,\;p=1\func{mod}3)  \tag{7.548}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,p^{2}a,pb-\omega ^{2}baaa-xbaaaa,\,\text{class }5\rangle
\;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}p,\;p=1%
\func{mod}3)  \tag{7.549}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,p^{2}a,pb-\omega ^{2}baaa-xbaaaa,\,\text{class }%
5\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}%
p,\;p=1\func{mod}3)  \tag{7.550}
\end{equation}

\subsection{Descendants of 6.445}

$2p-2+\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b\,|\,bab-baaa,p^{2}a-xbaaaa,pb-baaa,\,\text{class }5\rangle \;(%
\text{all }x,\;p=2\func{mod}3)  \tag{7.551}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,p^2a,pb-baaa-xbaaaa,\,\text{class }5\rangle \;(x
\neq 0,\;p=2 \func{mod} 3)  \tag{7.552}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,p^{2}a-xbaaaa,pb-baaa,\,\text{class }5\rangle \;(%
\text{all }x,\,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}p,\;p=1%
\func{mod}3)  \tag{7.553}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,p^2a,pb-baaa-xbaaaa,\,\text{class }5\rangle \;(x
\neq 0, x \sim x^{\prime }\text{ if }x^3=x^{\prime 3 }\func{mod} p,\;p=1 
\func{mod} 3)  \tag{7.554}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,p^{2}a-xbaaaa,pb-\omega baaa,\,\text{class }%
5\rangle \;(\text{all }x,\,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}%
\func{mod}p,\;p=1\func{mod}3)  \tag{7.555}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,p^{2}a,pb-\omega baaa-xbaaaa,\,\text{class }%
5\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}%
p,\;p=1\func{mod}3)  \tag{7.556}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,p^{2}a-xbaaaa,pb-\omega ^{2}baaa,\,\text{class }%
5\rangle \;(\text{all }x,\,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}%
\func{mod}p,\;p=1\func{mod}3)  \tag{7.557}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,p^{2}a,pb-\omega ^{2}baaa-xbaaaa,\,\text{class }%
5\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}%
p,\;p=1\func{mod}3)  \tag{7.558}
\end{equation}

\subsection{Descendants of 6.448}

$(p+1)/2$ algebras

\begin{equation}
\langle a,b\,|\,bab,p^{2}a,pb-baa-xbaaa,\,\text{class }5\rangle \;(\text{all 
}x,\,x\sim -x)  \tag{7.559}
\end{equation}

\subsection{Descendants of 6.451}

$(p+1)/2$ algebras

\begin{equation}
\langle a,b\,|\,bab,p^{2}a,pb-\omega baa-xbaaa,\,\text{class }5\rangle \;(%
\text{all }x,\,x\sim -x)  \tag{7.560}
\end{equation}

\subsection{Descendants of 6.454}

$8+2\gcd (p-1,3)+4\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b\,|\,babba,baa,p^2a,pb,\,\text{class }5\rangle  \tag{7.561}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa,p^2a,pb-babbb,\,\text{class }5\rangle  \tag{7.562}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa-babbb,p^2a,pb,\,\text{class }5\rangle  \tag{7.563}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa-babbb,p^2a,pb-babbb,\,\text{class }5\rangle 
\tag{7.564}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa-babbb,p^{2}a,pb-\omega babbb,\,\text{class }%
5\rangle  \tag{7.565}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa-babbb,p^{2}a,pb-xbabbb,\,\text{class }5\rangle
\;(x=\omega ^{2},\omega ^{3},\omega ^{4},\omega ^{5},\;p=1\func{mod}3) 
\tag{7.566}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa,p^2a-babbb,pb,\,\text{class }5\rangle  \tag{7.567}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa,p^{2}a-\omega babbb,pb,\,\text{class }5\rangle 
\tag{7.568}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa,p^{2}a-\omega ^{2}babbb,pb,\,\text{class }5\rangle
\;(p=1\func{mod}4)  \tag{7.569}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa,p^{2}a-\omega ^{3}babbb,pb,\,\text{class }5\rangle
\;(p=1\func{mod}4)  \tag{7.570}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa-babbb,p^2a-babbb,pb,\,\text{class }5\rangle 
\tag{7.571}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa-babbb,p^{2}a-\omega babbb,pb,\,\text{class }%
5\rangle  \tag{7.572}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa-babbb,p^{2}a-\omega ^{2}babbb,pb,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.573}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa-babbb,p^{2}a-\omega ^{3}babbb,pb,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.574}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb,baa,p^2a,pb,\,\text{class }5\rangle  \tag{7.575}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb,baa,p^2a-babba,pb,\,\text{class }5\rangle  \tag{7.576}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb,baa,p^2a,pb-babba,\,\text{class }5\rangle  \tag{7.577}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb,baa,p^{2}a,pb-\omega babba,\,\text{class }5\rangle 
\tag{7.578}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb-babba,baa,p^2a,pb,\,\text{class }5\rangle  \tag{7.579}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb-babba,baa,p^2a-babba,pb,\,\text{class }5\rangle 
\tag{7.580}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb-babba,baa,p^{2}a-\omega babba,pb,\,\text{class }%
5\rangle  \tag{7.581}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb-babba,baa,p^{2}a-\omega ^{2}babba,pb,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.582}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb-babba,baa,p^{2}a-\omega ^{3}babba,pb,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.583}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb-babba,baa,p^2a,pb-babba,\,\text{class }5\rangle 
\tag{7.584}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb-babba,baa,p^{2}a,pb-\omega babba,\,\text{class }%
5\rangle  \tag{7.585}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb-babba,baa,p^{2}a,pb-\omega ^{2}babba,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.586}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb-babba,baa,p^{2}a,pb-\omega ^{3}babba,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.587}
\end{equation}

\subsection{Descendants of 6.455}

4 algebras

\begin{equation}
\langle a,b\,|\,baa,p^2a,pb-babb,\,\text{class }5\rangle  \tag{7.588}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,p^2a-babba,pb-babb,\,\text{class }5\rangle  \tag{7.589}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,p^2a,pb-babb-babba,\,\text{class }5\rangle  \tag{7.590}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa,p^{2}a,pb-babb-\omega babba,\,\text{class }5\rangle 
\tag{7.591}
\end{equation}

\subsection{Descendants of 6.459}

$4p+2\gcd (p-1,3)+\gcd (p-1,4)+2\gcd (p-1,5)$ algebras

\begin{equation}
\langle a,b\,|\,babba,baa-babb,p^2a,pb,\,\text{class }5\rangle  \tag{7.592}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa-babb,p^2a-babbb,pb,\,\text{class }5\rangle 
\tag{7.593}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa-babb,p^{2}a-\omega babbb,pb,\,\text{class }5\rangle
\tag{7.594}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa-babb,p^{2}a-\omega ^{2}babbb,pb,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.595}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa-babb,p^{2}a-\omega ^{3}babbb,pb,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.596}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa-babb,p^2a,pb-babbb,\,\text{class }5\rangle 
\tag{7.597}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa-babb,p^{2}a,pb-xbabbb,\,\text{class }5\rangle
\;(x=\omega ,\omega ^{2},\omega ^{3},\omega ^{4},\;p=1\func{mod}5) 
\tag{7.598}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa-babb-babbb,p^2a-xbabbb,pb,\,\text{class }5\rangle
\;(\text{all }x)  \tag{7.599}
\end{equation}

\begin{equation}
\langle a,b\,|\,babba,baa-babb-babbb,p^2a,pb-xbabbb,\,\text{class }5\rangle
\;(x \neq 0)  \tag{7.600}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb,baa-babb,p^2a,pb,\,\text{class }5\rangle  \tag{7.601}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb,baa-babb,p^2a-babba,pb,\,\text{class }5\rangle 
\tag{7.602}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb,baa-babb,p^{2}a-xbabba,pb,\,\text{class }5\rangle
\;(x=\omega ,\omega ^{2},\omega ^{3},\omega ^{4},\;p=1\func{mod}5) 
\tag{7.603}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb,baa-babb,p^2a,pb-babba,\,\text{class }5\rangle 
\tag{7.604}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb,baa-babb,p^{2}a,pb-\omega babba,\,\text{class }5\rangle
\tag{7.605}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb,baa-babb,p^{2}a,pb-xbabba,\,\text{class }5\rangle
\;(x=\omega ^{2},\omega ^{3},\omega ^{4},\omega ^{5},\;p=1\func{mod}3) 
\tag{7.606}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb-babba,baa-babb,p^2a-xbabba,pb,\,\text{class }5\rangle
\;(\text{all }x)  \tag{7.607}
\end{equation}

\begin{equation}
\langle a,b\,|\,babbb-babba,baa-babb,p^2a,pb-xbabba,\,\text{class }5\rangle
\;(x \neq 0)  \tag{7.608}
\end{equation}

\subsection{Descendants of 6.460}

$3p-3+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b\,|\,baa-babb,p^{2}a-xbabba,pb-babb,\,\text{class }5\rangle \;(%
\text{all }x,\,x\sim -x,\;p=3\func{mod}4)  \tag{7.609}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,p^{2}a-xbabba,pb-\omega babb,\,\text{class }%
5\rangle \;(\text{all }x,\,x\sim -x,\;p=3\func{mod}4)  \tag{7.610}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,p^2a,pb-babb-xbabba,\,\text{class }5\rangle \;(x
\neq 0,\;p=3 \func{mod} 4)  \tag{7.611}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,p^{2}a,pb-\omega babb-xbabba,\,\text{class }%
5\rangle \;(x\neq 0,\;p=3\func{mod}4)  \tag{7.612}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,p^{2}a-xbabba,pb-babb,\,\text{class }5\rangle \;(%
\text{all }x,\,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}\func{mod}p,\;p=1%
\func{mod}4)  \tag{7.613}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,p^2a,pb-babb-xbabba,\,\text{class }5\rangle \;(x
\neq 0, x \sim -x,\;p=1 \func{mod} 4)  \tag{7.614}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,p^{2}a-xbabba,pb-\omega babb,\,\text{class }%
5\rangle \;(\text{all }x,\,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}\func{%
mod}p,\;p=1\func{mod}4)  \tag{7.615}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,p^{2}a,pb-\omega babb-xbabba,\,\text{class }%
5\rangle \;(x\neq 0,x\sim -x,\;p=1\func{mod}4)  \tag{7.616}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,p^{2}a-xbabba,pb-\omega ^{2}babb,\,\text{class }%
5\rangle \;(\text{all }x,\,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}\func{%
mod}p,\;p=1\func{mod}4)  \tag{7.617}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,p^{2}a,pb-\omega ^{2}babb-xbabba,\,\text{class }%
5\rangle \;(x\neq 0,x\sim -x,\;p=1\func{mod}4)  \tag{7.618}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,p^{2}a-xbabba,pb-\omega ^{3}babb,\,\text{class }%
5\rangle \;(\text{all }x,\,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}\func{%
mod}p,\;p=1\func{mod}4)  \tag{7.619}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-babb,p^{2}a,pb-\omega ^{3}babb-xbabba,\,\text{class }%
5\rangle \;(x\neq 0,x\sim -x,\;p=1\func{mod}4)  \tag{7.620}
\end{equation}

\subsection{Descendants of 6.467}

2 algebras

\begin{equation}
\langle a,b\,|\,baa,pb-ba,\,\text{class }5\rangle  \tag{7.621}
\end{equation}

\begin{equation}
\langle a,b\,|\,baa-p^4a,pb-ba,\,\text{class }5\rangle  \tag{7.622}
\end{equation}

\subsection{Descendants of 6.469}

$4+2\gcd (p-1,3)+3\gcd (p-1,5)+\gcd (p-1,8)$ algebras

\begin{equation}
\langle a,b\,|\,bab,baaab,pa,pb,\,\text{class }6\rangle  \tag{7.623}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,baaab,pa,pb-baaaaa,\,\text{class }6\rangle  \tag{7.624}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,baaab,pa,pb-xbaaaaa,\,\text{class }6\rangle \;(x=\omega
,\omega ^{2},\omega ^{3},\omega ^{4},\;p=1\func{mod}5)  \tag{7.625}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,baaab,pa-baaaaa,pb,\,\text{class }6\rangle  \tag{7.626}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,baaab-baaaaa,pa,pb,\,\text{class }6\rangle  \tag{7.627}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,baaab-baaaaa,pa,pb-baaaaa,\,\text{class }6\rangle 
\tag{7.628}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,baaab-baaaaa,pa,pb-xbaaaaa,\,\text{class }6\rangle
\;(x=\omega ,\omega ^{2},\omega ^{3},\omega ^{4},\;p=1\func{mod}5) 
\tag{7.629}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,baaab-baaaaa,pa-baaaaa,pb,\,\text{class }6\rangle 
\tag{7.630}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,baaab-baaaaa,pa-\omega baaaaa,pb,\,\text{class }6\rangle
\tag{7.631}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab,baaab-baaaaa,pa-xbaaaaa,pb,\,\text{class }6\rangle
\;(x=\omega ^{2},\omega ^{3},\omega ^{4},\omega ^{5},\;p=1\func{mod}3) 
\tag{7.632}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaaa,baaab,pa,pb,\,\text{class }6\rangle  \tag{7.633}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaaa,baaab,pa,pb-baaaaa,\,\text{class }6\rangle 
\tag{7.634}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaaa,baaab,pa,pb-xbaaaaa,\,\text{class }6\rangle
\;(x=\omega ,\omega ^{2},\omega ^{3},\omega ^{4},\;p=1\func{mod}5) 
\tag{7.635}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaaa,baaab,pa-baaaaa,pb,\,\text{class }6\rangle 
\tag{7.636}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaaa,baaab,pa-\omega baaaaa,pb,\,\text{class }6\rangle
\tag{7.637}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaaa,baaab,pa-\omega ^{2}baaaaa,pb,\,\text{class }%
6\rangle \;(p=1\func{mod}4)  \tag{7.638}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaaa,baaab,pa-\omega ^{3}baaaaa,pb,\,\text{class }%
6\rangle \;(p=1\func{mod}4)  \tag{7.639}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaaa,baaab,pa-xbaaaaa,pb,\,\text{class }6\rangle
\;(x=\omega ^{4},\omega ^{5},\omega ^{6},\omega ^{7},\;p=1\func{mod}8) 
\tag{7.640}
\end{equation}

\subsection{Descendants of 6.475}

$4p-1+\gcd (p-1,5)+\gcd (p-1,7)$ algebras

\begin{equation}
\langle a,b\,|\,bab-baaaa,baaab,pa,pb,\,\text{class }6\rangle  \tag{7.641}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,baaab,pa,pb-baaaaa,\,\text{class }6\rangle 
\tag{7.642}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,baaab,pa,pb-xbaaaaa,\,\text{class }6\rangle
\;(x=\omega ,\omega ^{2},\omega ^{3},\omega ^{4},\;p=1\func{mod}5) 
\tag{7.643}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,baaab,pa-baaaaa,pb,\,\text{class }6\rangle 
\tag{7.644}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,baaab,pa-xbaaaaa,pb,\,\text{class }6\rangle
\;(x=\omega ,\omega ^{2},\omega ^{3},\omega ^{4},\omega ^{5},\omega
^{6},\;p=1\func{mod}7)  \tag{7.645}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,baaab-baaaaa,pa,pb-xbaaaaa,\,\text{class }6\rangle
\;(\text{all }x)  \tag{7.646}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa,baaab-baaaaa,pa-xbaaaaa,pb,\,\text{class }6\rangle
\;(x \neq 0)  \tag{7.647}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa-baaaaa,baaab,pa,pb-xbaaaaa,\,\text{class }6\rangle
\;(\text{all }x)  \tag{7.648}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaaa-baaaaa,baaab,pa-xbaaaaa,pb,\,\text{class }6\rangle
\;(x \neq 0)  \tag{7.649}
\end{equation}

\subsection{Descendants of 6.507}

$2p^{2}+p+2p\gcd (p-1,3)+p\gcd (p-1,5)$ algebras

\begin{equation}
\langle a,b\,|\,bab-baaa,baaab-xbaaaaa,pa,pb,\,\text{class }6\rangle \;(%
\text{all }x)  \tag{7.650}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,baaab-xbaaaaa,pa-baaaaa,pb,\,\text{class }6\rangle
\;(\text{all }x)  \tag{7.651}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,baaab-xbaaaaa,pa-\omega baaaaa,pb,\,\text{class }%
6\rangle \;(\text{all }x)  \tag{7.652}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,baaab-xbaaaaa,pa-ybaaaaa,pb,\,\text{class }6\rangle
\;(\text{all }x,\,y=\omega ^{2},\omega ^{3},\omega ^{4},\omega ^{5},\;p=1%
\func{mod}3)  \tag{7.653}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa,baaab-xbaaaaa,pa-ybaaaaa,pb-baaaaa,\,\text{class }%
6\rangle \;(\text{all }x,\,y,\;p\neq 1\func{mod}5)  \tag{7.654}
\end{equation}

In the presentation below we have all $x,\,y,\;z=1,\omega ,\omega
^{2},\omega ^{3},\omega ^{4},\;y\sim y^{\prime }$ if $y^{5}=y^{\prime 5}%
\func{mod}p,\;p=1\func{mod}5$:

\begin{equation}
\langle a,b\,|\,bab-baaa,baaab-xbaaaaa,pa-ybaaaaa,pb-zbaaaaa,\,\text{class }%
6\rangle  \tag{7.655}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa-baaaaa,baaab-baaaaa,pa-xbaaaaa,pb-ybaaaaa,\,\text{%
class }6\rangle \;(\text{all }x,\,y,\;y\sim -y)  \tag{7.656}
\end{equation}

\begin{equation}
\langle a,b\,|\,bab-baaa-\omega baaaaa,baaab-baaaaa,pa-xbaaaaa,pb-ybaaaaa,\,%
\text{class }6\rangle \;(\text{all }x,\,y,\;y\sim -y)  \tag{7.657}
\end{equation}

\subsection{Descendants of 6.518}

2 algebras

\begin{equation}
\langle a,b\,|\,ba,pb,\,\text{class }6\rangle  \tag{7.658}
\end{equation}

\begin{equation}
\langle a,b\,|\,ba-p^5a,pb,\,\text{class }6\rangle  \tag{7.659}
\end{equation}

\subsection{Descendants of 3.1}

$p+14$ algebras

\begin{equation}
\langle a,b,c\,|\,ca,cb,\,\text{class }2\rangle  \tag{7.660}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,\,\text{class }2\rangle  \tag{7.661}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba,\,\text{class }2\rangle  \tag{7.662}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pb,\,\text{class }2\rangle  \tag{7.663}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pb-ba,\,\text{class }2\rangle  \tag{7.664}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pb-ca,\,\text{class }2\rangle  \tag{7.665}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pb,pc,\,\text{class }2\rangle  \tag{7.666}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pb-ba,pc,\,\text{class }2\rangle  \tag{7.667}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pb-ca,pc,\,\text{class }2\rangle  \tag{7.668}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pb-cb,pc,\,\text{class }2\rangle  \tag{7.669}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pb-ba,pc-cb,\,\text{class }2\rangle  \tag{7.670}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pb-ca,pc-cb,\,\text{class }2\rangle  \tag{7.671}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pb-ba,pc-ca,\,\text{class }2\rangle  \tag{7.672}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pb-ca,pc-ba,\,\text{class }2\rangle  \tag{7.673}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pb-\omega ca,pc-ba,\,\text{class }2\rangle  \tag{7.674}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,pb-xca,pc-ba-ca,\,\text{class }2\rangle \;(x \neq 0) 
\tag{7.675}
\end{equation}

\subsection{Descendants of 5.10}

$2p+7$ algebras

\begin{equation}
\langle a,b,c\,|\,ca,cb,pb-ba,pc,\,\text{class }3\rangle  \tag{7.676}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pb-ba,pc,\,\text{class }3\rangle  \tag{7.677}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-p^2a,cb,pb-ba,pc,\,\text{class }3\rangle  \tag{7.678}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-xp^2a,cb-baa,pb-ba,pc,\,\text{class }3\rangle \;(\text{%
all }x)  \tag{7.679}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa-p^2a,cb-baa,pb-ba,pc,\,\text{class }3\rangle 
\tag{7.680}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-xbaa-p^2a,cb-p^2a,pb-ba,pc,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.681}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-p^2a,pb-ba,pc,\,\text{class }3\rangle  \tag{7.682}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb-p^2a,pb-ba,pc,\,\text{class }3\rangle 
\tag{7.683}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-\omega baa,cb-p^{2}a,pb-ba,pc,\,\text{class }3\rangle 
\tag{7.684}
\end{equation}

\subsection{Descendants of 5.12}

$3p^{2}+17p+53+\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,pa,pb,\,\text{class }3\rangle  \tag{7.685}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,pa-baa,pb,\,\text{class }3\rangle  \tag{7.686}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,pa,pb-baa,\,\text{class }3\rangle  \tag{7.687}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,pa,pb-\omega baa,\,\text{class }3\rangle 
\tag{7.688}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca-p^2c,cb,pa,pb-xbaa,\,\text{class }3\rangle \;(\text{%
all }x)  \tag{7.689}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca-p^2c,cb,pa-baa,pb,\,\text{class }3\rangle 
\tag{7.690}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb,\,\text{class }3\rangle  \tag{7.691}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa-baa,pb,\,\text{class }3\rangle 
\tag{7.692}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb-baa,\,\text{class }3\rangle 
\tag{7.693}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb-\omega baa,\,\text{class }3\rangle 
\tag{7.694}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca-p^2c,cb-baa,pa,pb-xbaa,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.695}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca-p^2c,cb-baa,pa-baa,pb,\,\text{class }3\rangle 
\tag{7.696}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-p^2c,pa,pb,\,\text{class }3\rangle  \tag{7.697}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-p^2c,pa,pb-baa,\,\text{class }3\rangle 
\tag{7.698}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-p^{2}c,pa,pb-\omega baa,\,\text{class }3\rangle 
\tag{7.699}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-p^2c,pa-baa,pb-xbaa,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.700}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-p^2c,pa,pb,\,\text{class }3\rangle 
\tag{7.701}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-p^2c,pa,pb-baa,\,\text{class }3\rangle 
\tag{7.702}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-p^{2}c,pa,pb-\omega baa,\,\text{class }%
3\rangle  \tag{7.703}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-p^2c,pa-baa,pb-xbaa,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.704}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c,ca,cb,pa-baa,pb,\,\text{class }3\rangle  \tag{7.705}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c,ca,cb,pa-baa,pb-xbab,\,\text{class }3\rangle \;(x
\neq 0, x \sim x^{-1})  \tag{7.706}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c,ca,cb,pa-baa-bab,pb-bab,\,\text{class }3\rangle 
\tag{7.707}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c,ca,cb,pa-baa-\omega bab,pb-bab,\,\text{class }%
3\rangle  \tag{7.708}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c,ca,cb,pa,pb,\,\text{class }3\rangle  \tag{7.709}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c,ca,cb,pa-bab,pb,\,\text{class }3\rangle  \tag{7.710}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c,ca,cb,pa-\omega bab,pb,\,\text{class }3\rangle 
\tag{7.711}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c,ca,cb,pa-\omega bab,pb-baa,\,\text{class }3\rangle 
\tag{7.712}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c,ca,cb,pa-xbab,pb-baa-bab,\,\text{class }3\rangle \;(%
\text{all }x\text{ with }1+4x\text{ not a square }\func{\func{mod}}p) 
\tag{7.713}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c,ca,cb-baa,pa,pb,\,\text{class }3\rangle  \tag{7.714}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c,ca,cb-baa,pa,pb-baa,\,\text{class }3\rangle 
\tag{7.715}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c,ca,cb-baa,pa,pb-\omega baa,\,\text{class }3\rangle 
\tag{7.716}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c,ca,cb-baa,pa,pb-bab,\,\text{class }3\rangle 
\tag{7.717}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c,ca,cb-baa,pa-baa,pb-xbab,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.718}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c,ca,cb-baa,pa-baa,pb-baa-bab,\,\text{class }3\rangle 
\tag{7.719}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c,ca,cb-baa,pa-baa,pb-\omega baa-bab,\,\text{class }%
3\rangle  \tag{7.720}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c,ca,cb-baa,pa-bab,pb-xbaa-bab,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.721}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c,ca,cb-baa,pa-bab,pb,\,\text{class }3\rangle 
\tag{7.722}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c,ca,cb-baa,pa-bab,pb-baa,\,\text{class }3\rangle 
\tag{7.723}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c,ca,cb-baa,pa-bab,pb-\omega baa,\,\text{class }%
3\rangle  \tag{7.724}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c,ca,cb-baa,pa-\omega bab,pb-xbaa-bab,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.725}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c,ca,cb-baa,pa-\omega bab,pb,\,\text{class }3\rangle 
\tag{7.726}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c,ca,cb-baa,pa-\omega bab,pb-baa,\,\text{class }%
3\rangle  \tag{7.727}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c,ca,cb-baa,pa-\omega bab,pb-\omega baa,\,\text{class 
}3\rangle  \tag{7.728}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c,ca-bab,cb-baa,pa-xbaa-ybab,pb-zbaa-tbab,\,\text{%
class }3\rangle \;(\text{See Notes5.12})  \tag{7.729}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c,ca-\omega bab,cb-baa,pa-xbaa-ybab,pb-zbaa-tbab,\,%
\text{class }3\rangle \;(\text{See Notes5.12})  \tag{7.730}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c-baa,ca,cb,pa,pb,\,\text{class }3\rangle  \tag{7.731}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c-baa,ca,cb-baa,pa,pb,\,\text{class }3\rangle 
\tag{7.732}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c-baa,ca,cb-bab,pa,pb,\,\text{class }3\rangle 
\tag{7.733}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c-baa,ca-bab,cb,pa,pb,\,\text{class }3\rangle 
\tag{7.734}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c-baa,ca-bab,cb-baa,pa,pb,\,\text{class }3\rangle 
\tag{7.735}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca-bab,cb-\omega baa,pa,pb,\,\text{class }%
3\rangle  \tag{7.736}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c-baa,ca,cb,pa,pb-bab,\,\text{class }3\rangle 
\tag{7.737}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c-baa,ca,cb-baa,pa,pb-bab,\,\text{class }3\rangle 
\tag{7.738}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2c-baa,ca,cb-bab,pa,pb-bab,\,\text{class }3\rangle 
\tag{7.739}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca-bab,cb-xbaa,pa,pb-bab,\,\text{class }%
3\rangle \;(\text{all }x,\,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}%
\func{mod}p)  \tag{7.740}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca-\omega bab,cb-xbaa,pa,pb-bab,\,\text{class }%
3\rangle \;(\text{all }x,\,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}%
\func{mod}p,\;p=1\func{mod}3)  \tag{7.741}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca-\omega ^{2}bab,cb-xbaa,pa,pb-bab,\,\text{%
class }3\rangle \;(\text{all }x,\,x \sim x^{\prime }\text{ if }x^3=x^{\prime
3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.742}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca,cb-xbaa,pa-bab,pb,\,\text{class }3\rangle \;(%
\text{all }x,\,x\sim -x)  \tag{7.743}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca,cb-xbaa-bab,pa-bab,pb,\,\text{class }%
3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.744}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca-bab,cb-xbaa-ybab,pa-bab,pb,\,\text{class }%
3\rangle \;(\text{all }x,\,y,\;y\sim -y,\;p=3\func{mod}4)  \tag{7.745}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca-bab,cb-xbaa,pa-bab,pb,\,\text{class }%
3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.746}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca-\omega bab,cb-xbaa,pa-bab,pb,\,\text{class }%
3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.747}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca-bab,cb-xbaa-ybab,pa-bab,pb,\,\text{class }%
3\rangle \;(\text{all }x,\,y\neq 0,\;y\sim y^{\prime }\text{ if }%
y^{4}=y^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.748}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca-\omega bab,cb-xbaa-ybab,pa-bab,pb,\,\text{%
class }3\rangle \;(\text{all }x,\,y\neq 0,\;y\sim y^{\prime }\text{ if }%
y^{4}=y^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.749}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca,cb-xbaa,pa-\omega bab,pb,\,\text{class }%
3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.750}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca,cb-xbaa-bab,pa-\omega bab,pb,\,\text{class }%
3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.751}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca-bab,cb-xbaa-ybab,pa-\omega bab,pb,\,\text{%
class }3\rangle \;(\text{all }x,\,y,\;y\sim -y,\;p=3\func{mod}4)  \tag{7.752}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca-bab,cb-xbaa,pa-\omega bab,pb,\,\text{class }%
3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.753}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca-\omega bab,cb-xbaa,pa-\omega bab,pb,\,\text{%
class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.754}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca-bab,cb-xbaa-ybab,pa-\omega bab,pb,\,\text{%
class }3\rangle \;(\text{all }x,\,y\neq 0,\;y\sim y^{\prime }\text{ if }%
y^{4}=y^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.755}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}c-baa,ca-\omega bab,cb-xbaa-ybab,pa-\omega bab,pb,\,%
\text{class }3\rangle \;(\text{all }x,\,y\neq 0,y\sim y^{\prime}\text{ if }%
y^4=y^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.756}
\end{equation}

\subsection{Descendants of 5.14}

Algebra 5.14 has 
\begin{eqnarray*}
&&2p^{5}+7p^{4}+19p^{3}+49p^{2}+128p+256+(p^{2}+7p+29)\gcd (p-1,3) \\
&&\;\;\;+(p^{2}+7p+24)\gcd (p-1,4)+(p+3)\gcd (p-1,5)
\end{eqnarray*}%
immediate descendants of order $p^{7}$ and $p$-class $3$.

Algebra 5.14 has presentation 
\[
\langle a,b,c\,|\,cb,\,pa,\,pb,\,pc,\,\text{class }2\rangle , 
\]%
and so if $L$ is an immediate descendant of 5.14 of order $p^{7}$ then $%
L_{2} $ is generated by $ba$, $ca$ modulo $L_{3}$, and $L_{3}$ has order $%
p^{2}$ and is generated by $baa$, $bab$, $bac$, $caa$, $cab$. And $cb$, $pa$%
, $pb$, $pc\in L_{3}$. The commutator structure is given by the
classification of nilpotent Lie algebras over $\mathbb{Z}_{p}$. So we may
assume that one of the following holds:

\[
cb=caa=cab=cac=0, 
\]%
\[
caa=cab=cac=0,\,cb=baa, 
\]%
\[
cb=bab=bac=cab=cac=0, 
\]%
\[
cb=baa,\,bab=bac=cab=cac=0, 
\]%
\[
cb=bac=cac=0,\,caa=bab, 
\]%
\[
cb=baa,\,bac=cac=0,\,caa=bab, 
\]%
\[
cb=baa=bac=cac=0, 
\]%
\[
baa=bac=cac=0,\,cb=caa, 
\]%
\[
cb=bac=caa=0,\,cac=bab, 
\]%
\[
cb=bac=caa=0,\,cac=\omega bab, 
\]%
\[
bac=caa=0,\,cb=baa,\,cac=bab, 
\]%
\[
bac=caa=0,\,cb=baa,\,cac=\omega bab, 
\]%
\[
cb=bac=0,\,caa=baa,\,cac=-bab, 
\]%
\[
bac=0,\,cb=caa=baa,\,cac=-bab, 
\]%
\[
cb=baa=bac=caa=0, 
\]%
\[
cb=bac=caa=0,\,baa=cac, 
\]%
\[
cb=bac=0,\,baa=cac,\,caa=bab, 
\]%
\[
cb=bac=0,\,baa=cac,\,caa=\omega bab,\,(p=1\func{mod}3) 
\]%
\[
cb=baa=caa=cac=0, 
\]%
\[
cb=baa=cac=0,\,caa=bab, 
\]%
\[
cb=caa=cac=0,\,baa=bab, 
\]%
\[
cb=baa=caa=0,\,cac=\omega bab, 
\]%
\[
cb=baa=0,\,caa=bac,\,cac=\omega bab, 
\]%
\[
cb=baa=0,\,caa=kbab+bac,\,cac=\omega bab,\,(p=2\func{mod}3), 
\]%
where $k$ is any (one) integer which is not a value of 
\[
\frac{\lambda (\lambda ^{2}+3\omega \mu ^{2})}{\mu (3\lambda ^{2}+\omega \mu
^{2})}\func{mod}p. 
\]

Since the total number of descendants of 5.14 of order $p^{7}$ is of order $%
2p^{5}$, we need presentations with at least 5 parameters in some of these
cases. In each case the commutator structure is determined, and so to give a
presentation for the Lie rings we only need to specify $pa,pb,pc$. These
powers take values in $L_{3}$, which has order $p^{2}$, so we need 2
coefficients for each of $pa,pb,pc$. For the sake of simplicity I give a
single presentation with 6 parameters for each of the 24 different
commutator structures defined above, and I give the conditions for two sets
of parameters to define isomorphic Lie rings.

\subsubsection{Case 1}

\begin{equation}
\langle
a,b,c\,|%
\,cb,caa,cab,cac,pa-x_{1}baa-x_{2}bab,pb-x_{3}baa-x_{4}bab,pc-x_{5}baa-x_{6}bab\rangle .
\tag{7.757}
\end{equation}

Here $L_{3}$ is generated by $baa$ and $bab$, and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
baa \\ 
bab%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & \beta & \gamma \\ 
0 & \lambda & \mu \\ 
0 & 0 & \xi%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha ^{2}\lambda & \alpha \beta \lambda \\ 
0 & \alpha \lambda ^{2}%
\end{array}%
\right) ^{-1}. 
\]%
$\allowbreak $

There are $3p+22$ algebras in all in this case.

\subsubsection{Case 2}

\begin{equation}
\langle
a,b,c\,|%
\,cb-baa,caa,cab,cac,pa-x_{1}baa-x_{2}bab,pb-x_{3}baa-x_{4}bab,pc-x_{5}baa-x_{6}bab\rangle .
\tag{7.758}
\end{equation}%
Here $L_{3}$ is generated by $baa$ and $bab$, and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
baa \\ 
bab%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & \beta & \gamma \\ 
0 & \lambda & \mu \\ 
0 & 0 & \alpha ^{2}%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha ^{2}\lambda & \alpha \beta \lambda \\ 
0 & \alpha \lambda ^{2}%
\end{array}%
\right) ^{-1}. 
\]

The total number of algebras in this case is $5p+13+\gcd (p-1,3)+\gcd
(p-1,4) $.

\subsubsection{Case 3}

\begin{equation}
\langle
a,b,c\,|%
\,cb,bab,bac,cab,cac,pa-x_{1}baa-x_{2}caa,pb-x_{3}baa-x_{4}caa,pc-x_{5}baa-x_{6}caa\rangle .
\tag{7.759}
\end{equation}%
$L_{3}$ is generated by $baa$ and $caa$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
baa \\ 
caa%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & \beta & \gamma \\ 
0 & \lambda & \mu \\ 
0 & \nu & \xi%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha ^{2}\lambda & \alpha ^{2}\mu \\ 
\alpha ^{2}\nu & \alpha ^{2}\xi%
\end{array}%
\right) ^{-1}. 
\]%
$\allowbreak $

The total number of algebras in this case is $2p+8+\gcd (p-1,4)$.

\subsubsection{Case 4}

\begin{equation}
\langle
a,b,c\,|%
\,cb-baa,bab,bac,cab,cac,pa-x_{1}baa-x_{2}caa,pb-x_{3}baa-x_{4}caa,pc-x_{5}baa-x_{6}caa\rangle .
\tag{7.760}
\end{equation}%
$L_{3}$ is generated by $baa$ and $caa$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
baa \\ 
caa%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & \beta & \gamma \\ 
0 & \lambda & 0 \\ 
0 & \nu & \alpha ^{2}%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha ^{2}\lambda & 0 \\ 
\alpha ^{2}\nu & \alpha ^{4}%
\end{array}%
\right) ^{-1}. 
\]%
$\allowbreak $

The total number of algebras in this case is $6p+8+2\gcd (p-1,3)+\gcd
(p-1,4)+\gcd (p-1,5)$.

\subsubsection{Case 5}

\begin{equation}
\langle
a,b,c\,|%
\,cb,bac,caa-bab,cac,pa-x_{1}baa-x_{2}bab,pb-x_{3}baa-x_{4}bab,pc-x_{5}baa-x_{6}bab\rangle .
\tag{7.761}
\end{equation}%
$L_{3}$ is generated by $baa$ and $bab$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
baa \\ 
bab%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & \beta & \gamma \\ 
0 & \lambda & \mu \\ 
0 & 0 & \alpha ^{-1}\lambda ^{2}%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha ^{2}\lambda & \alpha ^{2}\mu +\alpha \beta \lambda \\ 
0 & \alpha \lambda ^{2}%
\end{array}%
\right) ^{-1}. 
\]%
$\allowbreak \allowbreak $

The total number of algebras in this case is $5p+13+2\gcd (p-1,3)+\gcd
(p-1,4)$.

\subsubsection{Case 6}

\begin{equation}
\langle
a,b,c\,|%
\,cb-baa,bac,caa-bab,cac,pa-x_{1}baa-x_{2}bab,pb-x_{3}baa-x_{4}bab,pc-x_{5}baa-x_{6}bab\rangle .
\tag{7.762}
\end{equation}%
$L_{3}$ is generated by $baa$ and $bab$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
baa \\ 
bab%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha ^{2} & \beta & \gamma \\ 
0 & \pm \alpha ^{3} & \mu \\ 
0 & 0 & \alpha ^{4}%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\pm \alpha ^{7} & \alpha ^{4}\mu \pm \alpha ^{5}\beta \\ 
0 & \alpha ^{8}%
\end{array}%
\right) ^{-1}. 
\]%
$\allowbreak $

The total number of algebras in this case is 
\[
p^2+3p-3+(p+2)\gcd (p-1,3)+(p+1)\gcd (p-1,4)+(p+1)\gcd (p-1,5). 
\]

\subsubsection{Case 7}

\begin{equation}
\langle
a,b,c\,|%
\,cb,baa,bac,cac,pa-x_{1}bab-x_{2}caa,pb-x_{3}bab-x_{4}caa,pc-x_{5}bab-x_{6}caa\rangle .
\tag{7.763}
\end{equation}%
$L_{3}$ is generated by $bab$ and $caa$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
bab \\ 
caa%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & 0 & \gamma \\ 
0 & \lambda & 0 \\ 
0 & 0 & \xi%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha \lambda ^{2} & 0 \\ 
0 & \alpha ^{2}\xi%
\end{array}%
\right) ^{-1}. 
\]%
$\allowbreak $

The total number of algebras in this case is $2p^2+11p+43+\gcd (p-1,4)$.

\subsubsection{Case 8}

\begin{equation}
\langle
a,b,c\,|%
\,cb-caa,baa,bac,cac,pa-x_{1}bab-x_{2}caa,pb-x_{3}bab-x_{4}caa,pc-x_{5}bab-x_{6}caa\rangle .
\tag{7.764}
\end{equation}%
$L_{3}$ is generated by $bab$ and $caa$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
bab \\ 
caa%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & 0 & \gamma \\ 
0 & \alpha ^{2} & 0 \\ 
0 & 0 & \xi%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha ^{5} & 0 \\ 
0 & \alpha ^{2}\xi%
\end{array}%
\right) ^{-1}. 
\]%
$\allowbreak $

The total number of algebras in this case is 
\[
p^3+4p^2+6p+(p+5)\gcd (p-1,3)+3\gcd (p-1,4)+\gcd (p-1,5). 
\]

\subsubsection{Cases 9 and 10}

\begin{equation}
\langle
a,b,c\,|%
\,cb,bac,caa,cac-bab,pa-x_{1}baa-x_{2}bab,pb-x_{3}baa-x_{4}bab,pc-x_{5}baa-x_{6}bab\rangle .
\tag{7.765}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bac,caa,cac-\omega
bab,pa-x_{1}baa-x_{2}bab,pb-x_{3}baa-x_{4}bab,pc-x_{5}baa-x_{6}bab\rangle . 
\tag{7.766}
\end{equation}%
$L_{3}$ is generated by $baa$ and $bab$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
baa \\ 
bab%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & \beta & 0 \\ 
0 & \lambda & 0 \\ 
0 & 0 & \pm \lambda%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha ^{2}\lambda & \alpha \beta \lambda \\ 
0 & \alpha \lambda ^{2}%
\end{array}%
\right) ^{-1}. 
\]

The total number of algebras in both cases is 
\[
p^{3}+\frac{5}{2}p^{2}+7p+\frac{19}{2}+\frac{p+4}{2}\gcd (p-1,4). 
\]

\subsubsection{Cases 11 and 12}

\begin{equation}
\langle
a,b,c\,|%
\,cb-baa,bac,caa,cac-bab,pa-x_{1}baa-x_{2}bab,pb-x_{3}baa-x_{4}bab,pc-x_{5}baa-x_{6}bab\rangle .
\tag{7.767}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bac,caa,cac-\omega
bab,pa-x_{1}baa-x_{2}bab,pb-x_{3}baa-x_{4}bab,pc-x_{5}baa-x_{6}bab\rangle . 
\tag{7.768}
\end{equation}%
$L_{3}$ is generated by $baa$ and $bab$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
baa \\ 
bab%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & \beta & 0 \\ 
0 & \pm \alpha ^{2} & 0 \\ 
0 & 0 & \alpha ^{2}%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\pm \alpha ^{4} & \pm \alpha ^{3}\beta \\ 
0 & \alpha ^{5}%
\end{array}%
\right) ^{-1}. 
\]%
$\allowbreak \allowbreak $

The total number of algebras in both cases is 
\[
\allowbreak (p^{4}+p^{3}+4p^{2}+p-1+\allowbreak (p^{2}+2p+3)\gcd
(p-1,3)+(p+2)\gcd (p-1,4))/2 
\]

\subsubsection{Case 13}

\begin{equation}
\langle
a,b,c\,|%
\,cb,bac,caa-baa,cac+bab,pa-x_{1}baa-x_{2}bab,pb-x_{3}baa-x_{4}bab,pc-x_{5}baa-x_{6}bab\rangle .
\tag{7.769}
\end{equation}

$L_{3}$ is generated by $baa$ and $bab$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
baa \\ 
bab%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & \beta & -\beta \\ 
0 & \lambda & \mu \\ 
0 & \mu & \lambda%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha ^{2}(\lambda +\mu ) & \alpha \beta (\lambda +\mu ) \\ 
0 & \alpha (\lambda ^{2}-\mu ^{2})%
\end{array}%
\right) ^{-1}. 
\]%
$\allowbreak \allowbreak $

In this case there are $2p^{2}+11p+27+\gcd (p-1,4)$ immediate descendants of
order $p^{7}$ and $p$-class 3.

\subsubsection{Case 14}

\begin{equation}
\langle
a,b,c\,|%
\,cb-baa,bac,caa-baa,cac+bab,pa-x_{1}baa-x_{2}bab,pb-x_{3}baa-x_{4}bab,pc-x_{5}baa-x_{6}bab\rangle .
\tag{7.770}
\end{equation}%
$L_{3}$ is generated by $baa$ and $bab$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
baa \\ 
bab%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & \beta & -\beta \\ 
0 & \lambda & \lambda -\alpha ^{2} \\ 
0 & \lambda -\alpha ^{2} & \lambda%
\end{array}%
\right) A\left( 
\begin{array}{ll}
2\alpha ^{2}\lambda -\alpha ^{4} & 2\alpha \beta \lambda -\alpha ^{3}\beta
\\ 
0 & 2\alpha ^{3}\lambda -\alpha ^{5}%
\end{array}%
\right) ^{-1}. 
\]%
$\allowbreak \allowbreak $

In this case there are $p^{3}+2p^{2}+6p+10+(p+4)\gcd (p-1,3)$ algebras.

\subsubsection{Case 15}

\begin{equation}
\langle
a,b,c\,|%
\,cb,baa,bac,caa,pa-x_{1}bab-x_{2}cac,pb-x_{3}bab-x_{4}cac,pc-x_{5}bab-x_{6}cac\rangle .
\tag{7.771}
\end{equation}%
$L_{3}$ is generated by $bab$ and $cac$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
bab \\ 
cac%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & 0 & 0 \\ 
0 & \beta & 0 \\ 
0 & 0 & \gamma%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha \beta ^{2} & 0 \\ 
0 & \alpha \gamma ^{2}%
\end{array}%
\right) ^{-1} 
\]%
and 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & 0 & 0 \\ 
0 & 0 & \beta \\ 
0 & \gamma & 0%
\end{array}%
\right) A\left( 
\begin{array}{ll}
0 & \alpha \beta ^{2} \\ 
\alpha \gamma ^{2} & 0%
\end{array}%
\right) ^{-1}. 
\]

The total number of algebras in this case is 
\[
\allowbreak p^{3}+\frac{7}{2}p^{2}+\frac{17}{2}p+\frac{59}{2}+\frac{5}{2}%
\gcd (p-1,3)+\frac{p+1}{2}\gcd (p-1,4). 
\]

\subsubsection{Case 16}

\begin{equation}
\langle
a,b,c\,|%
\,cb,bac,caa,cac-baa,pa-x_{1}baa-x_{2}bab,pb-x_{3}baa-x_{4}bab,pc-x_{5}baa-x_{6}bab\rangle .
\tag{7.772}
\end{equation}%
$L_{3}$ is generated by $baa$ and $bab$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
baa \\ 
bab%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & 0 & 0 \\ 
0 & \alpha ^{-1}\gamma ^{2} & 0 \\ 
0 & 0 & \gamma%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha \gamma ^{2} & 0 \\ 
0 & \alpha ^{-1}\gamma ^{4}%
\end{array}%
\right) ^{-1}. 
\]

The total number of algebras here is 
\[
2p^{4}+4p^{3}+8p^{2}+14p+11+4\gcd (p-1,3)+3\gcd (p-1,4). 
\]

\subsubsection{Case 17}

\begin{equation}
\langle
a,b,c\,|%
\,cb,bac,caa-bab,cac-baa,pa-x_{1}baa-x_{2}bab,pb-x_{3}baa-x_{4}bab,pc-x_{5}baa-x_{6}bab\rangle .
\tag{7.773}
\end{equation}%
$L_{3}$ is generated by $baa$ and $bab$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
baa \\ 
bab%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & 0 & 0 \\ 
0 & \alpha ^{-1}\gamma ^{2} & 0 \\ 
0 & 0 & \gamma%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha \gamma ^{2} & 0 \\ 
0 & \alpha ^{2}\gamma%
\end{array}%
\right) ^{-1} 
\]%
or 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & 0 & 0 \\ 
0 & 0 & \alpha ^{-1}\gamma ^{2} \\ 
0 & \gamma & 0%
\end{array}%
\right) A\left( 
\begin{array}{ll}
0 & \alpha \gamma ^{2} \\ 
\alpha ^{2}\gamma & 0%
\end{array}%
\right) ^{-1} 
\]%
with $\alpha ^{3}=\gamma ^{3}$.

If $p\neq 1\func{mod}3$ then $\alpha =\gamma $ and the number of orbits is 
\[
p^5+p^4+p^3+p^2+p+2+(p^2+p+1)\gcd (p-1,4)/2. 
\]

If $p=1\func{mod}3$ then $\alpha =\gamma $ or $\xi \gamma $ or $\xi ^2\gamma 
$ where $\xi ^3=1$. The number of orbits is then

\[
(p^{5}+p^{4}+p^{3}+p^{2}+7p+10)/3+(p^{2}+p+1)\gcd (p-1,4)/2. 
\]

So in general the number of orbits is 
\[
\allowbreak (p^{4}+2p^{3}+3p^{2}+4p+2)\frac{p-1}{\gcd (p-1,3)}%
+3p+4+(p^{2}+p+1)\gcd (p-1,4)/2. 
\]

\subsubsection{Case 18}

\begin{equation}
\langle a,b,c\,|\,cb,bac,caa-\omega
bab,cac-baa,pa-x_{1}baa-x_{2}bab,pb-x_{3}baa-x_{4}bab,pc-x_{5}baa-x_{6}bab%
\rangle \;(p=1\func{mod}3).  \tag{7.774}
\end{equation}%
This case is very similar to Case 17, though we do not have as many
automorphisms. $L_{3}$ is generated by $baa$ and $bab$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
baa \\ 
bab%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & 0 & 0 \\ 
0 & \alpha ^{-1}\gamma ^{2} & 0 \\ 
0 & 0 & \gamma%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha \gamma ^{2} & 0 \\ 
0 & \alpha ^{2}\gamma%
\end{array}%
\right) ^{-1} 
\]%
with $\alpha ^{3}=\gamma ^{3}$.

The number of algebras is 
\[
(2p^{5}+2p^{4}+2p^{3}+2p^{2}+14p+17)/3. 
\]

Combining Case 17 and Case 18, the total number of algebras in the two cases
is 
\[
\allowbreak p^{5}+p^{4}+p^{3}+p^{2}-2p-\frac{3}{2}+(3p+\frac{7}{2})\gcd
(p-1,3)+(p^{2}+p+1)\gcd (p-1,4)/2. 
\]

\subsubsection{Case 19}

\begin{equation}
\langle
a,b,c\,|%
\,cb,baa,caa,cac,pa-x_{1}bab-x_{2}bac,pb-x_{3}bab-x_{4}bac,pc-x_{5}bab-x_{6}bac\rangle .
\tag{7.775}
\end{equation}%
$L_{3}$ is generated by $bab$ and $bac$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
bab \\ 
bac%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & 0 & 0 \\ 
0 & \beta & \gamma \\ 
0 & 0 & \delta%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha \beta ^{2} & 2\alpha \beta \gamma \\ 
0 & \alpha \beta \delta%
\end{array}%
\right) ^{-1} 
\]%
$\allowbreak $

The total number of algebras in this case is $2p^{2}+11p+27+\gcd (p-1,4)$.

\subsubsection{Case 20}

\begin{equation}
\langle
a,b,c\,|%
\,cb,baa,caa-bab,cac,pa-x_{1}bab-x_{2}bac,pb-x_{3}bab-x_{4}bac,pc-x_{5}bab-x_{6}bac\rangle .
\tag{7.776}
\end{equation}%
$L_{3}$ is generated by $bab$ and $bac$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
bab \\ 
bac%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & 0 & 0 \\ 
0 & \beta & 0 \\ 
0 & 0 & \alpha ^{-1}\beta ^{2}%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha \beta ^{2} & 0 \\ 
0 & \beta ^{3}%
\end{array}%
\right) ^{-1} 
\]

The total number of algebras here is 
\[
2p^{4}+4p^{3}+6p^{2}+11p+11+2\gcd (p-1,3)+(p+1)\gcd (p-1,4). 
\]

\subsubsection{Case 21}

\begin{equation}
\langle
a,b,c\,|%
\,cb,bab-baa,caa,cac,pa-x_{1}baa-x_{2}bac,pb-x_{3}baa-x_{4}bac,pc-x_{5}baa-x_{6}bac\rangle .
\tag{7.777}
\end{equation}%
$L_{3}$ is generated by $baa$ and $bac$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
baa \\ 
bac%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & 0 & 2\beta \\ 
0 & \alpha & \beta \\ 
0 & 0 & \gamma%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha ^{3} & 2\alpha ^{2}\beta \\ 
0 & \alpha ^{2}\gamma%
\end{array}%
\right) ^{-1}. 
\]%
$\allowbreak $

The total number of algebras in this case is 
\[
2p^{3}+6p^{2}+7p+7+(p+1)\gcd (p-1,4). 
\]

\subsubsection{Case 22}

\begin{equation}
\langle a,b,c\,|\,cb,baa,caa,cac-\omega
bab,pa-x_{1}bab-x_{2}bac,pb-x_{3}bab-x_{4}bac,pc-x_{5}bab-x_{6}bac\rangle . 
\tag{7.778}
\end{equation}%
$L_{3}$ is generated by $bab$ and $bac$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
bab \\ 
bac%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & 0 & 0 \\ 
0 & \omega \beta & \pm \gamma \\ 
0 & \omega \gamma & \pm \omega \beta%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\omega \alpha (\omega \beta ^{2}+\gamma ^{2}) & \pm 2\omega \alpha \beta
\gamma \\ 
2\omega ^{2}\alpha \beta \gamma & \pm \omega \alpha (\omega \beta
^{2}+\gamma ^{2})%
\end{array}%
\right) ^{-1}. 
\]

The total number of algebras in Case 22 is 
\[
(2p^{3}+3p^{2}+3p+13-\gcd (p-1,3)+(p+1)\gcd (p-1,4))/2. 
\]

\subsubsection{Case 23}

\begin{equation}
\langle a,b,c\,|\,cb,baa,caa-bac,cac-\omega
bab,pa-x_{1}bab-x_{2}bac,pb-x_{3}bab-x_{4}bac,pc-x_{5}bab-x_{6}bac\rangle . 
\tag{7.779}
\end{equation}%
$L_{3}$ is generated by $bab$ and $bac$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
bab \\ 
bac%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & 0 & 0 \\ 
0 & \alpha & 0 \\ 
0 & 0 & \pm \alpha%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha ^{3} & 0 \\ 
0 & \pm \alpha ^{3}%
\end{array}%
\right) ^{-1} 
\]%
or when $p=2\func{mod}3$ and $12\omega \beta ^{2}=-1$, 
\[
A\rightarrow \left( 
\begin{array}{lll}
4\omega \alpha \beta & -3\omega \alpha \beta & \frac{\alpha }{2} \\ 
0 & -2\omega \alpha \beta & \alpha \\ 
0 & \pm \omega \alpha & \mp 2\omega \alpha \beta%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\frac{8}{3}\omega ^{2}\alpha ^{3}\beta & \frac{4}{3}\omega \alpha ^{3} \\ 
\pm \frac{4}{3}\omega ^{2}\alpha ^{3} & \pm \frac{8}{3}\omega ^{2}\alpha
^{3}\beta%
\end{array}%
\right) ^{-1}. 
\]

If $p=1\func{mod}3$ there are $p^{5}+p^{4}+p^{3}+p^{2}+p+2+(p^{2}+p+1)\gcd
(p-1,4)/2$ algebras.

When $p=2\func{mod}3$ the number of algebras here is 
\[
\allowbreak \frac{1}{3}p^{5}+\frac{1}{3}p^{4}+\frac{1}{3}p^{3}+\frac{1}{3}%
p^{2}+p+2+(p^{2}+p+1)\gcd (p-1,4)/2. 
\]

\subsubsection{Case 24}

\begin{equation}
\langle a,b,c\,|\,cb,baa,caa-xbab-bac,cac-\omega
bab,pa-x_{1}bab-x_{2}bac,pb-x_{3}bab-x_{4}bac,pc-x_{5}bab-x_{6}bac\rangle
\;(p=2\func{mod}3).  \tag{7.780}
\end{equation}%
where $x$ is any (fixed) integer which is not a value of 
\[
\frac{\lambda (\lambda ^{2}+3\omega \mu ^{2})}{\mu (3\lambda ^{2}+\omega \mu
^{2})}\func{mod}p. 
\]%
$L_{3}$ is generated by $bab$ and $bac$ and if we let 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =A\left( 
\begin{array}{l}
bab \\ 
bac%
\end{array}%
\right) 
\]%
then the isomorphism classes of algebras satisfying these commutator
relations correspond to the orbits of $3\times 2$ matrices $A$ under the
action 
\[
A\rightarrow \left( 
\begin{array}{lll}
\alpha & 0 & 0 \\ 
0 & \alpha & 0 \\ 
0 & 0 & \alpha%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\alpha ^{3} & 0 \\ 
0 & \alpha ^{3}%
\end{array}%
\right) ^{-1} 
\]%
and 
\[
A\rightarrow \left( 
\begin{array}{lll}
-4\alpha & k\alpha \beta +3\alpha & 3k\omega ^{-1}\alpha +\alpha \beta \\ 
0 & 2\alpha & 2\alpha \beta \\ 
0 & 2\omega \alpha \beta & 2\alpha%
\end{array}%
\right) A\left( 
\begin{array}{ll}
32\alpha ^{3} & -32\alpha ^{3}\beta \\ 
-32\omega \alpha ^{3}\beta & 32\alpha ^{3}%
\end{array}%
\right) ^{-1} 
\]%
where $\omega \beta ^{2}=-3$.

The number of orbits is 
\[
\allowbreak \frac{2}{3}p^{5}+\frac{2}{3}p^{4}+\frac{2}{3}p^{3}+\frac{2}{3}%
p^{2}+2p+3. 
\]

The total number of algebras from Case 23 and Case 24 is 
\[
p^{5}+p^{4}+p^{3}+p^{2}+4p+\frac{13}{2}-(p+\frac{3}{2})\gcd
(p-1,3)+(p^{2}+p+1)\gcd (p-1,4)/2. 
\]

The total number of algebras from cases 17, 18, 23 and 24 is 
\[
p^{5}+p^{4}+p^{3}+p^{2}+2p+5+(2p+2)\gcd (p-1,3)+(p^{2}+p+1)\gcd (p-1,4). 
\]

\subsection{Descendants of 5.15}

$3p^{2}+12p+14+(p+2)\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba,pb,pc,\,\text{class }3\rangle  \tag{7.781}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-caa,pb,pc,\,\text{class }3\rangle  \tag{7.782}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-cac,pb,pc,\,\text{class }3\rangle  \tag{7.783}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-\omega cac,pb,pc,\,\text{class }3\rangle 
\tag{7.784}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba,pb,pc-caa,\,\text{class }3\rangle  \tag{7.785}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-cac,pb,pc-caa,\,\text{class }3\rangle  \tag{7.786}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-\omega cac,pb,pc-caa,\,\text{class }3\rangle 
\tag{7.787}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-caa-xcac,pb,pc-caa,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.788}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba,pb,pc-\omega caa,\,\text{class }3\rangle 
\tag{7.789}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-cac,pb,pc-\omega caa,\,\text{class }3\rangle 
\tag{7.790}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-\omega cac,pb,pc-\omega caa,\,\text{class }%
3\rangle  \tag{7.791}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-caa-xcac,pb,pc-\omega caa,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.792}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-xcaa,pb,pc-cac,\,\text{class }3\rangle \;(\text{%
all }x)  \tag{7.793}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-caa-cac,pb,pc-cac,\,\text{class }3\rangle 
\tag{7.794}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-caa-\omega cac,pb,pc-cac,\,\text{class }3\rangle 
\tag{7.795}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba,pb-caa,pc,\,\text{class }3\rangle  \tag{7.796}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-cac,pb-caa,pc,\,\text{class }3\rangle  \tag{7.797}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-\omega cac,pb-caa,pc,\,\text{class }3\rangle 
\tag{7.798}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-\omega ^{2}cac,pb-caa,pc,\,\text{class }3\rangle
\;(p=1\func{mod}4)  \tag{7.799}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-\omega ^{3}cac,pb-caa,pc,\,\text{class }3\rangle
\;(p=1\func{mod}4)  \tag{7.800}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-xcac,pb-caa,pc-cac,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.801}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba,pb-cac,pc,\,\text{class }3\rangle  \tag{7.802}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba-caa,pb-cac,pc,\,\text{class }3\rangle  \tag{7.803}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba,pb-cac,pc-caa,\,\text{class }3\rangle  \tag{7.804}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba,pb-cac,pc-\omega caa,\,\text{class }3\rangle 
\tag{7.805}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba,pb-cac,pc-\omega ^{2}caa,\,\text{class }3\rangle
\;(p=1\func{mod}4)  \tag{7.806}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-ba,pb-cac,pc-\omega ^{3}caa,\,\text{class }3\rangle
\;(p=1\func{mod}4)  \tag{7.807}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-ba,pb,pc,\,\text{class }3\rangle  \tag{7.808}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-ba-cac,pb,pc,\,\text{class }3\rangle  \tag{7.809}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-ba-\omega cac,pb,pc,\,\text{class }3\rangle 
\tag{7.810}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-ba,pb,pc-xcaa,\,\text{class }3\rangle \;(x \neq
0)  \tag{7.811}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-ba-cac,pb,pc-xcaa,\,\text{class }3\rangle \;(x
\neq 0)  \tag{7.812}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-ba-\omega cac,pb,pc-xcaa,\,\text{class }3\rangle
\;(x\neq 0)  \tag{7.813}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-ba-caa-ycac,pb,pc-xcaa,\,\text{class }3\rangle
\;(x\neq 0,\text{ all }y)  \tag{7.814}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-ba-xcac,pb,pc-cac,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.815}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-ba-ycac,pb-caa,pc-xcac,\,\text{class }3\rangle
\;(\text{all }x,\,y,x\sim -x)  \tag{7.816}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-ba,pb-cac,pc-xcaa,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.817}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-ba-xcaa,pb-cac,pc+2caa,\,\text{class }3\rangle
\;(x \neq 0, x \sim x^{\prime}\text{ if }x^4=x^{\prime 4 }\func{mod} p) 
\tag{7.818}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-ba,pb-\omega cac,pc-xcaa,\,\text{class }3\rangle
\;(\text{all }x,\;p=1\func{mod}4)  \tag{7.819}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-ba-xcaa,pb-\omega cac,pc+2caa,\,\text{class }%
3\rangle \;(x\neq 0,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}\func{mod}%
p,\;p=1\func{mod}4)  \tag{7.820}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,pa-ba,pb,pc,\,\text{class }3\rangle 
\tag{7.821}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,pa-ba-cac,pb,pc,\,\text{class }3\rangle 
\tag{7.822}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,pa-ba-\omega cac,pb,pc,\,\text{class }%
3\rangle  \tag{7.823}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,pa-ba,pb,pc-xcaa,\,\text{class }3\rangle
\;(x\neq 0)  \tag{7.824}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,pa-ba-cac,pb,pc-xcaa,\,\text{class }3\rangle
\;(x\neq 0)  \tag{7.825}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,pa-ba-\omega cac,pb,pc-xcaa,\,\text{class }%
3\rangle \;(x\neq 0)  \tag{7.826}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,pa-ba-caa-ycac,pb,pc-xcaa,\,\text{class }%
3\rangle \;(x\neq 0,\text{ all }y)  \tag{7.827}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,pa-ba-xcac,pb,pc-cac,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.828}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,pa-ba-ycac,pb-caa,pc-xcac,\,\text{class }%
3\rangle \;(\text{all }x,\,y,\;x\sim -x)  \tag{7.829}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,pa-ba,pb-cac,pc-xcaa,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.830}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,pa-ba-xcaa,pb-cac,pc+(2\ast \omega )caa,\,%
\text{class }3\rangle \;(x\neq 0,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}%
\func{mod}p)  \tag{7.831}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,pa-ba,pb-\omega cac,pc-xcaa,\,\text{class }%
3\rangle \;(\text{all }x,\;p=1\func{mod}4)  \tag{7.832}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,pa-ba-xcaa,pb-\omega cac,pc+(2\ast \omega
)caa,\,\text{class }3\rangle \;(x\neq 0,x\sim x^{\prime}\text{ if }%
x^4=x^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.833}
\end{equation}

\subsection{Descendants of 5.16}

$p^{4}+2p^{3}+5p^{2}+14p$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ba,pc,\,\text{class }3\rangle  \tag{7.834}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ba-caa,pc,\,\text{class }3\rangle  \tag{7.835}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ba-cac,pc,\,\text{class }3\rangle  \tag{7.836}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-caa,pb-ba,pc,\,\text{class }3\rangle  \tag{7.837}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-caa,pb-ba-cac,pc,\,\text{class }3\rangle  \tag{7.838}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-caa,pb-ba-caa-xcac,pc,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.839}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-cac,pb-ba,pc,\,\text{class }3\rangle  \tag{7.840}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-cac,pb-ba-caa,pc,\,\text{class }3\rangle  \tag{7.841}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-cac,pb-ba-cac,pc,\,\text{class }3\rangle  \tag{7.842}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-\omega cac,pb-ba,pc,\,\text{class }3\rangle 
\tag{7.843}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-\omega cac,pb-ba-caa,pc,\,\text{class }3\rangle 
\tag{7.844}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-\omega cac,pb-ba-cac,pc,\,\text{class }3\rangle 
\tag{7.845}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ba,pc-xcaa,\,\text{class }3\rangle \;(x \neq 0) 
\tag{7.846}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ba-cac,pc-xcaa,\,\text{class }3\rangle \;(x \neq
0)  \tag{7.847}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ba-caa,pc-xcaa,\,\text{class }3\rangle \;(x \neq
0)  \tag{7.848}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ba-caa-cac,pc-xcaa,\,\text{class }3\rangle \;(x
\neq 0)  \tag{7.849}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-cac,pb-ba,pc-xcaa,\,\text{class }3\rangle \;(x \neq
0)  \tag{7.850}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-cac,pb-ba-caa,pc-xcaa,\,\text{class }3\rangle \;(x
\neq 0)  \tag{7.851}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-cac,pb-ba-ycaa-cac,pc-xcaa,\,\text{class }3\rangle
\;(x\neq 0,\text{ all }y,\;y\sim -y)  \tag{7.852}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-\omega cac,pb-ba,pc-xcaa,\,\text{class }3\rangle
\;(x\neq 0)  \tag{7.853}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-\omega cac,pb-ba-caa,pc-xcaa,\,\text{class }3\rangle
\;(x\neq 0)  \tag{7.854}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-\omega cac,pb-ba-ycaa-cac,pc-xcaa,\,\text{class }%
3\rangle \;(x\neq 0,\text{ all }y,\;y\sim -y)  \tag{7.855}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-caa-zcac,pb-ba,pc-xcaa,\,\text{class }3\rangle
\;(x\neq 0,\;\text{all }z)  \tag{7.856}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-caa-zcac,pb-ba-cac,pc-xcaa,\,\text{class }3\rangle
\;(x\neq 0,\;\text{all }z)  \tag{7.857}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-caa-zcac,pb-ba-caa-ycac,pc-xcaa,\,\text{class }%
3\rangle \;(x\neq 0,\text{ all }y,z)  \tag{7.858}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-xcaa,pb-ba,pc-cac,\,\text{class }3\rangle \;(x \neq
1)  \tag{7.859}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-xcaa,pb-ba-cac,pc-cac,\,\text{class }3\rangle \;(x
\neq 1)  \tag{7.860}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-xcaa,pb-ba-caa-ycac,pc-cac,\,\text{class }3\rangle
\;(x\neq 1,\text{ all }y)  \tag{7.861}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-caa-xcac,pb-ba,pc-cac,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.862}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-caa-xcac,pb-ba-cac,pc-cac,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.863}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-caa-xcac,pb-ba-caa,pc-cac,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.864}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa,pb-ba,pc-xcaa,\,\text{class }3\rangle \;(\text{%
all }x)  \tag{7.865}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-cac,pb-ba,pc-xcaa,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.866}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-\omega cac,pb-ba,pc-xcaa,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.867}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-ycac,pb-ba,pc-xcaa-cac,\,\text{class }3\rangle
\;(\text{all }x,\,y)  \tag{7.868}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-caa-ycac,pb-ba,pc-xcaa-zcac,\,\text{class }%
3\rangle \;(\text{all }x,\,y,z)  \tag{7.869}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-ycac,pb-ba-cac,pc-zcaa-tcac,\,\text{class }%
3\rangle \;(\text{all }y,z,t,\;t\sim -t)  \tag{7.870}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-xcaa-ycac,pb-ba-cac,pc-zcaa-tcac,\,\text{class }%
3\rangle \;(x\neq 0,x\sim -x,\text{ all }y,z,t)  \tag{7.871}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-ycac,pb-ba-\omega cac,pc-zcaa-tcac,\,\text{class 
}3\rangle \;(\text{all }y,z,t,\;t\sim -t)  \tag{7.872}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,pa-xcaa-ycac,pb-ba-\omega cac,pc-zcaa-tcac,\,\text{%
class }3\rangle \;(x\neq 0,\;x\sim -x,\text{ all }y,z,t)  \tag{7.873}
\end{equation}

\subsection{Descendants of 5.18}

$3p^{3}+6p^{2}+6p+11+(p+7)\gcd (p-1,3)+(p+1)\gcd (p-1,4)+\gcd (p-1,5)$
algebras

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca,pc,\,\text{class }3\rangle  \tag{7.874}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca-baa,pc,\,\text{class }3\rangle  \tag{7.875}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca-\omega baa,pc,\,\text{class }3\rangle 
\tag{7.876}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-bab,pb-ca,pc,\,\text{class }3\rangle  \tag{7.877}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-bab,pb-ca-baa,pc,\,\text{class }3\rangle  \tag{7.878}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-bab,pb-ca-\omega baa,pc,\,\text{class }3\rangle 
\tag{7.879}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-\omega bab,pb-ca,pc,\,\text{class }3\rangle 
\tag{7.880}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-\omega bab,pb-ca-baa,pc,\,\text{class }3\rangle 
\tag{7.881}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-\omega bab,pb-ca-\omega baa,pc,\,\text{class }%
3\rangle  \tag{7.882}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca-bab,pc,\,\text{class }3\rangle  \tag{7.883}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca-baa-bab,pc,\,\text{class }3\rangle  \tag{7.884}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca-\omega baa-bab,pc,\,\text{class }3\rangle 
\tag{7.885}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-bab,pb-ca-xbaa-bab,pc,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.886}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-\omega bab,pb-ca-xbaa-bab,pc,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.887}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-baa,pb-ca-xbab,pc,\,\text{class }3\rangle \;(\text{%
all }x)  \tag{7.888}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-baa,pb-ca-baa-xbab,pc,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.889}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-baa,pb-ca-\omega baa-xbab,pc,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.890}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-baa-bab,pb-ca-xbaa-ybab,pc,\,\text{class }3\rangle
\;(\text{all }x,\,y)  \tag{7.891}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-baa-\omega bab,pb-ca-xbaa-ybab,pc,\,\text{class }%
3\rangle \;(\text{all }x,\,y)  \tag{7.892}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca,pc-baa,\,\text{class }3\rangle  \tag{7.893}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-bab,pb-ca,pc-baa,\,\text{class }3\rangle  \tag{7.894}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-\omega bab,pb-ca,pc-baa,\,\text{class }3\rangle 
\tag{7.895}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca-bab,pc-baa,\,\text{class }3\rangle  \tag{7.896}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-xbab,pb-ca-bab,pc-baa,\,\text{class }3\rangle \;(x
\neq 0, x \sim x^{\prime }\text{ if }x^3=x^{\prime 3 }\func{mod} p) 
\tag{7.897}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca,pc-\omega baa,\,\text{class }3\rangle \;(p=1%
\func{mod}3)  \tag{7.898}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-bab,pb-ca,pc-\omega baa,\,\text{class }3\rangle
\;(p=1\func{mod}3)  \tag{7.899}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-\omega bab,pb-ca,pc-\omega baa,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.900}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca-bab,pc-\omega baa,\,\text{class }3\rangle
\;(p=1\func{mod}3)  \tag{7.901}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-xbab,pb-ca-bab,pc-\omega baa,\,\text{class }3\rangle
\;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}p,\;p=1%
\func{mod}3)  \tag{7.902}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca,pc-\omega ^{2}baa,\,\text{class }3\rangle
\;(p=1\func{mod}3)  \tag{7.903}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-bab,pb-ca,pc-\omega ^{2}baa,\,\text{class }3\rangle
\;(p=1\func{mod}3)  \tag{7.904}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-\omega bab,pb-ca,pc-\omega ^{2}baa,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.905}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca-bab,pc-\omega ^{2}baa,\,\text{class }3\rangle
\;(p=1\func{mod}3)  \tag{7.906}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-xbab,pb-ca-bab,pc-\omega ^{2}baa,\,\text{class }%
3\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}%
p,\;p=1\func{mod}3)  \tag{7.907}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca,pc-bab,\,\text{class }3\rangle  \tag{7.908}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca-baa,pc-bab,\,\text{class }3\rangle  \tag{7.909}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca-\omega baa,pc-bab,\,\text{class }3\rangle 
\tag{7.910}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-baa,pb-ca-xbaa,pc-bab,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.911}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-xbaa,pb-ca-ybaa,pc-baa-bab,\,\text{class }3\rangle
\;(\text{all }x,\,y,\;p=2\func{mod}3)  \tag{7.912}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca,pc-baa-bab,\,\text{class }3\rangle \;(p=1 
\func{mod} 3)  \tag{7.913}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca,pc-\omega baa-bab,\,\text{class }3\rangle
\;(p=1\func{mod}3)  \tag{7.914}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca,pc-\omega ^{2}baa-bab,\,\text{class }3\rangle
\;(p=1\func{mod}3)  \tag{7.915}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca-xbaa,pc-baa-bab,\,\text{class }3\rangle \;(x
\neq 0, x \sim x^{\prime }\text{ if }x^3=x^{\prime 3 }\func{mod} p,\;p=1 
\func{mod} 3)  \tag{7.916}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca-xbaa,pc-\omega baa-bab,\,\text{class }3\rangle
\;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}p,\;p=1%
\func{mod}3)  \tag{7.917}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa,pb-ca-xbaa,pc-\omega ^{2}baa-bab,\,\text{class }%
3\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}%
p,\;p=1\func{mod}3)  \tag{7.918}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-xbaa,pb-ca-ybaa,pc-baa-bab,\,\text{class }3\rangle
\;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}p,\text{
all }y,\;p=1\func{mod}3)  \tag{7.919}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-xbaa,pb-ca-ybaa,pc-\omega baa-bab,\,\text{class }%
3\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}p,%
\text{ all }y,\;p=1\func{mod}3)  \tag{7.920}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,pa-xbaa,pb-ca-ybaa,pc-\omega ^{2}baa-bab,\,\text{class }%
3\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}p,%
\text{ all }y,\;p=1\func{mod}3)  \tag{7.921}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa,pb-ca,pc,\,\text{class }3\rangle \;(p=1 \func{mod%
} 4)  \tag{7.922}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-bab,pb-ca,pc,\,\text{class }3\rangle \;(p=1 
\func{mod} 4)  \tag{7.923}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-\omega bab,pb-ca,pc,\,\text{class }3\rangle
\;(p=1\func{mod}4)  \tag{7.924}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-\omega ^{2}bab,pb-ca,pc,\,\text{class }3\rangle
\;(p=1\func{mod}4,\;p=1\func{mod}3)  \tag{7.925}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-\omega ^{3}bab,pb-ca,pc,\,\text{class }3\rangle
\;(p=1\func{mod}4,\;p=1\func{mod}3)  \tag{7.926}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-\omega ^{4}bab,pb-ca,pc,\,\text{class }3\rangle
\;(p=1\func{mod}4,\;p=1\func{mod}3)  \tag{7.927}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-\omega ^{5}bab,pb-ca,pc,\,\text{class }3\rangle
\;(p=1\func{mod}4,\;p=1\func{mod}3)  \tag{7.928}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbab,pb-ca-bab,pc,\,\text{class }3\rangle \;(%
\text{all }x,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.929}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbab,pb-ca-\omega bab,pc,\,\text{class }3\rangle
\;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.930}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbab,pb-ca-\omega ^{2}bab,pc,\,\text{class }%
3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.931}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbab,pb-ca-\omega ^{3}bab,pc,\,\text{class }%
3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.932}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-baa-xbab,pb-ca-ybab,pc,\,\text{class }3\rangle
\;(\text{all }x,\,x\sim -x,\text{ all }y,\;p=1\func{mod}4)  \tag{7.933}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-\omega baa-xbab,pb-ca-ybab,pc,\,\text{class }%
3\rangle \;(\text{all }x,\,y,x\sim -x,\;p=1\func{mod}4)  \tag{7.934}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-\omega ^{2}baa-xbab,pb-ca-ybab,pc,\,\text{class }%
3\rangle \;(\text{all }x,\,y,x\sim -x,\;p=1\func{mod}4)  \tag{7.935}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-\omega ^{3}baa-xbab,pb-ca-ybab,pc,\,\text{class }%
3\rangle \;(\text{all }x,\,y,x\sim -x,\;p=1\func{mod}4)  \tag{7.936}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbaa-ybab,pb-ca-baa-zbab,pc,\,\text{class }%
3\rangle \;(\text{all }x,\,y,z,\;p=1\func{mod}4)  \tag{7.937}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbaa-ybab,pb-ca-\omega baa-zbab,pc,\,\text{class 
}3\rangle \;(\text{all }x,\,y,z,\;p=1\func{mod}4)  \tag{7.938}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa,pb-ca,pc,\,\text{class }3\rangle \;(p=3 \func{mod%
} 4)  \tag{7.939}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-bab,pb-ca,pc,\,\text{class }3\rangle \;(p=3 
\func{mod} 4)  \tag{7.940}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-\omega bab,pb-ca,pc,\,\text{class }3\rangle
\;(p=3\func{mod}4)  \tag{7.941}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-\omega ^{2}bab,pb-ca,pc,\,\text{class }3\rangle
\;(p=3\func{mod}4,\;p=1\func{mod}3)  \tag{7.942}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-\omega ^{3}bab,pb-ca,pc,\,\text{class }3\rangle
\;(p=3\func{mod}4,\;p=1\func{mod}3)  \tag{7.943}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-\omega ^{4}bab,pb-ca,pc,\,\text{class }3\rangle
\;(p=3\func{mod}4,\;p=1\func{mod}3)  \tag{7.944}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-\omega ^{5}bab,pb-ca,pc,\,\text{class }3\rangle
\;(p=3\func{mod}4,\;p=1\func{mod}3)  \tag{7.945}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbab,pb-ca-bab,pc,\,\text{class }3\rangle \;(%
\text{all }x,\;p=3\func{mod}4)  \tag{7.946}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbab,pb-ca-\omega bab,pc,\,\text{class }3\rangle
\;(\text{all }x,\;p=3\func{mod}4)  \tag{7.947}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-baa-xbab,pb-ca-ybab,pc,\,\text{class }3\rangle
\;(\text{all }x,\,y,\;p=3\func{mod}4)  \tag{7.948}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-\omega baa-xbab,pb-ca-ybab,pc,\,\text{class }%
3\rangle \;(\text{all }x,\,y,\;p=3\func{mod}4)  \tag{7.949}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbaa-ybab,pb-ca-baa-zbab,pc,\,\text{class }%
3\rangle \;(\text{all }x,\,y,z,\;p=3\func{mod}4)  \tag{7.950}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbaa-ybab,pb-ca-\omega baa-zbab,pc,\,\text{class 
}3\rangle \;(\text{all }x,\,y,z,\;p=3\func{mod}4)  \tag{7.951}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa,pb-ca,pc-bab,\,\text{class }3\rangle  \tag{7.952}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa,pb-ca,pc-\omega bab,\,\text{class }3\rangle
\;(p=1\func{mod}5)  \tag{7.953}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa,pb-ca,pc-\omega ^{2}bab,\,\text{class }3\rangle
\;(p=1\func{mod}5)  \tag{7.954}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa,pb-ca,pc-\omega ^{3}bab,\,\text{class }3\rangle
\;(p=1\func{mod}5)  \tag{7.955}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa,pb-ca,pc-\omega ^{4}bab,\,\text{class }3\rangle
\;(p=1\func{mod}5)  \tag{7.956}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa,pb-ca-xbaa,pc-bab,\,\text{class }3\rangle
\;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod}p) 
\tag{7.957}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa,pb-ca-xbaa,pc-\omega bab,\,\text{class }3\rangle
\;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod}p,\;p=1%
\func{mod}5)  \tag{7.958}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa,pb-ca-xbaa,pc-\omega ^{2}bab,\,\text{class }%
3\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod%
}p,\;p=1\func{mod}5)  \tag{7.959}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa,pb-ca-xbaa,pc-\omega ^{3}bab,\,\text{class }%
3\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod%
}p,\;p=1\func{mod}5)  \tag{7.960}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa,pb-ca-xbaa,pc-\omega ^{4}bab,\,\text{class }%
3\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod%
}p,\;p=1\func{mod}5)  \tag{7.961}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbaa,pb-ca-ybaa,pc-bab,\,\text{class }3\rangle
\;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod}p,\text{
all }y)  \tag{7.962}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbaa,pb-ca-ybaa,pc-\omega bab,\,\text{class }%
3\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod%
}p,\text{ all }y,\;p=1\func{mod}5)  \tag{7.963}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbaa,pb-ca-ybaa,pc-\omega ^{2}bab,\,\text{class }%
3\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod%
}p,\text{ all }y,\;p=1\func{mod}5)  \tag{7.964}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbaa,pb-ca-ybaa,pc-\omega ^{3}bab,\,\text{class }%
3\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod%
}p,\text{ all }y,\;p=1\func{mod}5)  \tag{7.965}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbaa,pb-ca-ybaa,pc-\omega ^{4}bab,\,\text{class }%
3\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod%
}p,\text{ all }y,\;p=1\func{mod}5)  \tag{7.966}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbab,pb-ca,pc-baa,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.967}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbab,pb-ca-ybab,pc-baa,\,\text{class }3\rangle
\;(y\neq 0,\;y\sim y^{\prime }\text{ if }y^{3}=y^{\prime 3}\func{mod}p,\;%
\text{all }x)  \tag{7.968}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbab,pb-ca,pc-\omega baa,\,\text{class }3\rangle
\;(\text{all }x,\;p=1\func{mod}3)  \tag{7.969}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbab,pb-ca-ybab,pc-\omega baa,\,\text{class }%
3\rangle \;(y\neq 0,\;y\sim y^{\prime }\text{ if }y^{3}=y^{\prime 3}\func{mod%
}p,\;\text{all }x,\;p=1\func{mod}3)  \tag{7.970}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbab,pb-ca,pc-\omega ^{2}baa,\,\text{class }%
3\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.971}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-xbab,pb-ca-ybab,pc-\omega ^{2}baa,\,\text{class }%
3\rangle \;(y\neq 0,\;y\sim y^{\prime }\text{ if }y^{3}=y^{\prime 3}\func{mod%
}p,\;\text{all }x,\;p=1\func{mod}3)  \tag{7.972}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-ybab,pb-ca-zbab,pc-baa-xbab,\,\text{class }%
3\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}p,%
\text{ all }y,z)  \tag{7.973}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-ybab,pb-ca-zbab,pc-\omega baa-xbab,\,\text{class 
}3\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}\func{%
mod}p,\text{ all }y,z,\;p=1\func{mod}3)  \tag{7.974}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,pa-ybab,pb-ca-zbab,pc-\omega ^{2}baa-xbab,\,\text{%
class }3\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}%
\func{mod}p,\text{ all }y,z,\;p=1\func{mod}3)  \tag{7.975}
\end{equation}

\subsection{Descendants of 5.8}

$p+8$ algebras

\begin{equation}
\langle a,b,c\,|\,ba,ca-p^2a,cb-p^2b,pc,\,\text{class }3\rangle  \tag{7.976}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba,ca-p^2b,cb-p^2a,pc,\,\text{class }3\rangle  \tag{7.977}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba,ca-\omega p^{2}b,cb-p^{2}a,pc,\,\text{class }3\rangle 
\tag{7.978}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba,ca-xp^2b,cb-p^2a-p^2b,pc,\,\text{class }3\rangle \;(x
\neq 0)  \tag{7.979}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba,ca,cb-p^2a,pc,\,\text{class }3\rangle  \tag{7.980}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba-p^2b,ca,cb-p^2a,pc,\,\text{class }3\rangle  \tag{7.981}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba,ca,cb-p^2b,pc,\,\text{class }3\rangle  \tag{7.982}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba-p^2a,ca,cb-p^2b,pc,\,\text{class }3\rangle  \tag{7.983}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba,ca,cb,pc,\,\text{class }3\rangle  \tag{7.984}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba-p^2a,ca,cb,pc,\,\text{class }3\rangle  \tag{7.985}
\end{equation}

\subsection{Descendants of 5.9}

$4p^{2}+26p+107+5\gcd (p-1,3)+(p+4)\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,ca,cb,pb,pc,\,\text{class }3\rangle  \tag{7.986}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab,cb,pb,pc,\,\text{class }3\rangle  \tag{7.987}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-p^2a,cb,pb,pc,\,\text{class }3\rangle  \tag{7.988}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab-p^2a,cb,pb,pc,\,\text{class }3\rangle 
\tag{7.989}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-\omega bab-p^{2}a,cb,pb,pc,\,\text{class }3\rangle 
\tag{7.990}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca,cb-p^2a,pb,pc,\,\text{class }3\rangle  \tag{7.991}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab,cb-p^2a,pb,pc,\,\text{class }3\rangle 
\tag{7.992}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-p^2a,cb-p^2a,pb,pc,\,\text{class }3\rangle 
\tag{7.993}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab-p^2a,cb-p^2a,pb,pc,\,\text{class }3\rangle 
\tag{7.994}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-\omega bab-p^{2}a,cb-p^{2}a,pb,pc,\,\text{class }%
3\rangle  \tag{7.995}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca,cb,pb,pc-bab,\,\text{class }3\rangle  \tag{7.996}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab,cb,pb,pc-bab,\,\text{class }3\rangle 
\tag{7.997}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-p^2a,cb,pb,pc-bab,\,\text{class }3\rangle 
\tag{7.998}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab-p^2a,cb,pb,pc-bab,\,\text{class }3\rangle 
\tag{7.999}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-\omega bab-p^{2}a,cb,pb,pc-bab,\,\text{class }%
3\rangle  \tag{7.1000}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca,cb-p^2a,pb,pc-bab,\,\text{class }3\rangle 
\tag{7.1001}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab,cb-p^2a,pb,pc-bab,\,\text{class }3\rangle 
\tag{7.1002}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-xbab-p^{2}a,cb-p^{2}a,pb,pc-bab,\,\text{class }%
3\rangle \;(\text{all }x,\,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}%
\func{mod}p)  \tag{7.1003}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca,cb-\omega p^{2}a,pb,pc-bab,\,\text{class }3\rangle
\;(p=1\func{mod}3)  \tag{7.1004}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab,cb-\omega p^{2}a,pb,pc-bab,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1005}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-xbab-p^{2}a,cb-\omega p^{2}a,pb,pc-bab,\,\text{%
class }3\rangle \;(\text{all }x,\,x \sim x^{\prime }\text{ if }x^3=x^{\prime
3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.1006}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca,cb-\omega ^{2}p^{2}a,pb,pc-bab,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1007}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab,cb-\omega ^{2}p^{2}a,pb,pc-bab,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1008}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-xbab-p^{2}a,cb-\omega ^{2}p^{2}a,pb,pc-bab,\,\text{%
class }3\rangle \;(\text{all }x,\,x \sim x^{\prime }\text{ if }x^3=x^{\prime
3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.1009}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca,cb,pb-bab,pc,\,\text{class }3\rangle  \tag{7.1010}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab,cb,pb-bab,pc,\,\text{class }3\rangle 
\tag{7.1011}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-p^2a,cb,pb-bab,pc,\,\text{class }3\rangle 
\tag{7.1012}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab-p^2a,cb,pb-bab,pc,\,\text{class }3\rangle 
\tag{7.1013}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-\omega bab-p^{2}a,cb,pb-bab,pc,\,\text{class }%
3\rangle  \tag{7.1014}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca,cb-p^2a,pb-bab,pc,\,\text{class }3\rangle 
\tag{7.1015}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab,cb-p^2a,pb-bab,pc,\,\text{class }3\rangle 
\tag{7.1016}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-\omega bab,cb-p^{2}a,pb-bab,pc,\,\text{class }%
3\rangle  \tag{7.1017}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-\omega ^{2}bab,cb-p^{2}a,pb-bab,pc,\,\text{class }%
3\rangle \;(p=1\func{mod}4)  \tag{7.1018}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-\omega ^{3}bab,cb-p^{2}a,pb-bab,pc,\,\text{class }%
3\rangle \;(p=1\func{mod}4)  \tag{7.1019}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-xbab-p^2a,cb-p^2a,pb-bab,pc,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.1020}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-xbab-\omega p^{2}a,cb-p^{2}a,pb-bab,pc,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.1021}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,pb,pc,\,\text{class }3\rangle  \tag{7.1022}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca-p^2a,cb,pb,pc,\,\text{class }3\rangle  \tag{7.1023}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pb,pc,\,\text{class }3\rangle  \tag{7.1024}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca-p^2a,cb-baa,pb,pc,\,\text{class }3\rangle 
\tag{7.1025}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca-\omega p^{2}a,cb-baa,pb,pc,\,\text{class }3\rangle 
\tag{7.1026}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-p^2a,pb,pc,\,\text{class }3\rangle  \tag{7.1027}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-p^2a,pb,pc,\,\text{class }3\rangle 
\tag{7.1028}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,pb,pc-baa,\,\text{class }3\rangle  \tag{7.1029}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca-p^2a,cb,pb,pc-baa,\,\text{class }3\rangle 
\tag{7.1030}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pb,pc-baa,\,\text{class }3\rangle 
\tag{7.1031}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca-p^2a,cb-baa,pb,pc-baa,\,\text{class }3\rangle 
\tag{7.1032}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca-\omega p^{2}a,cb-baa,pb,pc-baa,\,\text{class }%
3\rangle  \tag{7.1033}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-p^2a,pb,pc-baa,\,\text{class }3\rangle 
\tag{7.1034}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-p^2a,pb,pc-baa,\,\text{class }3\rangle 
\tag{7.1035}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,pb-baa,pc,\,\text{class }3\rangle  \tag{7.1036}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca-p^2a,cb,pb-baa,pc,\,\text{class }3\rangle 
\tag{7.1037}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca-xp^2a,cb-baa,pb-baa,pc,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.1038}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-p^2a,pb-baa,pc,\,\text{class }3\rangle 
\tag{7.1039}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-p^2a,pb-baa,pc,\,\text{class }3\rangle 
\tag{7.1040}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,pb-\omega baa,pc,\,\text{class }3\rangle 
\tag{7.1041}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca-p^{2}a,cb,pb-\omega baa,pc,\,\text{class }3\rangle 
\tag{7.1042}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca-xp^{2}a,cb-baa,pb-\omega baa,pc,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.1043}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-p^{2}a,pb-\omega baa,pc,\,\text{class }3\rangle 
\tag{7.1044}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-p^{2}a,pb-\omega baa,pc,\,\text{class }%
3\rangle  \tag{7.1045}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb,pb,pc,\,\text{class }3\rangle  \tag{7.1046}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb,pb-baa,pc,\,\text{class }3\rangle  \tag{7.1047}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca,cb,pb-\omega baa,pc,\,\text{class }3\rangle 
\tag{7.1048}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb,pb-bab,pc,\,\text{class }3\rangle  \tag{7.1049}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb,pb,pc-baa,\,\text{class }3\rangle  \tag{7.1050}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb,pb-bab,pc-baa,\,\text{class }3\rangle 
\tag{7.1051}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb,pb,pc-bab,\,\text{class }3\rangle  \tag{7.1052}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb,pb-baa,pc-bab,\,\text{class }3\rangle 
\tag{7.1053}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca,cb,pb-\omega baa,pc-bab,\,\text{class }3\rangle 
\tag{7.1054}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-bab,pb,pc,\,\text{class }3\rangle  \tag{7.1055}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-bab,pb-baa,pc,\,\text{class }3\rangle 
\tag{7.1056}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca,cb-bab,pb-\omega baa,pc,\,\text{class }3\rangle 
\tag{7.1057}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-bab,pb-bab,pc,\,\text{class }3\rangle 
\tag{7.1058}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-bab,pb-baa-bab,pc,\,\text{class }3\rangle 
\tag{7.1059}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca,cb-bab,pb-\omega baa-bab,pc,\,\text{class }%
3\rangle  \tag{7.1060}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-bab,pb,pc-bab,\,\text{class }3\rangle 
\tag{7.1061}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-bab,pb-baa,pc-bab,\,\text{class }3\rangle 
\tag{7.1062}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca,cb-bab,pb-\omega baa,pc-bab,\,\text{class }%
3\rangle  \tag{7.1063}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-bab,pb,pc-baa,\,\text{class }3\rangle 
\tag{7.1064}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-bab,pb-bab,pc-baa,\,\text{class }3\rangle 
\tag{7.1065}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-bab,pb-xbab,pc-baa-bab,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.1066}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-baa,pb,pc,\,\text{class }3\rangle  \tag{7.1067}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-baa,pb-baa,pc,\,\text{class }3\rangle 
\tag{7.1068}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca,cb-baa,pb-\omega baa,pc,\,\text{class }3\rangle 
\tag{7.1069}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-baa,pb-bab,pc,\,\text{class }3\rangle 
\tag{7.1070}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-baa,pb-baa-bab,pc,\,\text{class }3\rangle 
\tag{7.1071}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca,cb-baa,pb-\omega baa-bab,pc,\,\text{class }%
3\rangle  \tag{7.1072}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-baa,pb,pc-bab,\,\text{class }3\rangle 
\tag{7.1073}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-baa,pb-baa,pc-bab,\,\text{class }3\rangle 
\tag{7.1074}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca,cb-baa,pb-\omega baa,pc-bab,\,\text{class }%
3\rangle  \tag{7.1075}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca,cb-baa,pb-\omega ^{2}baa,pc-bab,\,\text{class }%
3\rangle \;(p=1\func{mod}4)  \tag{7.1076}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca,cb-baa,pb-\omega ^{3}baa,pc-bab,\,\text{class }%
3\rangle \;(p=1\func{mod}4)  \tag{7.1077}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-baa,pb,pc-baa,\,\text{class }3\rangle 
\tag{7.1078}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-baa,pb-bab,pc-baa,\,\text{class }3\rangle 
\tag{7.1079}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-baa,pb-xbab,pc-baa-bab,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.1080}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-baa-bab,pb,pc,\,\text{class }3\rangle 
\tag{7.1081}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-baa-bab,pb-bab,pc,\,\text{class }3\rangle 
\tag{7.1082}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca,cb-baa-bab,pb-\omega bab,pc,\,\text{class }%
3\rangle \;(p=1\func{mod}4)  \tag{7.1083}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca,cb-baa-bab,pb-baa-xbab,pc,\,\text{class }%
3\rangle \;(\text{all }x,\,x\sim 1-x)  \tag{7.1084}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca,cb-baa-bab,pb-\omega baa-xbab,pc,\,\text{class }%
3\rangle \;(\text{all }x,\,x\sim \omega -x)  \tag{7.1085}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca,cb-baa-bab,pb-xbaa,pc-bab,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.1086}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca,cb-baa-bab,pb-xbab,pc-baa-ybab,\,\text{class }%
3\rangle \;(\text{all }x,\,y\neq \frac{1}{2},\;y\sim 1-y)  \tag{7.1087}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca,cb-baa-bab,pb-xbab,pc-baa-\frac{1}{2}bab,\,\text{%
class }3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1088}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca-bab,cb,pb,pc,\,\text{class }3\rangle  \tag{7.1089}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca-bab,cb,pb-baa,pc,\,\text{class }3\rangle 
\tag{7.1090}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca-bab,cb,pb-\omega baa,pc,\,\text{class }3\rangle 
\tag{7.1091}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca-bab,cb,pb-bab,pc,\,\text{class }3\rangle 
\tag{7.1092}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca-bab,cb,pb,pc-baa,\,\text{class }3\rangle 
\tag{7.1093}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca-bab,cb,pb-bab,pc-baa,\,\text{class }3\rangle 
\tag{7.1094}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca-bab,cb,pb-\omega bab,pc-baa,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1095}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca-bab,cb,pb-\omega ^{2}bab,pc-baa,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1096}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,ca-bab,cb,pb-xbaa,pc-bab,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.1097}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca-bab,cb-\omega baa,pb,pc,\,\text{class }3\rangle 
\tag{7.1098}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca-bab,cb-\omega baa,pb-bab,pc,\,\text{class }%
3\rangle  \tag{7.1099}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca-bab,cb-\omega baa,pb-\omega bab,pc,\,\text{class 
}3\rangle \;(p=1\func{mod}4)  \tag{7.1100}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca-bab,cb-\omega baa,pb-baa-xbab,pc,\,\text{class }%
3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1101}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca-bab,cb-\omega baa,pb-\omega baa-xbab,pc,\,\text{%
class }3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1102}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca-bab,cb-\omega baa,pb-xbaa,pc-bab,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.1103}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca-bab,cb-\omega baa,pb-xbab,pc-baa,\,\text{class }%
3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1104}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,ca-bab,cb-\omega baa,pb-ybab,pc-xbaa-bab,\,\text{%
class }3\rangle \;(x\neq 0,\text{ all }y,\;x\sim -x)  \tag{7.1105}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-baa,ca,cb,pb,pc-bab,\,\text{class }3\rangle 
\tag{7.1106}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-baa,ca,cb,pb-xbab,pc,\,\text{class }3\rangle \;(\text{%
all }x)  \tag{7.1107}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-baa,ca,cb-baa,pb,pc-bab,\,\text{class }3\rangle 
\tag{7.1108}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-baa,ca,cb-baa,pb,pc-\omega bab,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1109}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-baa,ca,cb-baa,pb,pc-\omega ^{2}bab,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1110}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-baa,ca,cb-baa,pb-xbab,pc,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.1111}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-baa,ca,cb-bab,pb,pc-bab,\,\text{class }3\rangle 
\tag{7.1112}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-baa,ca,cb-bab,pb-xbab,pc,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.1113}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-baa,ca,cb-baa-bab,pb,pc-xbab,\,\text{class }3\rangle
\;(x \neq 0, x \sim -x)  \tag{7.1114}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-baa,ca,cb-baa-bab,pb-xbab,pc,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.1115}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-baa,ca,cb-\omega baa-bab,pb,pc-xbab,\,\text{class }%
3\rangle \;(x\neq 0,\;x\sim -x)  \tag{7.1116}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-baa,ca,cb-\omega baa-bab,pb-xbab,pc,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.1117}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-baa,ca-bab,cb,pb,pc-bab,\,\text{class }3\rangle 
\tag{7.1118}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-baa,ca-bab,cb,pb-xbab,pc,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.1119}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-baa,ca-bab,cb-baa,pb,pc-xbab,\,\text{class }%
3\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{4}=x^{\prime 4}\func{mod%
}p)  \tag{7.1120}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-baa,ca-bab,cb-\omega baa,pb,pc-xbab,\,\text{class }%
3\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{4}=x^{\prime 4}\func{mod%
}p)  \tag{7.1121}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-baa,ca-bab,cb-\omega ^{2}baa,pb,pc-xbab,\,\text{%
class }3\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{4}=x^{\prime 4}%
\func{mod}p,\;p=1\func{mod}4)  \tag{7.1122}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-baa,ca-bab,cb-\omega ^{3}baa,pb,pc-xbab,\,\text{%
class }3\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{4}=x^{\prime 4}%
\func{mod}p,\;p=1\func{mod}4)  \tag{7.1123}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-baa,ca-bab,cb-baa,pb-xbab,pc,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.1124}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-baa,ca-bab,cb-\omega baa,pb-xbab,pc,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.1125}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-baa,ca-bab,cb-\omega ^{2}baa,pb-xbab,pc,\,\text{%
class }3\rangle \;(\text{all }x,\;p=1\func{mod}4)  \tag{7.1126}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-baa,ca-bab,cb-\omega ^{3}baa,pb-xbab,pc,\,\text{%
class }3\rangle \;(\text{all }x,\;p=1\func{mod}4)  \tag{7.1127}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-baa,ca-bab,cb-xbaa-bab,pb,pc-ybab,\,\text{class }%
3\rangle \;(\text{all }x,\;y\neq 0,\;y\sim -y)  \tag{7.1128}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-baa,ca-bab,cb-xbaa-bab,pb-ybab,pc,\,\text{class }%
3\rangle \;(\text{all }x,\,y)  \tag{7.1129}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-baa,ca-bab,cb-xbaa-\omega bab,pb,pc-ybab,\,\text{%
class }3\rangle \;(\text{all }x,\;y\neq 0,\;y\sim -y)  \tag{7.1130}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-baa,ca-bab,cb-xbaa-\omega bab,pb-ybab,pc,\,\text{%
class }3\rangle \;(\text{all }x,\,y)  \tag{7.1131}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-bab,ca,cb,pb,pc,\,\text{class }3\rangle  \tag{7.1132}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-bab,ca,cb,pb-baa,pc,\,\text{class }3\rangle 
\tag{7.1133}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-bab,ca,cb,pb-\omega baa,pc,\,\text{class }3\rangle 
\tag{7.1134}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-bab,ca,cb,pb,pc-baa,\,\text{class }3\rangle 
\tag{7.1135}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-bab,ca-baa,cb,pb,pc,\,\text{class }3\rangle 
\tag{7.1136}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-bab,ca-baa,cb,pb-baa,pc,\,\text{class }3\rangle 
\tag{7.1137}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-bab,ca-baa,cb,pb-\omega baa,pc,\,\text{class }%
3\rangle  \tag{7.1138}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-bab,ca-baa,cb,pb,pc-baa,\,\text{class }3\rangle 
\tag{7.1139}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-bab,ca-bab,cb,pb,pc,\,\text{class }3\rangle 
\tag{7.1140}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-bab,ca-bab,cb,pb-baa,pc,\,\text{class }3\rangle 
\tag{7.1141}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-bab,ca-bab,cb,pb-\omega baa,pc,\,\text{class }%
3\rangle  \tag{7.1142}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-bab,ca-bab,cb,pb,pc-baa,\,\text{class }3\rangle 
\tag{7.1143}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-bab,ca-bab,cb,pb,pc-\omega baa,\,\text{class }%
3\rangle \;(p=1\func{mod}4)  \tag{7.1144}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-bab,ca,cb-baa,pb,pc,\,\text{class }3\rangle 
\tag{7.1145}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-bab,ca,cb-baa,pb-baa,pc,\,\text{class }3\rangle 
\tag{7.1146}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-bab,ca,cb-baa,pb-\omega baa,pc,\,\text{class }%
3\rangle  \tag{7.1147}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-bab,ca,cb-baa,pb,pc-xbaa,\,\text{class }3\rangle
\;(x\neq 0,\;x\sim -x)  \tag{7.1148}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-bab,ca-bab,cb-baa,pb-xbaa,pc,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.1149}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a-bab,ca-bab,cb-baa,pb,pc-xbaa,\,\text{class }3\rangle
\;(x \neq 0, x \sim -x)  \tag{7.1150}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-bab,ca-\omega bab,cb-baa,pb-xbaa,pc,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.1151}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-bab,ca-\omega bab,cb-baa,pb,pc-xbaa,\,\text{class }%
3\rangle \;(x\neq 0,\;x\sim -x)  \tag{7.1152}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca,cb,pb,pc,\,\text{class }3\rangle 
\tag{7.1153}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca,cb,pb-baa,pc,\,\text{class }3\rangle 
\tag{7.1154}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca,cb,pb-\omega baa,pc,\,\text{class }%
3\rangle  \tag{7.1155}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca,cb,pb,pc-baa,\,\text{class }3\rangle 
\tag{7.1156}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca-baa,cb,pb,pc,\,\text{class }3\rangle 
\tag{7.1157}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca-baa,cb,pb-baa,pc,\,\text{class }%
3\rangle  \tag{7.1158}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca-baa,cb,pb-\omega baa,pc,\,\text{class 
}3\rangle  \tag{7.1159}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca-baa,cb,pb,pc-baa,\,\text{class }%
3\rangle  \tag{7.1160}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca-bab,cb,pb,pc,\,\text{class }3\rangle 
\tag{7.1161}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca-bab,cb,pb-baa,pc,\,\text{class }%
3\rangle  \tag{7.1162}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca-bab,cb,pb-\omega baa,pc,\,\text{class 
}3\rangle  \tag{7.1163}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca-bab,cb,pb,pc-baa,\,\text{class }%
3\rangle  \tag{7.1164}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca-bab,cb,pb,pc-\omega baa,\,\text{class 
}3\rangle \;(p=1\func{mod}4)  \tag{7.1165}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca,cb-baa,pb,pc,\,\text{class }3\rangle 
\tag{7.1166}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca,cb-baa,pb-baa,pc,\,\text{class }%
3\rangle  \tag{7.1167}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca,cb-baa,pb-\omega baa,pc,\,\text{class 
}3\rangle  \tag{7.1168}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca,cb-baa,pb,pc-xbaa,\,\text{class }%
3\rangle \;(x\neq 0,\;x\sim -x)  \tag{7.1169}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca-bab,cb-baa,pb-xbaa,pc,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.1170}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca-bab,cb-baa,pb,pc-xbaa,\,\text{class }%
3\rangle \;(x\neq 0,\;x\sim -x)  \tag{7.1171}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca-\omega bab,cb-baa,pb-xbaa,pc,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.1172}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a-\omega bab,ca-\omega bab,cb-baa,pb,pc-xbaa,\,\text{%
class }3\rangle \;(x\neq 0,\;x\sim -x)  \tag{7.1173}
\end{equation}

\subsection{Descendants of 6.100}

3 algebras

\begin{equation}
\langle a,b,c\,|\,baa,caa,cb,pb-\omega ca,pc-ba,\,\text{class }3\rangle 
\tag{7.1174}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,cb-p^{2}a,pb-\omega ca,pc-ba,\,\text{class }%
3\rangle  \tag{7.1175}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^{2}a,cb,pb-\omega ca,pc-ba,\,\text{class }3\rangle 
\tag{7.1176}
\end{equation}

\subsection{Descendants of 6.101}

$3(p-1)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,caa,cb,pb-xca,pc-ba-ca,\,\text{class }3\rangle \;(%
\text{all }x\text{ with }1+4x\text{ not a square }\func{mod}p)  \tag{7.1177}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,cb-p^{2}a,pb-xca,pc-ba-ca,\,\text{class }3\rangle
\;(\text{all }x\text{ with }1+4x\text{ not a square }\func{mod}p) 
\tag{7.1178}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^{2}a,cb,pb-xca,pc-ba-ca,\,\text{class }3\rangle \;(%
\text{all }x\text{ with }1+4x\text{ not a square }\func{mod}p)  \tag{7.1179}
\end{equation}

\subsection{Descendants of 6.102}

$p+3$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,cb-xp^2b,pa-ca,pc-ba,\,\text{class }3\rangle \;(\text{%
all }x)  \tag{7.1180}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2b,cb,pa-ca,pc-ba,\,\text{class }3\rangle  \tag{7.1181}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2b-baa,cb,pa-ca,pc-ba,\,\text{class }3\rangle 
\tag{7.1182}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}b-\omega baa,cb,pa-ca,pc-ba,\,\text{class }3\rangle 
\tag{7.1183}
\end{equation}

\subsection{Descendants of 6.103}

$p+3$ algebras

\begin{equation}
\langle a,b,c\,|\,bab,cb,pa-ba,pc-ca,\,\text{class }3\rangle  \tag{7.1184}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cb-p^2b,pa-ba,pc-ca,\,\text{class }3\rangle 
\tag{7.1185}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2b-xbab,cb,pa-ba,pc-ca,\,\text{class }3\rangle \;(\text{%
all }x)  \tag{7.1186}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2b+bab,cb,pa-ba,pc-ca-bab,\,\text{class }3\rangle 
\tag{7.1187}
\end{equation}

\subsection{Descendants of 6.104}

$5p+24+3\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cab,cac,cbb,cbc,pa-baa,pb,pc,\,\text{class }%
3\rangle  \tag{7.1188}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,caa,cab+bac,cac,cbb,cbc,pa-bac,pb,pc,\,\text{class 
}3\rangle  \tag{7.1189}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cab,cac+bab,cbb,cbc,pa-bab,pb,pc,\,\text{class 
}3\rangle  \tag{7.1190}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cab,cac+\omega bab,cbb,cbc,pa-bab,pb,pc,\,%
\text{class }3\rangle  \tag{7.1191}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cab,cac,cbb,cbc,pa-bab,pb,pc,\,\text{class }%
3\rangle  \tag{7.1192}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cab,cac,cbb,cbc,pa-\omega bab,pb,pc,\,\text{%
class }3\rangle  \tag{7.1193}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa-bab,cab,cac,cbb,cbc,pa-bab,pb,pc,\,\text{class 
}3\rangle  \tag{7.1194}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa-bab,cab,cac,cbb,cbc,pa-\omega bab,pb,pc,\,%
\text{class }3\rangle  \tag{7.1195}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cab-bab,cac-xbab,cbb,cbc,pa-bab,pb,pc,\,\text{%
class }3\rangle \;(x \neq -2)  \tag{7.1196}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cab-bab,cac-\frac{1}{4}bab,cbb,cbc,pa-\omega
bab,pb,pc,\,\text{class }3\rangle  \tag{7.1197}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cab-bab,cac+2bab,cbb,cbc,pa-bab,pb,pc,\,\text{%
class }3\rangle  \tag{7.1198}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa-bab,cab-bab,cac+2bab,cbb,cbc,pa-bab,pb,pc,\,%
\text{class }3\rangle  \tag{7.1199}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cab,cac,cbc,pa-cbb,pb,pc,\,\text{class }%
3\rangle  \tag{7.1200}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cab,cac,cbb-baa,cbc,pa-baa,pb,pc,\,\text{class 
}3\rangle  \tag{7.1201}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cab,cac,cbb-caa,cbc,pa-caa,pb,pc,\,\text{class 
}3\rangle  \tag{7.1202}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cab,cac,cbb-\omega caa,cbc,pa-caa,pb,pc,\,%
\text{class }3\rangle  \tag{7.1203}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,caa-xbac,cab,cac,cbb-bac,cbc,pa-bac,pb,pc,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.1204}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac-baa,caa+2baa,cab,cac,cbb-baa,cbc,pa-baa,pb,pc,\,%
\text{class }3\rangle  \tag{7.1205}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac-baa,caa+2baa,cab,cac,cbb-baa,cbc,pa-\omega
baa,pb,pc,\,\text{class }3\rangle  \tag{7.1206}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cab,cbb-cac,cbc,pa-cac,pb,pc,\,\text{class 
}3\rangle  \tag{7.1207}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cab,cbb-cac,cbc,pa-\omega cac,pb,pc,\,%
\text{class }3\rangle  \tag{7.1208}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cab,cac+bab,cbb+bab,cbc,pa-bab,pb,pc,\,\text{%
class }3\rangle  \tag{7.1209}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cab,cac+bab,cbb+bab,cbc,pa-\omega bab,pb,pc,\,%
\text{class }3\rangle  \tag{7.1210}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cab,cac+\omega bab,cbb+\omega
bab,cbc,pa-bab,pb,pc,\,\text{class }3\rangle  \tag{7.1211}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cab,cac+\omega bab,cbb+\omega
bab,cbc,pa-\omega bab,pb,pc,\,\text{class }3\rangle  \tag{7.1212}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-xbaa,bac,caa,cab,cac-baa,cbb-baa,cbc,pa-baa,pb,pc,\,%
\text{class }3\rangle \;(\text{all }x,\,x \sim x^{\prime }\text{ if }%
x^3=x^{\prime 3}\func{mod}p)  \tag{7.1213}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-xbaa,bac,caa,cab,cac-\omega baa,cbb-\omega
baa,cbc,pa-baa,pb,pc,\,\text{class }3\rangle \;(\text{all }x,\,x\sim
x^{\prime}\text{ if }x^3=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3) 
\tag{7.1214}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-xbaa,bac,caa,cab,cac-\omega ^{2}baa,cbb-\omega
^{2}baa,cbc,pa-baa,pb,pc,\,\text{class }3\rangle \;(\text{all }x,\,x\sim
x^{\prime}\text{ if }x^3=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3) 
\tag{7.1215}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-xbaa,bac,caa,cab,cac-baa,cbb-baa,cbc,pa-\omega
baa,pb,pc,\,\text{class }3\rangle \;(\text{all }x,\,x\sim x^{\prime }\text{
if }x^{3}=x^{\prime 3}\func{mod}p)  \tag{7.1216}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-xbaa,bac,caa,cab,cac-\omega baa,cbb-\omega
baa,cbc,pa-\omega baa,pb,pc,\,\text{class }3\rangle \;(\text{all }%
x,\,x^{\prime} \text{ if }x^3=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3) 
\tag{7.1217}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-xbaa,bac,caa,cab,cac-\omega ^{2}baa,cbb-\omega
^{2}baa,cbc,pa-\omega baa,pb,pc,\,\text{class }3\rangle \;(\text{all }x,\,x
\sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3) 
\tag{7.1218}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cab,cac,cbb,cbc,pa,pb,pc,\,\text{class }%
3\rangle  \tag{7.1219}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cab,cac-bab,cbb,cbc,pa,pb,pc,\,\text{class }%
3\rangle  \tag{7.1220}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cab,cac-\omega bab,cbb,cbc,pa,pb,pc,\,\text{%
class }3\rangle  \tag{7.1221}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa-bab,cab,cac,cbb,cbc,pa,pb,pc,\,\text{class }%
3\rangle  \tag{7.1222}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cab,cac-baa,cbb-baa,cbc-xbaa,pa,pb,pc,\,\text{%
class }3\rangle \;(\text{all }x,\,x \sim x^{\prime }\text{ if }x^3=x^{\prime
3}\func{mod}p)  \tag{7.1223}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cab,cac-baa,cbb-\omega baa,cbc-xbaa,pa,pb,pc,\,%
\text{class }3\rangle \;(\text{all }x,\,x \sim x^{\prime }\text{ if }%
x^3=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.1224}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cab,cac-baa,cbb-\omega
^{2}baa,cbc-xbaa,pa,pb,pc,\,\text{class }3\rangle \;(\text{all }x,\,x \sim
x^{\prime} \text{ if }x^3=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3) 
\tag{7.1225}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa-baa,cab,cac+baa,cbb+baa,cbc-baa,pa,pb,pc,\,%
\text{class }3\rangle  \tag{7.1226}
\end{equation}

\subsection{Descendants of 6.105}

$11+5\gcd (p-1,3)+\gcd (p-1,5)$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,pa-cb,pb,pc,\,\text{class }3\rangle 
\tag{7.1227}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,pa-cb-caa,pb,pc,\,\text{class }3\rangle 
\tag{7.1228}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,pa-cb,pb,pc-caa,\,\text{class }3\rangle 
\tag{7.1229}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,pa-cb,pb,pc-\omega caa,\,\text{class }%
3\rangle  \tag{7.1230}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,pa-cb,pb-caa,pc,\,\text{class }3\rangle 
\tag{7.1231}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,pa-cb,pb,pc,\,\text{class }3\rangle 
\tag{7.1232}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,pa-cb,pb,pc-cac,\,\text{class }3\rangle 
\tag{7.1233}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,pa-cb,pb-cac,pc,\,\text{class }3\rangle 
\tag{7.1234}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,pa-cb,pb-\omega cac,pc,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1235}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,pa-cb,pb-\omega ^{2}cac,pc,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1236}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,pa-cb,pb,pc,\,\text{class }3\rangle 
\tag{7.1237}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,pa-cb,pb,pc-baa,\,\text{class }3\rangle
\tag{7.1238}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,pa-cb,pb,pc-xbaa,\,\text{class }%
3\rangle \;(x=\omega ,\omega ^{2},\omega ^{3},\omega ^{4},\;p=1\func{mod}5) 
\tag{7.1239}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,pa-cb,pb-baa,pc,\,\text{class }3\rangle
\tag{7.1240}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,pa-cb,pb-\omega baa,pc,\,\text{class }%
3\rangle  \tag{7.1241}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,pa-cb,pb-xbaa,pc,\,\text{class }%
3\rangle \;(x=\omega ^{2},\omega ^{3},\omega ^{4},\omega ^{5},\;p=1\func{mod}%
3)  \tag{7.1242}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+bab,pa-cb,pb,pc,\,\text{class }3\rangle 
\tag{7.1243}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+bab,pa-cb,pb,pc-bab,\,\text{class }3\rangle
\tag{7.1244}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+bab,pa-cb,pb,pc-\omega bab,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1245}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+bab,pa-cb,pb,pc-\omega ^{2}bab,\,\text{%
class }3\rangle \;(p=1\func{mod}3)  \tag{7.1246}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+bab,pa-cb,pb-bab,pc-bab,\,\text{class }%
3\rangle  \tag{7.1247}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+\omega bab,pa-cb,pb,pc,\,\text{class }%
3\rangle  \tag{7.1248}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+\omega bab,pa-cb,pb,pc-bab,\,\text{class }%
3\rangle  \tag{7.1249}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+\omega bab,pa-cb,pb,pc-\omega bab,\,\text{%
class }3\rangle \;(p=1\func{mod}3)  \tag{7.1250}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+\omega bab,pa-cb,pb,pc-\omega ^{2}bab,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.1251}
\end{equation}

\subsection{Descendants of 6.106}

$p^{2}+10p+32+\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,caa,cab,cac,cbc,pa-ba,pb,pc-xcbb,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.1252}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cac,cbc,pa-ba,pb-cbb,pc-xcbb,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.1253}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cbb-cac,cbc,pa-ba,pb,pc-xcac,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.1254}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cbb-cac,cbc,pa-ba,pb-cac,pc-xcac,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.1255}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac,cbb-caa,cbc,pa-ba,pb,pc-xcaa,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.1256}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac,cbb-caa,cbc,pa-ba,pb-caa,pc-xcaa,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.1257}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac,cbb-\omega caa,cbc,pa-ba,pb,pc-xcaa,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.1258}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac,cbb-\omega caa,cbc,pa-ba,pb-caa,pc-xcaa,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.1259}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac-caa,cbb-caa,cbc,pa-ba,pb-ycaa,pc-xcaa,\,\text{%
class }3\rangle \;(\text{all }x,\,y,\;y\sim -y)  \tag{7.1260}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac-caa,cbb-\omega caa,cbc,pa-ba,pb-ycaa,pc-xcaa,\,%
\text{class }3\rangle \;(\text{all }x,\,y,\;y\sim -y)  \tag{7.1261}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac,cbb,cbc,pa-ba,pb,pc,\,\text{class }3\rangle 
\tag{7.1262}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac,cbb,cbc,pa-ba,pb,pc-cab,\,\text{class }3\rangle 
\tag{7.1263}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac,cbb,cbc,pa-ba,pb-cab,pc,\,\text{class }3\rangle 
\tag{7.1264}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac,cbb,cbc,pa-ba,pb-cab,pc-cab,\,\text{class }3\rangle
\tag{7.1265}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac,cbb,cbc-cab,pa-ba,pb,pc,\,\text{class }3\rangle 
\tag{7.1266}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac,cbb,cbc-cab,pa-ba,pb-cab,pc,\,\text{class }3\rangle
\tag{7.1267}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac,cbb,cbc-cab,pa-ba,pb-\omega cab,pc,\,\text{class }%
3\rangle  \tag{7.1268}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac,cbb,cbc-cab,pa-ba,pb-xcab,pc-cab,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.1269}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cac,cbb,pa-ba,pb,pc,\,\text{class }3\rangle 
\tag{7.1270}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cac,cbb,pa-ba,pb,pc-cbc,\,\text{class }3\rangle 
\tag{7.1271}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cac,cbb,pa-ba,pb-cbc,pc,\,\text{class }3\rangle 
\tag{7.1272}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cac,cbb,pa-ba,pb-\omega cbc,pc,\,\text{class }%
3\rangle  \tag{7.1273}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cac,cbb,pa-ba-cbc,pb,pc,\,\text{class }3\rangle 
\tag{7.1274}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cac,cbb,pa-ba-cbc,pb,pc-cbc,\,\text{class }3\rangle
\tag{7.1275}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cac,cbb,pa-ba-cbc,pb-cbc,pc,\,\text{class }3\rangle
\tag{7.1276}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cac,cbb,pa-ba-cbc,pb-\omega cbc,pc,\,\text{class }%
3\rangle  \tag{7.1277}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac,cbb,cbc-caa,pa-ba,pb,pc,\,\text{class }3\rangle 
\tag{7.1278}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac,cbb,cbc-caa,pa-ba,pb,pc-caa,\,\text{class }3\rangle
\tag{7.1279}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac,cbb,cbc-caa,pa-ba,pb,pc-\omega caa,\,\text{class }%
3\rangle  \tag{7.1280}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac,cbb,cbc-caa,pa-ba,pb-caa,pc,\,\text{class }3\rangle
\tag{7.1281}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac,cbb,cbc-caa,pa-ba,pb-\omega caa,pc,\,\text{class }%
3\rangle  \tag{7.1282}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac,cbb,cbc-caa,pa-ba,pb-\omega ^{2}caa,pc,\,\text{%
class }3\rangle \;(p=1\func{mod}4)  \tag{7.1283}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac,cbb,cbc-caa,pa-ba,pb-\omega ^{3}caa,pc,\,\text{%
class }3\rangle \;(p=1\func{mod}4)  \tag{7.1284}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cbb,cbc,pa-ba,pb,pc,\,\text{class }3\rangle 
\tag{7.1285}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cbb,cbc,pa-ba-cac,pb,pc,\,\text{class }3\rangle 
\tag{7.1286}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cbb,cbc,pa-ba-\omega cac,pb,pc,\,\text{class }%
3\rangle  \tag{7.1287}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cbb,cbc,pa-ba,pb,pc-cac,\,\text{class }3\rangle 
\tag{7.1288}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cbb,cbc,pa-ba-cac,pb,pc-cac,\,\text{class }3\rangle
\tag{7.1289}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cbb,cbc,pa-ba-\omega cac,pb,pc-cac,\,\text{class }%
3\rangle  \tag{7.1290}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cab,cbb,cbc,pa-ba,pb-cac,pc,\,\text{class }3\rangle 
\tag{7.1291}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac,cbb,cbc,pa-ba,pb,pc,\,\text{class }3\rangle 
\tag{7.1292}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac,cbb,cbc,pa-ba,pb,pc-caa,\,\text{class }3\rangle 
\tag{7.1293}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac,cbb,cbc,pa-ba,pb,pc-\omega caa,\,\text{class }%
3\rangle  \tag{7.1294}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac,cbb,cbc,pa-ba,pb-caa,pc,\,\text{class }3\rangle 
\tag{7.1295}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac-caa,cbb,cbc,pa-ba,pb,pc,\,\text{class }3\rangle 
\tag{7.1296}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac-caa,cbb,cbc,pa-ba,pb,pc-caa,\,\text{class }3\rangle
\tag{7.1297}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac-caa,cbb,cbc,pa-ba,pb,pc-\omega caa,\,\text{class }%
3\rangle  \tag{7.1298}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac-caa,cbb,cbc,pa-ba,pb-caa,pc,\,\text{class }3\rangle
\tag{7.1299}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac-caa,cbb,cbc,pa-ba,pb-\omega caa,pc,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1300}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cab,cac-caa,cbb,cbc,pa-ba,pb-\omega ^{2}caa,pc,\,\text{%
class }3\rangle \;(p=1\func{mod}3)  \tag{7.1301}
\end{equation}

\subsection{Descendants of 6.108}

$(p-1)(12+\gcd (p-1,3))/2+\gcd (p-1,4)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,bab,pa-ca,pb-cb,pc,\,\text{class }3\rangle 
\tag{7.1302}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,pa-ca,pb-cb,pc,\,\text{class }3\rangle 
\tag{7.1303}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,pa-ca-bab,pb-cb,pc,\,\text{class }3\rangle 
\tag{7.1304}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,pa-ca-\omega bab,pb-cb,pc,\,\text{class }3\rangle 
\tag{7.1305}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,pa-ca,pb-cb,pc-bab,\,\text{class }3\rangle 
\tag{7.1306}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,pa-ca,pb-xcb,pc,\,\text{class }3\rangle \;(x\neq
0,1,-1,\;x\sim x^{-1})  \tag{7.1307}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac-baa,pa-ca,pb+2cb,pc,\,\text{class }3\rangle 
\tag{7.1308}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,pa-ca,pb-xcb,pc,\,\text{class }3\rangle \;(x\neq
0,1,-1,\;x\sim x^{-1})  \tag{7.1309}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,pa-ca-bab,pb-xcb,pc,\,\text{class }3\rangle
\;(x\neq 0,1,-1,\;x\sim x^{-1})  \tag{7.1310}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,pa-ca-\omega bab,pb-xcb,pc,\,\text{class }3\rangle
\;(x\neq 0,1,-1,\;x\sim x^{-1})  \tag{7.1311}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,pa-ca,pb-xcb,pc-bab,\,\text{class }3\rangle
\;(x\neq 0,1,-1,\;x\sim x^{-1})  \tag{7.1312}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,pa-ca,pb-xcb,pc,\,\text{class }3\rangle \;(x\neq
0,1,-1,\;x\sim x^{-1})  \tag{7.1313}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,pa-ca,pb-xcb-baa,pc,\,\text{class }3\rangle
\;(x\neq 0,1,-1,\;x\sim x^{-1})  \tag{7.1314}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,pa-ca,pb-xcb-\omega baa,pc,\,\text{class }3\rangle
\;(x\neq 0,1,-1,\;x\sim x^{-1})  \tag{7.1315}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,pa-ca,pb-xcb,pc-baa,\,\text{class }3\rangle
\;(x\neq 0,1,-1,\;x\sim x^{-1})  \tag{7.1316}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,pa-ca,pb-xcb,pc,\,\text{class }3\rangle
\;(x\neq 0,1,-1,\;x\sim x^{-1})  \tag{7.1317}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,pa-ca,pb-xcb-baa,pc,\,\text{class }3\rangle
\;(x\neq 0,1,-1,\;x\sim x^{-1})  \tag{7.1318}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,pa-ca,pb-xcb-\omega baa,pc,\,\text{class }%
3\rangle \;(x\neq 0,1,-1,\;x\sim x^{-1})  \tag{7.1319}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,pa-ca,pb-xcb,pc-baa,\,\text{class }3\rangle
\;(x\neq 0,1,-1,\;x\sim x^{-1})  \tag{7.1320}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,pa-ca,pb-xcb,pc-\omega baa,\,\text{class }%
3\rangle \;(x\neq 0,1,-1,\;x\sim x^{-1},\;p=1\func{mod}3)  \tag{7.1321}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,pa-ca,pb-xcb,pc-\omega ^{2}baa,\,\text{class }%
3\rangle \;(x\neq 0,1,-1,\;x\sim x^{-1},\;p=1\func{mod}3)  \tag{7.1322}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,pa-ca,pb+cb,pc,\,\text{class }3\rangle 
\tag{7.1323}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,cab,pa-ca,pb+cb,pc,\,\text{class }3\rangle 
\tag{7.1324}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,cab,pa-ca-bab,pb+cb,pc,\,\text{class }3\rangle 
\tag{7.1325}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,cab,pa-ca-\omega bab,pb+cb,pc,\,\text{class }3\rangle 
\tag{7.1326}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,cab,pa-ca,pb+cb,pc-bab,\,\text{class }3\rangle 
\tag{7.1327}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,cab,pa-ca,pb+cb,pc,\,\text{class }3\rangle 
\tag{7.1328}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,cab,pa-ca,pb+cb-baa,pc,\,\text{class }3\rangle 
\tag{7.1329}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,cab,pa-ca,pb+cb-\omega baa,pc,\,\text{class }%
3\rangle \;(p=1\func{mod}4)  \tag{7.1330}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,cab,pa-ca,pb+cb,pc-baa,\,\text{class }3\rangle 
\tag{7.1331}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,cab,pa-ca,pb+cb,pc-\omega baa,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1332}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,cab,pa-ca,pb+cb,pc-\omega ^{2}baa,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1333}
\end{equation}

\subsection{Descendants of 6.109}

$7+2\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,bab,pa-ca-cb,pb-cb,pc,\,\text{class }3\rangle 
\tag{7.1334}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,pa-ca-cb,pb-cb,pc,\,\text{class }3\rangle 
\tag{7.1335}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,pa-ca-cb,pb-cb-bab,pc,\,\text{class }3\rangle 
\tag{7.1336}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,pa-ca-cb,pb-cb-\omega bab,pc,\,\text{class }%
3\rangle  \tag{7.1337}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,pa-ca-cb,pb-cb,pc-bab,\,\text{class }3\rangle 
\tag{7.1338}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,pa-ca-cb,pb-cb,pc-\omega bab,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1339}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,pa-ca-cb,pb-cb,pc-\omega ^{2}bab,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1340}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,pa-ca-cb,pb-cb,pc,\,\text{class }3\rangle 
\tag{7.1341}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,pa-ca-cb,pb-cb-baa,pc,\,\text{class }3\rangle 
\tag{7.1342}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,pa-ca-cb,pb-cb-\omega baa,pc,\,\text{class }%
3\rangle  \tag{7.1343}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,pa-ca-cb,pb-cb,pc-baa,\,\text{class }3\rangle 
\tag{7.1344}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,pa-ca-cb,pb-cb,pc-\omega baa,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1345}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,pa-ca-cb,pb-cb,pc-\omega ^{2}baa,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1346}
\end{equation}

\subsection{Descendants of 6.110}

$2+\gcd (p-1,3)+\gcd (p-1,4)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,bab,pa-\omega cb,pb-ca,pc,\,\text{class }3\rangle 
\tag{7.1347}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,cbb,pa-\omega cb,pb-ca,pc,\,\text{class }3\rangle 
\tag{7.1348}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,cbb,pa-\omega cb,pb-ca,pc-bab,\,\text{class }3\rangle 
\tag{7.1349}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,cbb,pa-\omega cb,pb-ca,pc-\omega bab,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1350}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,cbb,pa-\omega cb,pb-ca,pc-\omega ^{2}bab,\,\text{class 
}3\rangle \;(p=1\func{mod}3)  \tag{7.1351}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,cbb,pa-\omega cb,pb-ca-bab,pc,\,\text{class }3\rangle 
\tag{7.1352}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,cbb,pa-\omega cb,pb-ca-\omega bab,pc,\,\text{class }%
3\rangle \;(p=1\func{mod}4)  \tag{7.1353}
\end{equation}

\subsection{Descendants of 6.111}

$(p-1)(4+\gcd (p-1,3))/2$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,bab,pa-xcb,pb-ca-cb,pc,\,\text{class }3\rangle \;(%
\text{all }x\text{ with }1+4x\text{ not a square }\func{mod}p)  \tag{7.1354}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,pa-xcb,pb-ca-cb,pc,\,\text{class }3\rangle \;(%
\text{all }x\text{ with }1+4x\text{ not a square }\func{mod}p)  \tag{7.1355}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,pa-xcb,pb-ca-cb-bab,pc,\,\text{class }3\rangle \;(%
\text{all }x\text{ with }1+4x\text{ not a square }\func{mod}p)  \tag{7.1356}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,pa-xcb,pb-ca-cb-\omega bab,pc,\,\text{class }%
3\rangle \;(\text{all }x\text{ with }1+4x\text{ not a square }\func{mod}p) 
\tag{7.1357}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,pa-xcb,pb-ca-cb,pc-bab,\,\text{class }3\rangle \;(%
\text{all }x\text{ with }1+4x\text{ not a square }\func{mod}p)  \tag{7.1358}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,pa-xcb,pb-ca-cb,pc-\omega bab,\,\text{class }%
3\rangle \;(\text{all }x\text{ with }1+4x\text{ not a square }\func{mod}%
p,\;p=1\func{mod}3)  \tag{7.1359}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,pa-xcb,pb-ca-cb,pc-\omega ^{2}bab,\,\text{class }%
3\rangle \;(\text{all }x\text{ with }1+4x\text{ not a square }\func{mod}%
p,\;p=1\func{mod}3)  \tag{7.1360}
\end{equation}

\subsection{Descendants of 6.112}

$p^{2}+4p+3+2\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c\,|\,bab,cbb,pa-ba,pb-ca,pc,\,\text{class }3\rangle 
\tag{7.1361}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cbb,pa-ba-cbc,pb-ca,pc,\,\text{class }3\rangle 
\tag{7.1362}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cbb,pa-ba-\omega cbc,pb-ca,pc,\,\text{class }3\rangle
\;(p=1\func{mod}3)  \tag{7.1363}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cbb,pa-ba-\omega ^{2}cbc,pb-ca,pc,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1364}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cbb,pa-ba-xcbc,pb-ca,pc-cbc,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.1365}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cbc,pa-ba,pb-ca,pc-xcbb,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.1366}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cbc,pa-ba,pb-ca-cbb,pc-xcbb,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.1367}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cbc-cbb,pa-ba,pb-ca-xcbb,pc-ycbb,\,\text{class }%
3\rangle \;(\text{all }x,\,y)  \tag{7.1368}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cbb,cbc,pa-ba,pb-ca,pc,\,\text{class }3\rangle 
\tag{7.1369}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cbb,cbc-bab,pa-ba,pb-ca,pc,\,\text{class }3\rangle 
\tag{7.1370}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cbb,cbc-\omega bab,pa-ba,pb-ca,pc,\,\text{class }3\rangle
\;(p=1\func{mod}3)  \tag{7.1371}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cbb,cbc-\omega ^{2}bab,pa-ba,pb-ca,pc,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.1372}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cbb-bab,cbc-xbab,pa-ba,pb-ca,pc,\,\text{class }3\rangle \;(%
\text{all }x,\,x\sim -x)  \tag{7.1373}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cbb-\omega bab,cbc-xbab,pa-ba,pb-ca,pc,\,\text{class }%
3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1374}
\end{equation}

\subsection{Descendants of 6.113}

$5p+4$ algebras

\begin{equation}
\langle a,b,c\,|\,bab,cac,pa-ba,pb-cb,pc,\,\text{class }3\rangle 
\tag{7.1375}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cac,pa-ba,pb-cb,pc-caa,\,\text{class }3\rangle 
\tag{7.1376}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cac,pa-ba,pb-cb,pc-\omega caa,\,\text{class }3\rangle 
\tag{7.1377}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cac,pa-ba,pb-cb-caa,pc-xcaa,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.1378}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cac,pa-ba,pb-cb-\omega caa,pc-xcaa,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.1379}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,pa-ba-xcac,pb-cb,pc,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.1380}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,pa-ba-xcac,pb-cb,pc-cac,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.1381}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac,pa-ba,pb-cb,pc,\,\text{class }3\rangle 
\tag{7.1382}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac-xbab,pa-ba,pb-cb,pc,\,\text{class }3\rangle \;(x
\neq 0)  \tag{7.1383}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa-bab,cac+2bab,pa-ba,pb-cb,pc,\,\text{class }3\rangle 
\tag{7.1384}
\end{equation}

\subsection{Descendants of 6.114}

$4p-4$ algebras

\begin{equation}
\langle a,b,c\,|\,bac,pa-ba,pb-cb,pc+ba-ca,\,\text{class }3\rangle 
\tag{7.1385}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bac-bab,pa-ba,pb-cb,pc+ba-ca-xbab,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.1386}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bac,pa-ba,pb-cb,pc-3ba-ca,\,\text{class }3\rangle 
\tag{7.1387}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bac+bab,pa-ba,pb-cb,pc-3ba-ca,\,\text{class }3\rangle 
\tag{7.1388}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bac-zbab,pa-ba,pb-cb,pc-xba-ca,\,\text{class }3\rangle
\;(x\neq -1,3)  \tag{7.1389}
\end{equation}%
In algebra 7.1389, (for a given $x\neq -1,3$) $z$ and $z^{\prime }$ define
isomorphic algebras if the ratios $1:z$ and $1:z^{\prime }$ are in the same
orbit of ratios $\alpha :\beta $ under the action%
\[
\left( 
\begin{array}{c}
\alpha \\ 
\beta%
\end{array}%
\right) \rightarrow A\left( 
\begin{array}{c}
\alpha \\ 
\beta%
\end{array}%
\right) 
\]%
where $A$ equals%
\[
\left( 
\begin{array}{cc}
x-1 & 1 \\ 
-1 & 0%
\end{array}%
\right) \text{ or }\left( 
\begin{array}{cc}
x^{2}-2x & x-1 \\ 
1-x & -1%
\end{array}%
\right) \text{ or }\left( 
\begin{array}{cc}
\allowbreak \left( 1+\gamma x\right) \left( \gamma x-2\gamma +1\right) & 
\allowbreak \gamma \left( \gamma x+2-\gamma \right) \\ 
-\gamma \left( \gamma x+2-\gamma \right) & -\left( -1+\gamma \right) \left(
\gamma +1\right)%
\end{array}%
\right) 
\]%
with $\gamma \neq -1$ and $\gamma $ is not a root of $\allowbreak \gamma
^{2}+(x-1)\gamma +1=0$. (Note that the ratio $1:0$ is in the same orbit as
the ratio $0:1$.)

\subsection{Descendants of 6.115}

3 algebras

\begin{equation}
\langle a,b,c\,|\,bac,cac,pa-ba,pb-ca,pc-cb,\,\text{class }3\rangle 
\tag{7.1390}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bac,cac-bab,pa-ba,pb-ca,pc-cb,\,\text{class }3\rangle 
\tag{7.1391}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bac,cac-\omega bab,pa-ba,pb-ca,pc-cb,\,\text{class }%
3\rangle  \tag{7.1392}
\end{equation}

\subsection{Descendants of 6.116}

3 algebras

\begin{equation}
\langle a,b,c\,|\,bab,pa-ba,pb-ca,pc+cb,\,\text{class }3\rangle  \tag{7.1393}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,pa-ba-cac,pb-ca,pc+cb,\,\text{class }3\rangle 
\tag{7.1394}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,pa-ba-\omega cac,pb-ca,pc+cb,\,\text{class }3\rangle 
\tag{7.1395}
\end{equation}

\subsection{Descendants of 6.117}

$p+1$ algebras

\begin{equation}
\langle a,b,c\,|\,bab,pa-ba-xcac,pb-ca,pc-\omega ba+cb,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.1396}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,pa-ba,pb-ca,pc-\omega ba+cb,\,\text{class }3\rangle 
\tag{7.1397}
\end{equation}

\subsection{Descendants of 6.118}

$11+4\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c\,|\,ca,cb,baaa,baab,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.1398}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,baaa,baab,pa,pb-babb,pc,\,\text{class }4\rangle 
\tag{7.1399}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,baaa,baab,pa-babb,pb,pc,\,\text{class }4\rangle 
\tag{7.1400}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,baaa,baab,pa-\omega babb,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1401}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,baaa,baab,pa-\omega ^{2}babb,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1402}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,baaa,baab,pa,pb,pc-babb,\,\text{class }4\rangle 
\tag{7.1403}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-babb,cb,baaa,baab,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.1404}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-babb,cb,baaa,baab,pa,pb-babb,pc,\,\text{class }4\rangle
\tag{7.1405}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-babb,cb,baaa,baab,pa-babb,pb,pc,\,\text{class }4\rangle
\tag{7.1406}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-babb,cb,baaa,baab,pa-\omega babb,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1407}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-babb,cb,baaa,baab,pa-\omega ^{2}babb,pb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1408}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-babb,cb,baaa,baab,pa,pb,pc-babb,\,\text{class }4\rangle
\tag{7.1409}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,baab,babb+baaa,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.1410}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,baab,babb+baaa,pa,pb-baaa,pc,\,\text{class }4\rangle
\tag{7.1411}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,baab,babb+baaa,pa,pb-\omega baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1412}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,baab,babb+baaa,pa,pb-\omega ^{2}baaa,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1413}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,baab,babb+baaa,pa-baaa,pb-baaa,pc,\,\text{class }%
4\rangle  \tag{7.1414}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,baab,babb+baaa,pa,pb,pc-baaa,\,\text{class }4\rangle
\tag{7.1415}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,baab,babb+\omega baaa,pa,pb,pc,\,\text{class }%
4\rangle  \tag{7.1416}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,baab,babb+\omega baaa,pa,pb-baaa,pc,\,\text{class }%
4\rangle  \tag{7.1417}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,baab,babb+\omega baaa,pa,pb-\omega baaa,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1418}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,baab,babb+\omega baaa,pa,pb-\omega ^{2}baaa,pc,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.1419}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,baab,babb+\omega baaa,pa,pb,pc-baaa,\,\text{class }%
4\rangle  \tag{7.1420}
\end{equation}

\subsection{Descendants of 6.119}

$(p-1)/2+3+2\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-bab,pb,pc,\,\text{class }4\rangle  \tag{7.1421}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-bab-baaa,pb,pc,\,\text{class }4\rangle 
\tag{7.1422}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-bab-\omega baaa,pb,pc,\,\text{class }4\rangle
\;(p=1\func{mod}4)  \tag{7.1423}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-bab,pb-baaa,pc,\,\text{class }4\rangle 
\tag{7.1424}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-bab,pb-\omega baaa,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.1425}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-bab,pb-\omega ^{2}baaa,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.1426}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-bab,pb,pc-baaa,\,\text{class }4\rangle 
\tag{7.1427}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa-bab,pb,pc-xbaaa,\,\text{class }4\rangle \;(%
\text{all }x,\,x\sim -x)  \tag{7.1428}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa-bab-baaa,pb,pc,\,\text{class }4\rangle 
\tag{7.1429}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa-bab-\omega baaa,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.1430}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa-bab,pb-baaa,pc,\,\text{class }4\rangle 
\tag{7.1431}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa-bab,pb-\omega baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1432}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa-bab,pb-\omega ^{2}baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1433}
\end{equation}

\subsection{Descendants of 6.120}

$(p-1)/2+3+2\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-\omega bab,pb,pc,\,\text{class }4\rangle 
\tag{7.1434}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-\omega bab-baaa,pb,pc,\,\text{class }4\rangle 
\tag{7.1435}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-\omega bab-\omega baaa,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.1436}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-\omega bab,pb-baaa,pc,\,\text{class }4\rangle 
\tag{7.1437}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-\omega bab,pb-\omega baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1438}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-\omega bab,pb-\omega ^{2}baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1439}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-\omega bab,pb,pc-baaa,\,\text{class }4\rangle 
\tag{7.1440}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa-\omega bab,pb,pc-xbaaa,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1441}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa-\omega bab-baaa,pb,pc,\,\text{class }4\rangle
\tag{7.1442}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa-\omega bab-\omega baaa,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.1443}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa-\omega bab,pb-baaa,pc,\,\text{class }4\rangle
\tag{7.1444}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa-\omega bab,pb-\omega baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1445}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa-\omega bab,pb-\omega ^{2}baaa,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1446}
\end{equation}

\subsection{Descendants of 6.121}

$2p+4+2\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-baa,pb,pc,\,\text{class }4\rangle  \tag{7.1447}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-baa-babb,pb,pc,\,\text{class }4\rangle 
\tag{7.1448}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-baa-\omega babb,pb,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.1449}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-baa-\omega ^{2}babb,pb,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.1450}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-baa-xbabb,pb-babb,pc,\,\text{class }4\rangle \;(%
\text{all }x)  \tag{7.1451}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-baa,pb,pc-babb,\,\text{class }4\rangle 
\tag{7.1452}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-babb,cb,pa-baa,pb,pc,\,\text{class }4\rangle 
\tag{7.1453}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-babb,cb,pa-baa-babb,pb,pc,\,\text{class }4\rangle 
\tag{7.1454}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-babb,cb,pa-baa-\omega babb,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1455}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-babb,cb,pa-baa-\omega ^{2}babb,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1456}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-babb,cb,pa-baa-xbabb,pb-babb,pc,\,\text{class }4\rangle
\;(\text{all }x)  \tag{7.1457}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-babb,cb,pa-baa,pb,pc-babb,\,\text{class }4\rangle 
\tag{7.1458}
\end{equation}

\subsection{Descendants of 6.122}

$p+2$ algebras

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-baa,pb+bab,pc,\,\text{class }4\rangle 
\tag{7.1459}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-baa,pb+bab,pc-baab,\,\text{class }4\rangle 
\tag{7.1460}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-baa,pb+bab-baab,pc,\,\text{class }4\rangle 
\tag{7.1461}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-baa-baab,pb+bab-xbaab,pc,\,\text{class }4\rangle
\;(x \neq 0)  \tag{7.1462}
\end{equation}

\subsection{Descendants of 6.125}

$p+1$ algebras

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa+bab,pb+\omega baa,pc-baaa,\,\text{class }4\rangle
\tag{7.1463}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa+bab-xbaaa,pb+\omega baa-ybaaa,pc,\,\text{class }%
4\rangle \;(\text{all }x,\,y,\;(x,y)\sim (x^{\prime },y^{\prime })\text{ if }%
y^{2}-\omega x^{2}=y^{\prime 2}-\omega x^{\prime 2}\func{mod}p)  \tag{7.1464}
\end{equation}

\subsection{Descendants of 6.127}

$3p+4+6\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa,pb,pc-bab,\,\text{class }4\rangle  \tag{7.1465}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-baaa,pb,pc-bab,\,\text{class }4\rangle 
\tag{7.1466}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa,pb-baaa,pc-bab,\,\text{class }4\rangle 
\tag{7.1467}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa,pb-\omega baaa,pc-bab,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.1468}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa,pb-\omega ^{2}baaa,pc-bab,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.1469}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa,pb,pc-bab-baaa,\,\text{class }4\rangle 
\tag{7.1470}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-baaa,pb,pc-bab-baaa,\,\text{class }4\rangle 
\tag{7.1471}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-\omega baaa,pb,pc-bab-baaa,\,\text{class }4\rangle
\tag{7.1472}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-\omega ^{2}baaa,pb,pc-bab-baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.1473}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa-\omega ^{3}baaa,pb,pc-bab-baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.1474}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa,pb-baaa,pc-bab-baaa,\,\text{class }4\rangle 
\tag{7.1475}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa,pb-\omega baaa,pc-bab-baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1476}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb,pa,pb-\omega ^{2}baaa,pc-bab-baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1477}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa,pb,pc-bab,\,\text{class }4\rangle 
\tag{7.1478}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa-baaa,pb,pc-bab,\,\text{class }4\rangle 
\tag{7.1479}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa-\omega baaa,pb,pc-bab,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1480}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa-\omega ^{2}baaa,pb,pc-bab,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1481}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa,pb-baaa,pc-bab,\,\text{class }4\rangle 
\tag{7.1482}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa,pb-\omega baaa,pc-bab,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1483}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa,pb-\omega ^{2}baaa,pc-bab,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1484}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa,pb-xbaaa,pc-bab-baaa,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.1485}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-baaa,pa-xbaaa,pb,pc-bab-baaa,\,\text{class }4\rangle
\;(x \neq 0)  \tag{7.1486}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-\omega baaa,pa,pb,pc-bab,\,\text{class }4\rangle 
\tag{7.1487}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-\omega baaa,pa-baaa,pb,pc-bab,\,\text{class }4\rangle
\tag{7.1488}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-\omega baaa,pa-\omega baaa,pb,pc-bab,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1489}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-\omega baaa,pa-\omega ^{2}baaa,pb,pc-bab,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1490}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-\omega baaa,pa,pb-baaa,pc-bab,\,\text{class }4\rangle
\tag{7.1491}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-\omega baaa,pa,pb-\omega baaa,pc-bab,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1492}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-\omega baaa,pa,pb-\omega ^{2}baaa,pc-bab,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1493}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-\omega baaa,pa,pb-xbaaa,pc-bab-baaa,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1494}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-\omega baaa,pa-xbaaa,pb,pc-bab-baaa,\,\text{class }%
4\rangle \;(x\neq 0)  \tag{7.1495}
\end{equation}

\subsection{Descendants of 6.131}

$15+(p+10)\gcd (p-1,3)+\gcd (p-1,4)+\gcd (p-1,7)$ algebras

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-bab,cb,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.1496}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-bab,cb,pa,pb,pc-babb,\,\text{class }4\rangle 
\tag{7.1497}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-bab,cb,pa,pb-babb,pc,\,\text{class }4\rangle 
\tag{7.1498}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-bab,cb,pa,pb-babb,pc-babb,\,\text{class }%
4\rangle  \tag{7.1499}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-bab,cb,pa-babb,pb,pc,\,\text{class }4\rangle 
\tag{7.1500}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-bab,cb,pa-\omega babb,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1501}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-bab,cb,pa-\omega ^{2}babb,pb,pc,\,\text{class 
}4\rangle \;(p=1\func{mod}3)  \tag{7.1502}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-bab,cb,pa-babb,pb-babb,pc,\,\text{class }%
4\rangle  \tag{7.1503}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-bab,cb,pa-\omega babb,pb-babb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1504}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-bab,cb,pa-\omega ^{2}babb,pb-babb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1505}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb,ca-bab,cb,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.1506}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb,ca-bab,cb,pa-baaa,pb,pc,\,\text{class }4\rangle 
\tag{7.1507}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb,ca-bab,cb,pa,pb-baaa,pc,\,\text{class }4\rangle 
\tag{7.1508}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb,ca-bab,cb,pa,pb-\omega baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1509}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb,ca-bab,cb,pa,pb-\omega ^{2}baaa,pc,\,\text{class 
}4\rangle \;(p=1\func{mod}3)  \tag{7.1510}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb,ca-bab,cb,pa,pb,pc-baaa,\,\text{class }4\rangle 
\tag{7.1511}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb,ca-bab,cb-baaa,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.1512}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb,ca-bab,cb-baaa,pa-baaa,pb,pc,\,\text{class }%
4\rangle  \tag{7.1513}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb,ca-bab,cb-baaa,pa-xbaaa,pb,pc,\,\text{class }%
4\rangle \;(x=\omega ^{i},\;i=1,2,3,4,5,6,\;p=1\func{mod}7)  \tag{7.1514}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb,ca-bab,cb-baaa,pa,pb-baaa,pc,\,\text{class }%
4\rangle  \tag{7.1515}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb,ca-bab,cb-baaa,pa,pb-\omega baaa,pc,\,\text{%
class }4\rangle  \tag{7.1516}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb,ca-bab,cb-baaa,pa,pb-xbaaa,pc,\,\text{class }%
4\rangle \;(x=\omega ^{i},\;i=2,3,4,5,\;p=1\func{mod}3)  \tag{7.1517}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb,ca-bab,cb-baaa,pa,pb,pc-baaa,\,\text{class }%
4\rangle  \tag{7.1518}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb,ca-bab,cb-baaa,pa,pb,pc-\omega baaa,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1519}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb,ca-bab,cb-baaa,pa,pb,pc-\omega ^{2}baaa,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1520}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-baaa,ca-bab,cb,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.1521}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-baaa,ca-bab,cb,pa,pb-baaa,pc,\,\text{class }%
4\rangle  \tag{7.1522}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-baaa,ca-bab,cb,pa,pb-\omega baaa,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1523}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-baaa,ca-bab,cb,pa,pb-\omega ^{2}baaa,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1524}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-baaa,ca-bab,cb,pa-baaa,pb-xbaaa,pc,\,\text{class 
}4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1525}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-baaa,ca-bab,cb,pa-\omega baaa,pb-xbaaa,pc,\,%
\text{class }4\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}3) 
\tag{7.1526}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-baaa,ca-bab,cb,pa-\omega ^{2}baaa,pb-xbaaa,pc,\,%
\text{class }4\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}3) 
\tag{7.1527}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-baaa,ca-bab,cb,pa,pb,pc-baaa,\,\text{class }%
4\rangle  \tag{7.1528}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-baaa,ca-bab,cb,pa,pb,pc-\omega baaa,\,\text{%
class }4\rangle \;(p=1\func{mod}4)  \tag{7.1529}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-\omega baaa,ca-bab,cb,pa,pb,pc,\,\text{class }%
4\rangle  \tag{7.1530}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-\omega baaa,ca-bab,cb,pa,pb-baaa,pc,\,\text{%
class }4\rangle  \tag{7.1531}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-\omega baaa,ca-bab,cb,pa,pb-\omega baaa,pc,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.1532}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-\omega baaa,ca-bab,cb,pa,pb-\omega ^{2}baaa,pc,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.1533}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-\omega baaa,ca-bab,cb,pa-baaa,pb-xbaaa,pc,\,%
\text{class }4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1534}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-\omega baaa,ca-bab,cb,pa-\omega
baaa,pb-xbaaa,pc,\,\text{class }4\rangle \;(\text{all }x,\,x\sim -x,\;p=1%
\func{mod}3)  \tag{7.1535}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-\omega baaa,ca-bab,cb,pa-\omega
^{2}baaa,pb-xbaaa,pc,\,\text{class }4\rangle \;(\text{all }x,\,x\sim -x,\;p=1%
\func{mod}3)  \tag{7.1536}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-\omega baaa,ca-bab,cb,pa,pb,pc-baaa,\,\text{%
class }4\rangle  \tag{7.1537}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab,babb-\omega baaa,ca-bab,cb,pa,pb,pc-\omega baaa,\,%
\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.1538}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,babb,ca-bab,cb,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.1539}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,babb,ca-bab,cb,pa,pb-baab,pc,\,\text{class }4\rangle 
\tag{7.1540}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,babb,ca-bab,cb,pa-baab,pb,pc,\,\text{class }4\rangle 
\tag{7.1541}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,babb,ca-bab,cb,pa-baab,pb-baab,pc,\,\text{class }%
4\rangle  \tag{7.1542}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,babb,ca-bab,cb,pa-baab,pb-\omega baab,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1543}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,babb,ca-bab,cb,pa-baab,pb-\omega ^{2}baab,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1544}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,babb,ca-bab,cb,pa,pb,pc-baab,\,\text{class }4\rangle 
\tag{7.1545}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,babb,ca-bab,cb,pa,pb,pc-\omega baab,\,\text{class }%
4\rangle  \tag{7.1546}
\end{equation}

\subsection{Descendants of 6.132}

$(p+1+3(p+1)\gcd (p-1,3)+\gcd (p-1,4))/2$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-bab,pb,pc,\,\text{class }4\rangle 
\tag{7.1547}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-bab-baaa,pb,pc,\,\text{class }4\rangle 
\tag{7.1548}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-bab-\omega baaa,pb,pc,\,\text{class }4\rangle
\;(p=1\func{mod}4)  \tag{7.1549}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-bab,pb-baaa,pc,\,\text{class }4\rangle 
\tag{7.1550}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-bab,pb-\omega baaa,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.1551}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-bab,pb-\omega ^{2}baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1552}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-bab,pb,pc-baaa,\,\text{class }4\rangle 
\tag{7.1553}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-bab,pb,pc-\omega baaa,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.1554}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-bab,pb,pc-\omega ^{2}baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1555}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-baaa,pa-bab,pb,pc-xbaaa,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.1556}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baaa,pa-bab,pb,pc-xbaaa,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}3)  \tag{7.1557}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega ^{2}baaa,pa-bab,pb,pc-xbaaa,\,\text{class 
}4\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}3)  \tag{7.1558}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-baaa,pa-bab,pb-xbaaa,pc,\,\text{class }4\rangle
\;(x \neq 0)  \tag{7.1559}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baaa,pa-bab,pb-xbaaa,pc,\,\text{class }%
4\rangle \;(x\neq 0,\;p=1\func{mod}3)  \tag{7.1560}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega ^{2}baaa,pa-bab,pb-xbaaa,pc,\,\text{class 
}4\rangle \;(x\neq 0,\;p=1\func{mod}3)  \tag{7.1561}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-baaa,pa-bab-xbaaa,pb,pc,\,\text{class }4\rangle
\;(x \neq 0, x \sim x^{\prime}\text{ if }x^6=x^{\prime 6 }\func{mod} p) 
\tag{7.1562}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baaa,pa-bab-xbaaa,pb,pc,\,\text{class }%
4\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^6=x^{\prime 6}\func{mod}%
p,\;p=1\func{mod}3)  \tag{7.1563}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega ^{2}baaa,pa-bab-xbaaa,pb,pc,\,\text{class 
}4\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^6=x^{\prime 6}\func{mod}%
p,\;p=1\func{mod}3)  \tag{7.1564}
\end{equation}

\subsection{Descendants of 6.133}

$(p+1+3(p+1)\gcd (p-1,3)+\gcd (p-1,4))/2$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega bab,pb,pc,\,\text{class }4\rangle 
\tag{7.1565}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega bab-baaa,pb,pc,\,\text{class }4\rangle
\tag{7.1566}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega bab-\omega baaa,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.1567}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega bab,pb-baaa,pc,\,\text{class }4\rangle
\tag{7.1568}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega bab,pb-\omega baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1569}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega bab,pb-\omega ^{2}baaa,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1570}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega bab,pb,pc-baaa,\,\text{class }4\rangle
\tag{7.1571}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega bab,pb,pc-\omega baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1572}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega bab,pb,pc-\omega ^{2}baaa,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1573}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-baaa,pa-\omega bab,pb,pc-xbaaa,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1574}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baaa,pa-\omega bab,pb,pc-xbaaa,\,\text{%
class }4\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}3)  \tag{7.1575}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega ^{2}baaa,pa-\omega bab,pb,pc-xbaaa,\,%
\text{class }4\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}3) 
\tag{7.1576}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-baaa,pa-\omega bab,pb-xbaaa,pc,\,\text{class }%
4\rangle \;(x\neq 0)  \tag{7.1577}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baaa,pa-\omega bab,pb-xbaaa,pc,\,\text{%
class }4\rangle \;(x\neq 0,\;p=1\func{mod}3)  \tag{7.1578}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega ^{2}baaa,pa-\omega bab,pb-xbaaa,pc,\,%
\text{class }4\rangle \;(x\neq 0,\;p=1\func{mod}3)  \tag{7.1579}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-baaa,pa-\omega bab-xbaaa,pb,pc,\,\text{class }%
4\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{6}=x^{\prime 6}\func{mod%
}p)  \tag{7.1580}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baaa,pa-\omega bab-xbaaa,pb,pc,\,\text{%
class }4\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{6}=x^{\prime 6}%
\func{mod}p,\;p=1\func{mod}3)  \tag{7.1581}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega ^{2}baaa,pa-\omega bab-xbaaa,pb,pc,\,%
\text{class }4\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^6=x^{\prime
6}\func{mod}p,\;p=1\func{mod}3)  \tag{7.1582}
\end{equation}

\subsection{Descendants of 6.134}

$3p-1+\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-baa,pb,pc-xbabb,\,\text{class }4\rangle \;(%
\text{all }x)  \tag{7.1583}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-baa,pb-babb,pc-xbabb,\,\text{class }4\rangle
\;(\text{all }x)  \tag{7.1584}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-baa-babb,pb,pc-babb,\,\text{class }4\rangle 
\tag{7.1585}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-baa-\omega babb,pb,pc-babb,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1586}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-baa-\omega ^{2}babb,pb,pc-babb,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1587}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-baa-xbabb,pb-babb,pc-babb,\,\text{class }%
4\rangle \;(x \neq 0)  \tag{7.1588}
\end{equation}

\subsection{Descendants of 6.135}

$p^{2}+2p+3+\gcd (p-1,3)+\gcd (p-1,4)+\gcd (p-1,5)$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-bab,pc,\,\text{class }4\rangle 
\tag{7.1589}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-bab-baaa,pc,\,\text{class }4\rangle 
\tag{7.1590}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-bab-\omega baaa,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.1591}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-bab-\omega ^{2}baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1592}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-baaa,pb-bab-xbaaa,pc,\,\text{class }4\rangle
\;(\text{all }x)  \tag{7.1593}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-baaa,pa,pb-bab,pc,\,\text{class }4\rangle 
\tag{7.1594}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-xbaaa,pa,pb-bab,pc,\,\text{class }4\rangle
\;(x=\omega ^{i},\;i=1,2,3,4,\;p=1\func{mod}5)  \tag{7.1595}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-baaa,pa,pb-bab-xbaaa,pc,\,\text{class }4\rangle
\;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod}p) 
\tag{7.1596}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-zbaaa,pa,pb-bab-xbaaa,pc,\,\text{class }4\rangle
\;(z=\omega ^{i},\;i=1,2,3,4,\;x\neq 0,\;x\sim x^{\prime }\text{ if }%
x^{5}=x^{\prime 5}\func{mod}p,\;p=1\func{mod}5)  \tag{7.1597}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-baaa,pa-xbaaa,pb-bab-ybaaa,pc,\,\text{class }%
4\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod%
}p,\text{ all }y)  \tag{7.1598}
\end{equation}

In the presentation below we have $z=\omega ^{i},\;i=1,2,3,4,\;x\neq
0,\;x\sim x^{\prime }$ if $x^{5}=x^{\prime 5}\func{mod}p,$ all $y,\;p=1\func{%
mod}5$:

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-zbaaa,pa-xbaaa,pb-bab-ybaaa,pc,\,\text{class }%
4\rangle  \tag{7.1599}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-xbaaa,pa,pb-bab,pc-baaa,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}\func{mod}p) 
\tag{7.1600}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-xbaaa,pa,pb-bab,pc-\omega baaa,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}\func{%
mod}p)  \tag{7.1601}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-xbaaa,pa,pb-bab,pc-\omega ^{2}baaa,\,\text{class 
}4\rangle \;(\text{all }x,\,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}\func{%
mod}p,\;p=1\func{mod}4)  \tag{7.1602}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-xbaaa,pa,pb-bab,pc-\omega ^{3}baaa,\,\text{class 
}4\rangle \;(\text{all }x,\,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}\func{%
mod}p,\;p=1\func{mod}4)  \tag{7.1603}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa+baa,pb-bab,pc,\,\text{class }4\rangle 
\tag{7.1604}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa+baa,pb-bab-baab,pc,\,\text{class }4\rangle 
\tag{7.1605}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa+baa,pb-bab,pc-baab,\,\text{class }4\rangle 
\tag{7.1606}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa+baa,pb-bab,pc-\omega baab,\,\text{class }%
4\rangle  \tag{7.1607}
\end{equation}

\subsection{Descendants of 6.138}

$p(p+1)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-baa-ybabb,pb-baa,pc-xbabb,\,\text{class }%
4\rangle \;(\text{all }x,\,y,\;y\sim -y)  \tag{7.1608}
\end{equation}

\subsection{Descendants of 6.139}

$(p+3)/2+\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-baa,pc,\,\text{class }4\rangle 
\tag{7.1609}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-babb,pb-baa,pc,\,\text{class }4\rangle 
\tag{7.1610}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega babb,pb-baa,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.1611}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega ^{2}babb,pb-baa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1612}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-xbabb,pb-baa,pc-babb,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.1613}
\end{equation}

\subsection{Descendants of 6.140}

$(3p+1)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-bab,pb-baa,pc-xbaaa,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.1614}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-bab,pb-baa-xbaaa,pc-ybaaa,\,\text{class }%
4\rangle \;(\text{If 2 is a square }\func{mod}p\text{ and }y^{2}=2,\;x\neq
0,\;x\sim -x,\,y\sim -y)  \tag{7.1615}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega bab,pb-baa,pc-xbaaa,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1616}
\end{equation}%
The following presentation only arises if $2$ is not a square modulo $p$:

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega bab,pb-baa-xbaaa,pc-ybaaa,\,\text{%
class }4\rangle \;(x\neq 0,\;x\sim -x,\,\omega y^{2}=2,\,y\sim -y) 
\tag{7.1617}
\end{equation}

\subsection{Descendants of 6.142}

$p(p+1/2$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-baa-ybabb,pb-\omega baa,pc-xbabb,\,\text{%
class }4\rangle \;(\text{all }x,\,y,\;y\sim -y)  \tag{7.1618}
\end{equation}

\subsection{Descendants of 6.143}

$(p+3)/2+\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-\omega baa,pc,\,\text{class }4\rangle 
\tag{7.1619}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-babb,pb-\omega baa,pc,\,\text{class }4\rangle
\tag{7.1620}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega babb,pb-\omega baa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1621}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega ^{2}babb,pb-\omega baa,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1622}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-xbabb,pb-\omega baa,pc-babb,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1623}
\end{equation}

\subsection{Descendants of 6.144}

$(3p+1)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-bab,pb-\omega baa,pc-xbaaa,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1624}
\end{equation}

The following presentation only arises if $2$ is not a square modulo $p$:

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-bab,pb-\omega baa-xbaaa,pc-ybaaa,\,\text{%
class }4\rangle \;(x\neq 0,\;x\sim -x,\,y^{2}=2\omega ^{3},\;y\sim -y) 
\tag{7.1625}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega bab,pb-\omega baa,pc-xbaaa,\,\text{%
class }4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1626}
\end{equation}

The following presentation only arises if $2$ is a square modulo $p$:

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega bab,pb-\omega baa-xbaaa,pc-ybaaa,\,%
\text{class }4\rangle \;(x\neq 0,\;x\sim -x,\,y^{2}=2\omega ^{2},\;y\sim -y)
\tag{7.1627}
\end{equation}

\subsection{Descendants of 6.146}

$2p^{2}+3p$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-xbaaa,pc-bab,\,\text{class }4\rangle \;(%
\text{all }x)  \tag{7.1628}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-xbaaa,pc-bab-baaa,\,\text{class }4\rangle
\;(\text{all }x)  \tag{7.1629}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-baaa,pb,pc-bab,\,\text{class }4\rangle 
\tag{7.1630}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-xbaaa,pb,pc-bab-baaa,\,\text{class }4\rangle
\;(x \neq 0)  \tag{7.1631}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-baaa,pa,pb-xbaaa,pc-bab-ybaaa,\,\text{class }%
4\rangle \;(\text{all }x,\,y,\;y\sim -y)  \tag{7.1632}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-baaa,pa-zbaaa,pb,pc-bab,\,\text{class }4\rangle
\;(z\neq 0,\;z\sim -z)  \tag{7.1633}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-baaa,pa-zbaaa,pb,pc-bab-ybaaa,\,\text{class }%
4\rangle \;(y,z\neq 0,\;y\sim -y)  \tag{7.1634}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baaa,pa,pb-xbaaa,pc-bab-ybaaa,\,\text{%
class }4\rangle \;(\text{all }x,\,y,\;y\sim -y)  \tag{7.1635}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baaa,pa-zbaaa,pb,pc-bab,\,\text{class }%
4\rangle \;(z\neq 0,\;z\sim -z)  \tag{7.1636}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baaa,pa-zbaaa,pb,pc-bab-ybaaa,\,\text{%
class }4\rangle \;(y,z\neq 0,\;y\sim -y)  \tag{7.1637}
\end{equation}

\subsection{Descendants of 6.148}

$p^{3}+p^{2}+p+(p+2)\gcd (p-1,3)+\gcd (p-1,5)$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb,pc-baa,\,\text{class }4\rangle 
\tag{7.1638}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb,pc-baa-babb,\,\text{class }4\rangle 
\tag{7.1639}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb,pc-baa-\omega babb,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.1640}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb,pc-baa-\omega ^{2}babb,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1641}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-babb,pc-baa,\,\text{class }4\rangle 
\tag{7.1642}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-\omega babb,pc-baa,\,\text{class }4\rangle
\;(p=1\func{mod}5)  \tag{7.1643}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-\omega ^{2}babb,pc-baa,\,\text{class }%
4\rangle \;(p=1\func{mod}5)  \tag{7.1644}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-\omega ^{3}babb,pc-baa,\,\text{class }%
4\rangle \;(p=1\func{mod}5)  \tag{7.1645}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-\omega ^{4}babb,pc-baa,\,\text{class }%
4\rangle \;(p=1\func{mod}5)  \tag{7.1646}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-babb,pc-baa-xbabb,\,\text{class }4\rangle
\;(x \neq 0, x \sim x^{\prime}\text{ if }x^5=x^{\prime 5 }\func{mod} p) 
\tag{7.1647}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-\omega babb,pc-baa-xbabb,\,\text{class }%
4\rangle \;(x\neq 0,x \sim x^{\prime}\text{ if }x^5=x^{\prime 5}\func{mod}%
p,\;p=1\func{mod}5)  \tag{7.1648}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-\omega ^{2}babb,pc-baa-xbabb,\,\text{class 
}4\rangle \;(x\neq 0,x \sim x^{\prime}\text{ if }x^5=x^{\prime 5}\func{mod}%
p,\;p=1\func{mod}5)  \tag{7.1649}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-\omega ^{3}babb,pc-baa-xbabb,\,\text{class 
}4\rangle \;(x\neq 0,x \sim x^{\prime}\text{ if }x^5=x^{\prime 5}\func{mod}%
p,\;p=1\func{mod}5)  \tag{7.1650}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa,pb-\omega ^{4}babb,pc-baa-xbabb,\,\text{class 
}4\rangle \;(x\neq 0,x \sim x^{\prime}\text{ if }x^5=x^{\prime 5}\func{mod}%
p,\;p=1\func{mod}5)  \tag{7.1651}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-babb,pb,pc-baa-xbabb,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.1652}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega babb,pb,pc-baa-xbabb,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1653}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega ^{2}babb,pb,pc-baa-xbabb,\,\text{class 
}4\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}3)  \tag{7.1654}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega ^{3}babb,pb,pc-baa-xbabb,\,\text{class 
}4\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}3)  \tag{7.1655}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega ^{4}babb,pb,pc-baa-xbabb,\,\text{class 
}4\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}3)  \tag{7.1656}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega ^{5}babb,pb,pc-baa-xbabb,\,\text{class 
}4\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}3)  \tag{7.1657}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-babb,pb-xbabb,pc-baa-ybabb,\,\text{class }%
4\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{6}=x^{\prime 6}\func{mod%
}p,\text{ all }y)  \tag{7.1658}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega babb,pb-xbabb,pc-baa-ybabb,\,\text{%
class }4\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{6}=x^{\prime 6}%
\func{mod}p,\text{ all }y)  \tag{7.1659}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega ^{2}babb,pb-xbabb,pc-baa-ybabb,\,\text{%
class }4\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{6}=x^{\prime 6}%
\func{mod}p,\text{ all }y,\;p=1\func{mod}3)  \tag{7.1660}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega ^{3}babb,pb-xbabb,pc-baa-ybabb,\,\text{%
class }4\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{6}=x^{\prime 6}%
\func{mod}p,\text{ all }y,\;p=1\func{mod}3)  \tag{7.1661}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega ^{4}babb,pb-xbabb,pc-baa-ybabb,\,\text{%
class }4\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{6}=x^{\prime 6}%
\func{mod}p,\text{ all }y,\;p=1\func{mod}3)  \tag{7.1662}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb,pa-\omega ^{5}babb,pb-xbabb,pc-baa-ybabb,\,\text{%
class }4\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{6}=x^{\prime 6}%
\func{mod}p,\text{ all }y,\;p=1\func{mod}3)  \tag{7.1663}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab-babb,cb,pa-xbabb,pb,pc-baa-ybabb,\,\text{class }%
4\rangle \;(\text{all }x,\,y,\;y\sim -y)  \tag{7.1664}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab-babb,cb,pa-xbabb,pb-ybabb,pc-baa-zbabb,\,\text{%
class }4\rangle \;(\text{all }x,z,\;y\neq 0,\;y\sim -y)  \tag{7.1665}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab-\omega babb,cb,pa-xbabb,pb,pc-baa-ybabb,\,\text{%
class }4\rangle \;(\text{all }x,\,y,\;y\sim -y)  \tag{7.1666}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab-\omega babb,cb,pa-xbabb,pb-ybabb,pc-baa-zbabb,\,%
\text{class }4\rangle \;(\text{all }x,z,\;y\neq 0,\;y\sim -y)  \tag{7.1667}
\end{equation}

\subsection{Descendants of 6.150}

$\bigskip $Algebra 6.150 has $4p+14+(p^{2}/2+2p+13/2)\gcd (p-1,3)+\gcd
(p-1,4)$ descendants of order $p^{7}$. It has presentation 
\[
\langle a,b,c\,|\,ca-baa,\,cb,\,pa,\,pb,\,pc,\,\text{class }3\rangle 
\]%
and so if $L$ is a descendant of 6.150 of order $p^{7}$ then the commutator
structure of $L$ is determined by the classification of nilpotent Lie
algebras over $\mathbb{Z}_{p}$. So we can assume that one of the following
sets of commutator relations holds. 
\begin{eqnarray*}
baaa &=&baab=ca-baa=cb=0, \\
baaa &=&baab=ca-baa-babb=cb=0, \\
baaa &=&baab=babb,\,ca-baa=cb=0, \\
babb &=&-baaa,\,baab=ca-baa=cb=0, \\
babb &=&-\omega baaa,\,baab=ca-baa=cb=0, \\
baaa &=&babb=ca-baa=cb=0, \\
babb &=&baab,\,baaa=ca-baa=cb=0, \\
baaa &=&baab,\,babb=xbaaa,\,ca-baa=cb=0\;(x\neq 0,1).
\end{eqnarray*}%
For each of these cases we obtain a generator $d$ for $L_{4}$ ($d$ equals
one of $baaa$, $baab$, $babb$) and we write 
\[
\left( 
\begin{array}{l}
pa \\ 
pb \\ 
pc%
\end{array}%
\right) =Ad 
\]%
where $A$ is a $3\times 1$ matrix over $\mathbb{Z}_{p}$. In each case the
isomorphism classes of algebras are given by the orbits of the matrices $A$
under a given action by a group of automorphisms. I was able to
\textquotedblleft solve\textquotedblright\ the problem in every case,
providing presentations with fewer parameters, and explicit relatively
simple equivalence relations on the parameter sets. However in four of the
cases the equivalence classes are slightly more complex than usual. For
example in one case the equivalence classes for a parameter $y$ were $\{\pm
y,\pm \frac{\omega }{y}\}$. These four cases are 3,4,5,8, and these are
described below.

\subsubsection{Case 3}

If $baaa=baab=babb,\,ca-baa=cb=0$ then $L_{4}$ is generated by $baaa$ and
the action on $A$ is 
\[
A\rightarrow \alpha ^{-4}\left( 
\begin{array}{lll}
\alpha & 0 & \gamma \\ 
0 & \alpha & -\gamma \\ 
0 & 0 & \alpha ^{2}%
\end{array}%
\right) A 
\]

and 
\[
A\rightarrow -\alpha ^{-4}\left( 
\begin{array}{lll}
0 & \alpha & \gamma \\ 
\alpha & 0 & -\gamma \\ 
0 & 0 & \alpha ^{2}%
\end{array}%
\right) A. 
\]

\subsubsection{Case 4}

If $babb+baaa=baab=ca-baa=cb=0$ then $L_{4}$ is generated by $baaa$ and the
action on $A$ is 
\[
A\rightarrow \alpha ^{-4}\left( 
\begin{array}{lll}
\pm \alpha & 0 & \gamma \\ 
0 & \alpha & \varepsilon \\ 
0 & 0 & \alpha ^{2}%
\end{array}%
\right) A 
\]

and 
\[
A\rightarrow \alpha ^{-4}\left( 
\begin{array}{lll}
0 & \pm \alpha & \gamma \\ 
\alpha & 0 & \varepsilon \\ 
0 & 0 & \alpha ^{2}%
\end{array}%
\right) A. 
\]

\subsubsection{Case 5}

If $babb+\omega baaa=baab=ca-baa=cb=0$ then $L_{4}$ is generated by $baaa$
and the action on $A$ is 
\[
A\rightarrow \alpha ^{-4}\left( 
\begin{array}{lll}
\pm \alpha & 0 & \gamma \\ 
0 & \alpha & \varepsilon \\ 
0 & 0 & \alpha ^{2}%
\end{array}%
\right) A 
\]

and 
\[
A\rightarrow \omega ^{-2}\alpha ^{-4}\left( 
\begin{array}{lll}
0 & \pm \alpha & \gamma \\ 
\omega \alpha & 0 & \varepsilon \\ 
0 & 0 & \omega \alpha ^{2}%
\end{array}%
\right) A. 
\]

\subsubsection{Case 8}

If $baaa=baab,\,babb=xbaaa,\,ca-baa=cb=0$ where $x\neq 0,1$ then $L_{4}$ is
generated by $baaa$. (If we set $x=0$ we have Case 7, and if we set $x=1$ we
have Case 3.) The action on $A$ is 
\[
A\rightarrow \alpha ^{-4}\left( 
\begin{array}{lll}
\alpha & 0 & \gamma \\ 
0 & \alpha & \varepsilon \\ 
0 & 0 & \alpha ^{2}%
\end{array}%
\right) A 
\]%
and 
\[
A\rightarrow x^{-2}\alpha ^{-4}\left( 
\begin{array}{lll}
0 & \alpha & \gamma \\ 
x\alpha & 0 & \varepsilon \\ 
0 & 0 & -x\alpha ^{2}%
\end{array}%
\right) A. 
\]

\subsubsection{The presentations}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa,cb,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.1668}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa,cb,pa-babb,pb,pc,\,\text{class }4\rangle 
\tag{7.1669}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa,cb,pa-\omega babb,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1670}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa,cb,pa-\omega ^{2}babb,pb,pc,\,\text{class 
}4\rangle \;(p=1\func{mod}3)  \tag{7.1671}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa,cb,pa,pb-babb,pc,\,\text{class }4\rangle 
\tag{7.1672}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa,cb,pa-babb,pb-babb,pc,\,\text{class }%
4\rangle  \tag{7.1673}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa,cb,pa-\omega babb,pb-babb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1674}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa,cb,pa-\omega ^{2}babb,pb-babb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1675}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa,cb,pa,pb,pc-babb,\,\text{class }4\rangle 
\tag{7.1676}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa,cb,pa,pb,pc-\omega babb,\,\text{class }%
4\rangle  \tag{7.1677}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa-babb,cb,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.1678}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa-babb,cb,pa,pb-babb,pc,\,\text{class }%
4\rangle  \tag{7.1679}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa-babb,cb,pa,pb-\omega babb,pc,\,\text{%
class }4\rangle  \tag{7.1680}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa-babb,cb,pa,pb-\omega ^{2}babb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}4)  \tag{7.1681}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa-babb,cb,pa,pb-\omega ^{3}babb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}4)  \tag{7.1682}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa-babb,cb,pa-babb,pb,pc,\,\text{class }%
4\rangle  \tag{7.1683}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa-babb,cb,pa-\omega babb,pb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1684}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa-babb,cb,pa-\omega ^{2}babb,pb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1685}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa-babb,cb,pa-babb,pb-xbabb,pc,\,\text{class 
}4\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}\func{%
mod}p)  \tag{7.1686}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa-babb,cb,pa-\omega babb,pb-xbabb,pc,\,%
\text{class }4\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime
3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.1687}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa-babb,cb,pa-\omega ^{2}babb,pb-xbabb,pc,\,%
\text{class }4\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }%
x^{3}=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.1688}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa-babb,cb,pa,pb,pc-babb,\,\text{class }%
4\rangle  \tag{7.1689}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,baab,ca-baa-babb,cb,pa,pb,pc-\omega babb,\,\text{%
class }4\rangle  \tag{7.1690}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baab-baaa,babb-baaa,ca-baa,cb,pa-xbaaa,pb-ybaaa,pc-zbaaa,\,%
\text{class }4\rangle \;(\text{Case 3 above})  \tag{7.1691}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,babb+baaa,baab,ca-baa,cb,pa-xbaaa,pb-ybaaa,pc-zbaaa,\,%
\text{class }4\rangle \;(\text{Case 4 above})  \tag{7.1692}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,babb+\omega baaa,baab,ca-baa,cb,pa-xbaaa,pb-ybaaa,pc,\,%
\text{class }4\rangle \;(\text{Case 5 above})  \tag{7.1693}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,babb,ca-baa,cb,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.1694}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,babb,ca-baa,cb,pa,pb-baab,pc,\,\text{class }4\rangle 
\tag{7.1695}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,babb,ca-baa,cb,pa-baab,pb-baab,pc,\,\text{class }%
4\rangle  \tag{7.1696}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,babb,ca-baa,cb,pa-baab,pb-\omega baab,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1697}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,babb,ca-baa,cb,pa-baab,pb-\omega ^{2}baab,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1698}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaa,babb,ca-baa,cb,pa,pb,pc-baab,\,\text{class }4\rangle 
\tag{7.1699}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,babb-baab,baaa,ca-baa,cb,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.1700}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,babb-baab,baaa,ca-baa,cb,pa,pb-baab,pc,\,\text{class }%
4\rangle  \tag{7.1701}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,babb-baab,baaa,ca-baa,cb,pa,pb-\omega baab,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1702}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,babb-baab,baaa,ca-baa,cb,pa,pb-\omega ^{2}baab,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1703}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,babb-baab,baaa,ca-baa,cb,pa-baab,pb-xbaab,pc,\,\text{class 
}4\rangle \;(\text{all }x)  \tag{7.1704}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,babb-baab,baaa,ca-baa,cb,pa-\omega baab,pb-xbaab,pc,\,%
\text{class }4\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.1705}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,babb-baab,baaa,ca-baa,cb,pa-\omega ^{2}baab,pb-xbaab,pc,\,%
\text{class }4\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.1706}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,babb-baab,baaa,ca-baa,cb,pa,pb,pc-baab,\,\text{class }%
4\rangle  \tag{7.1707}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,babb-baab,baaa,ca-baa,cb,pa,pb,pc-\omega baab,\,\text{%
class }4\rangle  \tag{7.1708}
\end{equation}

\begin{equation}
\langle
a,b,c\,|\,baab-baaa,babb-xbaaa,ca-baa,cb,pa-ybaaa,pb-zbaaa,pc-tbaaa,\,\text{%
class }4\rangle \;(\text{Case 8 above})  \tag{7.1709}
\end{equation}

\subsection{Descendants of 6.151}

$p^{2}+p+2+(p+1)\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-baa,pb,pc,\,\text{class }4\rangle 
\tag{7.1710}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-baa-babb,pb,pc,\,\text{class }4\rangle 
\tag{7.1711}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-baa-\omega babb,pb,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.1712}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-baa-\omega ^{2}babb,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1713}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa-babb,cb,pa-baa-xbabb,pb,pc,\,\text{class }4\rangle
\;(\text{all }x)  \tag{7.1714}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa-\omega babb,cb,pa-baa-xbabb,pb,pc,\,\text{class }%
4\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.1715}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa-\omega ^{2}babb,cb,pa-baa-xbabb,pb,pc,\,\text{class 
}4\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.1716}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa-xbabb,cb,pa-baa-ybabb,pb-babb,pc,\,\text{class }%
4\rangle \;(\text{all }x,\,y)  \tag{7.1717}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa-xbabb,cb,pa-baa,pb,pc-babb,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.1718}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa-xbabb,cb,pa-baa,pb,pc-\omega babb,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1719}
\end{equation}

\subsection{Descendants of 6.152}

$(p^{2}+2p+3+2\gcd (p-1,3)+(p+1)\gcd (p-1,4))/2$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-bab,pb,pc,\,\text{class }4\rangle 
\tag{7.1720}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-bab,pb-baaa,pc,\,\text{class }4\rangle 
\tag{7.1721}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-bab,pb-\omega baaa,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.1722}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-bab,pb-\omega ^{2}baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1723}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-bab-baaa,pb-xbaaa,pc,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}\func{mod}p) 
\tag{7.1724}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-bab-\omega baaa,pb-xbaaa,pc,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}\func{%
mod}p,\;p=1\func{mod}4)  \tag{7.1725}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb-baaa,pa-bab-xbaaa,pb-ybaaa,pc,\,\text{class }%
4\rangle \;(\text{all }x,\,y,\;y\sim y^{\prime }\text{ if }y^{4}=y^{\prime 4}%
\func{mod}p)  \tag{7.1726}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb-\omega baaa,pa-bab-xbaaa,pb-ybaaa,pc,\,\text{%
class }4\rangle \;(\text{all }x,\,y,\;y\sim y^{\prime }\text{ if }%
y^{4}=y^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.1727}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb-xbaaa,pa-bab,pb,pc-baaa,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.1728}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb-xbaaa,pa-bab,pb,pc-\omega baaa,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1729}
\end{equation}

\subsection{Descendants of 6.153}

$p(p+1)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-baa-bab-ybaaa,pb,pc-xbaaa,\,\text{class }%
4\rangle \;(x\neq 0,\text{ all }y,\;y\sim -y)  \tag{7.1730}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-baa-bab,pb-ybaaa,pc,\,\text{class }4\rangle
\;(\text{all }y,\;y\sim -y)  \tag{7.1731}
\end{equation}

\subsection{Descendants of 6.154}

$(p^{2}+2p+3+2\gcd (p-1,3)+(p+1)\gcd (p-1,4))/2$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-\omega bab,pb,pc,\,\text{class }4\rangle 
\tag{7.1732}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-\omega bab,pb-baaa,pc,\,\text{class }4\rangle
\tag{7.1733}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-\omega bab,pb-\omega baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1734}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-\omega bab,pb-\omega ^{2}baaa,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1735}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-\omega bab-baaa,pb-xbaaa,pc,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}\func{%
mod}p)  \tag{7.1736}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-\omega bab-\omega baaa,pb-xbaaa,pc,\,\text{%
class }4\rangle \;(\text{all }x,\,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}%
\func{mod}p,\;p=1\func{mod}4)  \tag{7.1737}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb-baaa,pa-\omega bab-xbaaa,pb-ybaaa,pc,\,\text{%
class }4\rangle \;(\text{all }x,\,y,\;y\sim y^{\prime }\text{ if }%
y^{4}=y^{\prime 4}\func{mod}p)  \tag{7.1738}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb-\omega baaa,pa-\omega bab-xbaaa,pb-ybaaa,pc,\,%
\text{class }4\rangle \;(\text{all }x,\,y,\;y\sim y^{\prime }\text{ if }%
y^{4}=y^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.1739}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb-xbaaa,pa-\omega bab,pb,pc-baaa,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1740}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb-xbaaa,pa-\omega bab,pb,pc-\omega baaa,\,\text{%
class }4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1741}
\end{equation}

\subsection{Descendants of 6.155}

$p(p+1)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-baa-\omega bab-ybaaa,pb,pc-xbaaa,\,\text{%
class }4\rangle \;(x\neq 0,\text{ all }y,\;y\sim -y)  \tag{7.1742}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-baa-\omega bab,pb-ybaaa,pc,\,\text{class }%
4\rangle \;(\text{all }y,\;y\sim -y)  \tag{7.1743}
\end{equation}

\subsection{Descendants of 6.156}

$p$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-bab,pb+baa,pc-xbaaa,\,\text{class }4\rangle
\;(x\neq 0,\;x\sim -x)  \tag{7.1744}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-bab-xbaaa,pb+baa,pc,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.1745}
\end{equation}

\subsection{Descendants of 6.157}

$2p-1$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-bab,pb+\omega baa,pc-xbaaa,\,\text{class }%
4\rangle \;(\text{all }x)  \tag{7.1746}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-bab-xbaaa,pb+\omega baa,pc,\,\text{class }%
4\rangle \;(x\neq 0,\;x\sim -x)  \tag{7.1747}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-bab,pb+\omega baa-xbaaa,pc+\omega baaa,\,%
\text{class }4\rangle \;(x\neq 0,\;x\sim -x)  \tag{7.1748}
\end{equation}

\subsection{Descendants of 6.158}

$p$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-\omega bab,pb+\omega baa,pc-xbaaa,\,\text{%
class }4\rangle \;(x\neq 0,-\omega ,\;x\sim -\omega -x)  \tag{7.1749}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-\omega bab-xbaaa,pb+\omega baa,pc,\,\text{%
class }4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1750}
\end{equation}

\subsection{Descendants of 6.159}

$(p^{3}+p^{2})/2$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-baa-xbabb,cb,pa-baa,pb-baa,pc-ybabb,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x,\,y\neq 0)  \tag{7.1751}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa-xbabb,cb,pa-baa-ybabb,pb-baa-zbabb,pc,\,\text{class 
}4\rangle \;(\text{all }x,\,y,z,\;(x,y,z)\sim (-x,-y,-z))  \tag{7.1752}
\end{equation}

\subsection{Descendants of 6.160}

$(p^{3}+p^{2})/2$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-baa-xbabb,cb,pa-baa,pb-\omega baa,pc-ybabb,\,\text{%
class }4\rangle \;(\text{all }x,\,x\sim -x,\,y\neq 0)  \tag{7.1753}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa-xbabb,cb,pa-baa-ybabb,pb-\omega baa-zbabb,pc,\,%
\text{class }4\rangle \;(\text{all }x,\,y,z,\;(x,y,z)\sim (-x,-y,-z)) 
\tag{7.1754}
\end{equation}

\subsection{Descendants of 6.160a}

$(p+3)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-baa,pb+bab,pc-xbaab,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.1755}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-baa-baab,pb+bab,pc-baab,\,\text{class }%
4\rangle  \tag{7.1756}
\end{equation}

\subsection{Descendants of 6.161}

$2p-1$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-baa-bab,pb+bab,pc-xbaaa,\,\text{class }%
4\rangle \;(\text{all }x)  \tag{7.1757}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-baa-bab,pb+bab-xbaaa,pc+baaa,\,\text{class }%
4\rangle \;(x\neq 0,\;x\sim -x)  \tag{7.1758}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-baa-bab,pb+bab-xbaaa,pc-2baaa,\,\text{class }%
4\rangle \;(x \neq 0, x \sim -x)  \tag{7.1759}
\end{equation}

\subsection{Descendants of 6.162}

$2p-1$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-baa-\omega bab,pb+bab,pc-xbaaa,\,\text{class }%
4\rangle \;(\text{all }x)  \tag{7.1760}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-baa-\omega bab,pb+bab-xbaaa,pc+\omega
^{-1}baaa,\,\text{class }4\rangle \;(x\neq 0,\;x\sim -x)  \tag{7.1761}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-baa-\omega bab,pb+bab-xbaaa,pc-2\omega
^{-1}baaa,\,\text{class }4\rangle \;(x\neq 0,x\sim -x)  \tag{7.1762}
\end{equation}

\subsection{Descendants of 6.163}

Algebras 6.163 -- 6.167 give a classification of algebras of order $p^{6}$
with presentations 
\[
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-\lambda baa-\mu bab,\,pb+\nu baa+\xi
bab,\,pc,\,\text{class }3\rangle 
\]%
with $\lambda ,\mu ,\nu ,\xi \neq 0$. Most of these algebras are terminal,
and we need a slightly different classification of these algebras from that
given in the classification of nilpotent Lie rings of order $p^{6}$, so as
to classify the capable ones. It turns out that $\frac{5}{2}p-\frac{9}{2}+%
\frac{1}{2}\gcd (p-1,4)$ of these algebras are capable, and that they have a
total of $\frac{1}{2}p^{3}+2p^{2}-5p+\frac{1}{2}+\frac{p}{2}\gcd (p-1,4)$
descendants of order $p^{7}$ and $p$-class 4.

If we write the parameters $\lambda ,\mu ,\nu ,\xi $ above in a matrix $%
A=\left( 
\begin{array}{cc}
\lambda & \mu \\ 
\nu & \xi%
\end{array}%
\right) $ then $A$ and $B$ give isomorphic algebras if%
\[
B=\left( 
\begin{array}{cc}
\frac{\lambda }{\alpha \beta } & \frac{\mu }{\beta ^{2}} \\ 
\frac{\nu }{\alpha ^{2}} & \frac{\xi }{\alpha \beta }%
\end{array}%
\right) \text{ or }\left( 
\begin{array}{cc}
\frac{\xi }{\alpha \beta } & \frac{\nu }{\beta ^{2}} \\ 
\frac{\mu }{\alpha ^{2}} & \frac{\lambda }{\alpha \beta }%
\end{array}%
\right) . 
\]

So we can take $\lambda =1$ and $\mu =1$ or $\omega $ (or any other fixed
integer which is not a square $\func{mod}p)$. Given these values of $\lambda
,\mu $ it turns out that the algebra is terminal unless $\xi =1$ or $\xi
=\mu \nu $. We choose a set of representatives for the equivalence classes
(as defined above) containing matrices of the form%
\[
\left( 
\begin{array}{cc}
1 & 1 \\ 
\nu & 1%
\end{array}%
\right) ,\,\left( 
\begin{array}{cc}
1 & \omega \\ 
\nu & 1%
\end{array}%
\right) ,\,\left( 
\begin{array}{cc}
1 & 1 \\ 
\nu & \nu%
\end{array}%
\right) ,\,\left( 
\begin{array}{cc}
1 & \omega \\ 
\nu & \omega \nu%
\end{array}%
\right) . 
\]%
Corresponding to these choices we obtain two families of descendants of
order $p^{7}$ with the following presentations involving parameters $y,z,t$:

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-\mu bab-ybaaa,\,pb+\nu
baa+bab-zbaaa,\,pc-tbaaa,\,\text{class }3\rangle \;(\mu =1,\omega ), 
\tag{7.1763}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,\,cb,\,pa-baa-\mu bab-ybaaa,\,pb+\nu baa+\mu \nu
bab-zbaaa,\,pc-tbaaa,\,\text{class }3\rangle \;(\mu =1,\omega ). 
\tag{7.1764}
\end{equation}

For the family 7.1763, $(y,z,t)$ and $(y^{\prime },z^{\prime },t^{\prime })$
gives isomorphic algebras if%
\[
(y^{\prime },z^{\prime },t^{\prime })=(\pm y+\gamma t+\gamma \mu
^{-1}+\varepsilon ,\pm z+\varepsilon t-\nu \gamma \mu ^{-1}+\nu \varepsilon
-2\varepsilon \mu ^{-1},t) 
\]%
for some $\gamma ,\varepsilon $, and where if $\mu \nu =1$ we need $\gamma
=\mu \varepsilon $, or if%
\[
(y^{\prime },z^{\prime },t^{\prime })=(-\alpha z-\gamma t+\varepsilon -\nu
\gamma +2\gamma \mu ^{-1},-\alpha ^{-1}y-\varepsilon t-\gamma \mu
^{-2}-\varepsilon \mu ^{-1},-t-\nu +\mu ^{-1}) 
\]%
for some $\gamma ,\varepsilon $, where $\alpha ^{2}\nu =\mu $, and where if $%
\mu \nu =1$ we need $\gamma =\mu \varepsilon $.

For the family 7.1764 we can assume that $\mu \nu \neq 1$. In this case $%
(y,z,t)$ and $(y^{\prime },z^{\prime },t^{\prime })$ gives isomorphic
algebras if%
\[
(y^{\prime },z^{\prime },t^{\prime })=(\pm y+\mu \varepsilon t+2\mu ,\pm
z+\varepsilon t-2\mu \varepsilon ,t), 
\]%
for some $\varepsilon $, or (when $\mu \nu =-1$ and $p=1\func{mod}4$, and $%
\alpha ^{2}=-\mu ^{2}$),%
\[
(y^{\prime },z^{\prime },t^{\prime })=(-\alpha z-\mu \varepsilon
t-2\varepsilon ,\alpha ^{-1}y-\varepsilon t+2\nu \varepsilon ,t) 
\]%
for some $\varepsilon $.

\subsection{Descendants of 6.168}

$p^{3}+2p^{2}+2p+2+\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa,pb,pc-baa,\,\text{class }4\rangle 
\tag{7.1765}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-babb,pb,pc-baa,\,\text{class }4\rangle 
\tag{7.1766}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-\omega babb,pb,pc-baa,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.1767}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-\omega ^{2}babb,pb,pc-baa,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1768}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-xbabb,pb,pc-baa-babb,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.1769}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-xbabb,pb,pc-baa-\omega babb,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1770}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-ybabb,pb-babb,pc-baa-xbabb,\,\text{class }%
4\rangle \;(\text{all }x,\,y,\;y\sim -y)  \tag{7.1771}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb,pa-ybabb,pb-\omega babb,pc-baa-xbabb,\,\text{%
class }4\rangle \;(\text{all }x,\,y,\;y\sim -y)  \tag{7.1772}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa-babb,cb,pa-zbabb,pb-xbabb,pc-baa-ybabb,\,\text{%
class }4\rangle \;(\text{all }x,\,y,z,\;z\sim -z)  \tag{7.1773}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa-\omega babb,cb,pa-zbabb,pb-xbabb,pc-baa-ybabb,\,%
\text{class }4\rangle \;(\text{all }x,\,y,z,\;z\sim -z)  \tag{7.1774}
\end{equation}

\subsection{Descendants of 6.172}

$(p^{4}+p^{2})/2$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb-ybaaa,pa-zbaaa,pb-tbaaa,pc-baa-bab-xbaaa,\,\text{%
class }4\rangle \;(\text{all }y,z,t,\;x\neq 0,\;x\sim -x)  \tag{7.1775}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-baa,cb-ybaaa,pa-zbaaa,pb-tbaaa,pc-baa-bab,\,\text{class 
}4\rangle \;(\text{all }y,z,t,\;(z,t)\sim (t,z))  \tag{7.1776}
\end{equation}

\subsection{Descendants of 6.173}

Algebra 6.173 has $3p+3+(p^{2}+2p+3)\gcd (p-1,3)/2$ descendants of order $%
p^{7}$. Algebra 6.173 has presentation 
\[
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,\,pa,\,pb,\,pc,\,\text{class }%
3\rangle . 
\]

If $L$ is a descendant of 6.173 of order $p^{7}$ then the commutator
structure of $L$ is given by the classification of nilpotent Lie algebras
over $\mathbb{Z}_{p}$ of order $p^{7}$. The commutator relations take the
form 
\[
ca=bab,\,cb=\omega baa,\,baab=\lambda baaa,\,babb=\mu baaa 
\]%
for some parameters $\lambda ,\mu $. Provided $\alpha ^{2}+2\alpha \beta
\lambda +\beta ^{2}\mu \neq 0$, the parameters $\lambda ,\mu $ give the same
algebra (up to isomorphism) as the pair%
\[
\frac{\pm (\omega \alpha \beta +\alpha ^{2}\lambda +\omega \beta ^{2}\lambda
+\alpha \beta \mu )}{\alpha ^{2}+2\alpha \beta \lambda +\beta ^{2}\mu },\;%
\frac{\omega ^{2}\beta ^{2}+2\omega \alpha \beta \lambda +\alpha ^{2}\mu }{%
\alpha ^{2}+2\alpha \beta \lambda +\beta ^{2}\mu }. 
\]%
There are $p+2$ orbits of pairs $\lambda ,\mu $ under this action.

We pick a set representative pairs $\lambda ,\mu $ for these orbits, and get
the following presentations for the descendants of 6.173 of order $p^{7}$:

\begin{equation}
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,baab-\lambda baaa,\,babb-\mu
baaa,\,pa-ybaaa,\,pb-zbaaa,\,pc-tbaaa,\,\text{class }4\rangle .  \tag{7.1777}
\end{equation}

For each pair $\lambda ,\mu $ we compute the set of pairs $(\alpha ,\beta )$
which fixes $\lambda ,\mu $ under one of \ the $\pm $ actions given above.
It turns out that we need to treat the pair $\lambda =\mu =0$ separately
from the other pairs.

If $\lambda =\mu =0$, then $(y,z,t)$ and $(y^{\prime },z^{\prime },t^{\prime
})$ give isomorphic algebras if%
\[
\left( 
\begin{array}{c}
y^{\prime } \\ 
z^{\prime } \\ 
t^{\prime }%
\end{array}%
\right) =\frac{1}{\alpha ^{4}}\left( 
\begin{array}{ccc}
\pm \alpha & 0 & 0 \\ 
0 & \alpha & \gamma \\ 
0 & 0 & \pm \alpha ^{2}%
\end{array}%
\right) \left( 
\begin{array}{c}
y \\ 
z \\ 
t%
\end{array}%
\right) 
\]%
for some $\alpha ,\gamma $ with $\alpha \neq 0$.

In all other cases we can assume that if $t\neq 0$ then $y=z=0$. For
non-zero $t$ we have%
\[
t\sim \frac{\pm t}{(\alpha ^{2}+2\alpha \beta \lambda +\beta ^{2}\mu )} 
\]%
for some $(\alpha ,\beta )$ which fixes $\lambda ,\mu $ under one of \ the $%
\pm $ actions given above. If $t=0$ then $(y,z)$ and $(y^{\prime },z^{\prime
})$ give isomorphic algebras if%
\[
\left( 
\begin{array}{c}
y^{\prime } \\ 
z^{\prime }%
\end{array}%
\right) =\frac{1}{(\alpha ^{2}-\omega \beta ^{2})(\alpha ^{2}+2\alpha \beta
\lambda +\beta ^{2}\mu )}\left( 
\begin{array}{cc}
\pm \alpha & \pm \beta \\ 
\omega \beta & \alpha%
\end{array}%
\right) \left( 
\begin{array}{c}
y \\ 
z%
\end{array}%
\right) 
\]%
for some $(\alpha ,\beta )$ which fixes $\lambda ,\mu $ under one of \ the $%
\pm $ actions given above.

\subsection{Descendants of 6.174}

$(p^{3}+p^{2}+p+1)/4$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-bab-ybaaa,pb,pc-xbaaa,\,\text{%
class }4\rangle \;(x\neq 0,\text{ all }y,\;x\sim -x,\;\,y\sim -y) 
\tag{7.1778}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-bab,pb-ybaaa,pc,\,\text{class }%
4\rangle \;(\text{all }y,\;y\sim -y)  \tag{7.1779}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-zbaa-bab-ybaaa,pb,pc-xbaaa,\,\text{%
class }4\rangle \;(x,z\neq 0,\text{ all }y,y\sim -y,z\sim -z)  \tag{7.1780}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-zbaa-bab,pb-ybaaa,pc,\,\text{class 
}4\rangle \;(\text{all }y,\;z\neq 0,\;y\sim -y,\;z\sim -z)  \tag{7.1781}
\end{equation}

\subsection{Descendants of 6.175}

$(p^{3}+p^{2}+p+1)/4$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-\omega bab-ybaaa,pb,pc-xbaaa,\,%
\text{class }4\rangle \;(x\neq 0,\text{ all }y,\;x\sim -x,\;y\sim -y) 
\tag{7.1782}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-\omega bab,pb-ybaaa,pc,\,\text{%
class }4\rangle \;(\text{all }y,\;y\sim -y)  \tag{7.1783}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-zbaa-\omega
bab-ybaaa,pb,pc-xbaaa,\,\text{class }4\rangle \;(x,z\neq 0,\text{ all }%
y,\;y\sim -y,\;z\sim -z)  \tag{7.1784}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-zbaa-\omega bab,pb-ybaaa,pc,\,%
\text{class }4\rangle \;(\text{all }y,\;z\neq 0,\;y\sim -y,\;z\sim -z) 
\tag{7.1785}
\end{equation}

\subsection{Descendants of 6.176}

$p(p-1)+p\gcd (p-1,4)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-baa,pb,pc-xbabb,\,\text{class }%
4\rangle \;(\text{all }x)  \tag{7.1786}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-baa,pb-ybabb,pc-xbabb,\,\text{%
class }4\rangle \;(\text{all }x,\,y\neq 0,\;y\sim y^{\prime }\text{ if }%
y^{4}=y^{\prime 4}\func{mod}p)  \tag{7.1787}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-baa-ybabb,pb,pc-babb,\,\text{class 
}4\rangle \;(y\neq 0,\;y\sim y^{\prime }\text{ if }y^{4}=y^{\prime 4}\func{%
mod}p)  \tag{7.1788}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-baa-zbabb,pb-ybabb,pc-babb,\,\text{%
class }4\rangle \;(y,z\neq 0,\;y\sim y^{\prime }\text{ if }y^{4}=y^{\prime 4}%
\func{mod}p)  \tag{7.1789}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-\omega baa,pb,pc-xbabb,\,\text{%
class }4\rangle \;(\text{all }x,\;p=1\func{mod}4)  \tag{7.1790}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-\omega baa,pb-ybabb,pc-xbabb,\,%
\text{class }4\rangle \;(\text{all }x,\;\,y\neq 0,\;y\sim y^{\prime }\text{
if }y^{4}=y^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.1791}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-\omega baa-ybabb,pb,pc-\omega
babb,\,\text{class }4\rangle \;(y\neq 0,\;y\sim y^{\prime }\text{ if }%
y^{4}=y^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.1792}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-\omega
baa-zbabb,pb-ybabb,pc-\omega babb,\,\text{class }4\rangle \;(y,z\neq
0,\;y\sim y^{\prime }\text{ if }y^{4}=y^{\prime 4}\func{mod}p,\;p=1\func{mod}%
4)  \tag{7.1793}
\end{equation}

\subsection{Descendants of 6.178}

Algebra 6.178 has $(3p^{2}-1)/2$ descendants of order $p^{7}$. Algebra 6.178
has presentation 
\[
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,\,pa-\lambda baa-\mu bab,\,pb-\nu
baa-\xi bab,\,pc,\,\text{class }3\rangle , 
\]%
where we write $A=\left( 
\begin{array}{ll}
\lambda & \mu \\ 
\nu & \xi%
\end{array}%
\right) $, and $A$ ranges over a set of representatives for the orbits of
non-singular $2\times 2$ matrices under the action 
\[
A\rightarrow \frac{1}{\det P}PAP^{-1} 
\]%
as $P$ ranges over non-singular matrices 
\[
P=\left( 
\begin{array}{ll}
\alpha & \beta \\ 
\pm \omega \beta & \pm \alpha%
\end{array}%
\right) . 
\]%
$\allowbreak $

These algebras are terminal unless $\xi =-\lambda $. The number of orbits of
non-singular matrices with $\xi =-\lambda $ is $(3p-1)/2$. The matrices
split up into one orbit of size $p-1$ (matrices $\left( 
\begin{array}{ll}
0 & y \\ 
\omega y & 0%
\end{array}%
\right) $), $p-1$ orbits of size $(p^{2}-1)/2$ (including two orbits of
elements $\left( 
\begin{array}{ll}
x & y \\ 
-\omega y & -x%
\end{array}%
\right) $), and $(p-1)/2$ orbits of size $p^{2}-1$. All orbits contain
matrices where $\lambda =0$ or $\lambda =1$.

It is possible to choose orbit representatives of the following 6 types:

\begin{enumerate}
\item $\left( 
\begin{array}{ll}
0 & 1 \\ 
\omega & 0%
\end{array}%
\right) $,

\item $\left( 
\begin{array}{ll}
1 & 0 \\ 
0 & -1%
\end{array}%
\right) $ when $p=1\func{mod}4$,

\item $\left( 
\begin{array}{ll}
0 & 1 \\ 
-\omega & 0%
\end{array}%
\right) $,

\item one representative $\left( 
\begin{array}{ll}
1 & \mu \\ 
-\omega \mu & -1%
\end{array}%
\right) $ $(\mu \neq 0)$ which is not in the same orbit as $\left( 
\begin{array}{ll}
0 & 1 \\ 
-\omega & 0%
\end{array}%
\right) $ when $p=3\func{mod}4$,

\item $p-3$ representatives $\left( 
\begin{array}{ll}
0 & \mu \\ 
\nu & 0%
\end{array}%
\right) $ $(\nu \neq \pm \omega \mu )$, and

\item $(p-1)/2\,$\ representatives of the form $\left( 
\begin{array}{ll}
1 & \mu \\ 
\nu & -1%
\end{array}%
\right) $ $(\nu \neq -\omega \mu )$.
\end{enumerate}

We then obtain the following presentations for the descendants of 6.178. The
first four cases of the matrix $A$ are straightforward.

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-bab,pb-\omega baa,pc-xbaaa,\,\text{%
class }4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1794}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-baa,pb+bab,pc-xbaab,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.1795}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-bab,pb+\omega baa,pc-xbaaa,\,\text{%
class }4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1796}
\end{equation}

For the one matrix $\left( 
\begin{array}{ll}
1 & \mu \\ 
-\omega \mu & -1%
\end{array}%
\right) $ $(\mu \neq 0)$ (when $p=3\func{mod}4$) we have:%
\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-baa-\mu bab,pb+\omega \mu
baa+bab,pc-xbaaa,\,\text{class }4\rangle \;(\text{all }x,\,x\sim -x,\;p=3%
\func{mod}4)  \tag{7.1797}
\end{equation}

For the $p-3$ matrices $\left( 
\begin{array}{ll}
0 & \mu \\ 
\nu & 0%
\end{array}%
\right) $ $(\nu \neq \pm \omega \mu )$ we have:

\[
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,\,pa-\mu bab,\,pb-\nu
baa,\,pc-xbaaa,\,\text{class }4\rangle \,(\text{all }x,\,x\sim -x).
\]%
But we have extra descendants if $(\omega \mu +2\nu )(2\omega \nu +\mu
^{-1}\nu ^{2})$ is a square. If $\omega \mu +2\nu =0$ then we have,

\[
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,\,pa-\mu bab-xbaaa,\,pb-\nu
baa,\,pc,\,\text{class }4\rangle \,(x\neq 0,\,x\sim -x), 
\]%
If $2\omega \nu +\mu ^{-1}\nu ^{2}=0$ we have

\[
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,\,pa-\mu bab,\,pb-\nu
baa-xbaaa,\,pc,\,\text{class }4\rangle \,(x\neq 0,\,x\sim -x), 
\]%
and if $(\omega \mu +2\nu )(2\omega \nu +\mu ^{-1}\nu ^{2})=y^{2}\neq 0$
then for one such value $y$ we have

\[
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,\,pa-\mu bab,\,pb-\nu
baa-xbaaa,\,pc-ybaaa,\,\text{class }4\rangle \,(x\neq 0,\,x\sim -x). 
\]%
We combine these four possibilities into a single presentation

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-\mu bab-xbaaa,pb-\nu
baa-ybaaa,pc-zbaaa,\,\text{class }4\rangle .  \tag{7.1798}
\end{equation}

For the $(p-1)/2$ matrices $\left( 
\begin{array}{ll}
1 & \mu \\ 
\nu & -1%
\end{array}%
\right) $ $(\nu \neq -\omega \mu )$ the situation is even more complicated.
First we have

\[
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,\,pa-baa-\mu bab,\,pb-\nu
baa+bab,\,pc-xbaab,\,\text{class }4\rangle \,(\text{all }x). 
\]%
But if $\left( 1+yz\right) \left( 2\left( \omega y+z\right) ^{2}+\omega
(1+yz)\right) $ is a square we have an additional $p-1$ descendants. It is
not that easy to prove, but $\left( 1+yz\right) \left( 2\left( \omega
y+z\right) ^{2}+\omega (1+yz)\right) $ cannot equal zero, under the
assumption that $A$ is not in the same orbit as a matrix with $(1,1)$ entry
equal to zero. If $\left( 1+yz\right) \left( 2\left( \omega y+z\right)
^{2}+\omega (1+yz)\right) =x^{2}\neq 0$, then if $x-\omega \mu -\nu =\omega
\mu ^{2}+2\mu \nu +1=0$ we have

\[
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,\,pa-baa-\mu bab-ybaab,\,pb-\nu
baa+bab,\,pc-xbaab,\,\text{class }4\rangle \,(y\neq 0,\,y\sim -y), 
\]%
but if one of $x-\omega \mu -\nu $, $\omega \mu ^{2}+2\mu \nu +1$ is
non-zero we have

\[
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,\,pa-baa-\mu bab,\,pb-\nu
baa+bab-ybaab,\,pc-xbaab,\,\text{class }4\rangle \,(y\neq 0,\,y\sim -y). 
\]%
And similarly for $-x$, $x+\omega \mu +\nu =\omega \mu ^{2}+2\mu \nu +1=0$
we have

\[
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,\,pa-baa-\mu bab-ybaab,\,pb-\nu
baa+bab,\,pc+xbaab,\,\text{class }4\rangle \,(y\neq 0,\,y\sim -y), 
\]%
but if one of $x+\omega \mu +\nu $, $\omega \mu ^{2}+2\mu \nu +1$ is
non-zero we have

\[
\langle a,b,c\,|\,ca-bab,\,cb-\omega baa,\,pa-baa-\mu bab,\,pb-\nu
baa+bab-ybaab,\,pc+xbaab,\,\text{class }4\rangle \,(y\neq 0,\,y\sim -y). 
\]%
We combine these five possibilities into a single presentation

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega baa,pa-baa-\mu bab-xbaab,pb-\nu
baa+bab-ybaab,pc-zbaab,\,\text{class }4\rangle .  \tag{7.1799}
\end{equation}

\subsection{Descendants of 6.179}

$(p^{4}+p^{2})/2$ algebras

\begin{equation}
\langle a,b,c\,|\,ca-bab,cb-\omega
baa-xbaaa,pa-ybaaa,pb-zbaaa,pc-bab-tbaaa,\,\text{class }4\rangle \;(\text{%
all }x,\,y,z,t,\;y\sim -y,\text{ if }y=0\text{ then }t\sim -t)  \tag{7.1800}
\end{equation}

\subsection{Descendants of 6.182}

4 algebras

\begin{equation}
\langle a,b,c\,|\,ba,ca,cb,p^2b,pc,\,\text{class }4\rangle  \tag{7.1801}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba,ca,cb-p^3a,p^2b,pc,\,\text{class }4\rangle  \tag{7.1802}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba,ca-p^3a,cb,p^2b,pc,\,\text{class }4\rangle  \tag{7.1803}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba-p^3a,ca,cb,p^2b,pc,\,\text{class }4\rangle  \tag{7.1804}
\end{equation}

\subsection{Descendants of 6.183}

2 algebras

\begin{equation}
\langle a,b,c\,|\,ba-p^2a,ca,cb,p^2b,pc,\,\text{class }4\rangle  \tag{7.1805}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba-p^2a,ca,cb-p^3a,p^2b,pc,\,\text{class }4\rangle 
\tag{7.1806}
\end{equation}

\subsection{Descendants of 6.184}

9 algebras

\begin{equation}
\langle a,b,c\,|\,baa,bab,ca,cb,pb,pc,\,\text{class }4\rangle  \tag{7.1807}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,ca-p^3a,cb,pb,pc,\,\text{class }4\rangle 
\tag{7.1808}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,ca,cb-p^3a,pb,pc,\,\text{class }4\rangle 
\tag{7.1809}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-p^3a,bab,ca,cb,pb,pc,\,\text{class }4\rangle 
\tag{7.1810}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-p^3a,bab,ca,cb-p^3a,pb,pc,\,\text{class }4\rangle 
\tag{7.1811}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab-p^3a,ca,cb,pb,pc,\,\text{class }4\rangle 
\tag{7.1812}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab-p^3a,ca-p^3a,cb,pb,pc,\,\text{class }4\rangle 
\tag{7.1813}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab-\omega p^{3}a,ca,cb,pb,pc,\,\text{class }4\rangle 
\tag{7.1814}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab-\omega p^{3}a,ca-p^{3}a,cb,pb,pc,\,\text{class }%
4\rangle  \tag{7.1815}
\end{equation}

\subsection{Descendants of 6.187}

$11+4\gcd (p-1,3)+2\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,p^2a-baaa,pb,pc,\,\text{class }4\rangle 
\tag{7.1816}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,p^2a-baaa,pb,pc,\,\text{class }4\rangle 
\tag{7.1817}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-\omega baaa,ca,cb,p^{2}a-baaa,pb,pc,\,\text{class }%
4\rangle  \tag{7.1818}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-\omega ^{2}baaa,ca,cb,p^{2}a-baaa,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.1819}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-\omega ^{3}baaa,ca,cb,p^{2}a-baaa,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.1820}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baaa,p^2a-baaa,pb,pc,\,\text{class }4\rangle 
\tag{7.1821}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baaa,p^2a-baaa,pb,pc,\,\text{class }4\rangle
\tag{7.1822}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-\omega baaa,ca,cb-baaa,p^{2}a-baaa,pb,pc,\,\text{class 
}4\rangle  \tag{7.1823}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-\omega ^{2}baaa,ca,cb-baaa,p^{2}a-baaa,pb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}4)  \tag{7.1824}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-\omega ^{3}baaa,ca,cb-baaa,p^{2}a-baaa,pb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}4)  \tag{7.1825}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,p^2a,pb,pc,\,\text{class }4\rangle  \tag{7.1826}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,p^2a,pb-baaa,pc,\,\text{class }4\rangle 
\tag{7.1827}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,p^{2}a,pb-\omega baaa,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.1828}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,p^{2}a,pb-\omega ^{2}baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1829}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,p^2a,pb,pc-baaa,\,\text{class }4\rangle 
\tag{7.1830}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baaa,p^2a,pb,pc,\,\text{class }4\rangle 
\tag{7.1831}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baaa,p^2a,pb-baaa,pc,\,\text{class }4\rangle 
\tag{7.1832}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baaa,p^{2}a,pb-\omega baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1833}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baaa,p^{2}a,pb-\omega ^{2}baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1834}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baaa,p^2a,pb,pc-baaa,\,\text{class }4\rangle 
\tag{7.1835}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,p^2a,pb,pc,\,\text{class }4\rangle 
\tag{7.1836}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,p^2a,pb-baaa,pc,\,\text{class }4\rangle 
\tag{7.1837}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,p^{2}a,pb-\omega baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1838}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,p^{2}a,pb-\omega ^{2}baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1839}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,p^2a,pb,pc-baaa,\,\text{class }4\rangle 
\tag{7.1840}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baaa,p^2a,pb,pc,\,\text{class }4\rangle 
\tag{7.1841}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baaa,p^2a,pb-baaa,pc,\,\text{class }4\rangle
\tag{7.1842}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baaa,p^{2}a,pb-\omega baaa,pc,\,\text{class 
}4\rangle \;(p=1\func{mod}3)  \tag{7.1843}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baaa,p^{2}a,pb-\omega ^{2}baaa,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1844}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baaa,p^2a,pb,pc-baaa,\,\text{class }4\rangle
\tag{7.1845}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baaa,p^{2}a,pb,pc-\omega baaa,\,\text{class 
}4\rangle  \tag{7.1846}
\end{equation}

\subsection{Descendants of 6.188}

$3+2\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,p^2a-baaa,pb,pc,\,\text{class }4\rangle 
\tag{7.1847}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,p^2a,pb,pc,\,\text{class }4\rangle 
\tag{7.1848}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,p^2a,pb-baaa,pc,\,\text{class }4\rangle 
\tag{7.1849}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,p^{2}a,pb-\omega baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1850}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,p^{2}a,pb-\omega ^{2}baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1851}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,p^2a,pb,pc-baaa,\,\text{class }4\rangle 
\tag{7.1852}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,p^2a,pb-baaa,pc-baaa,\,\text{class }4\rangle
\tag{7.1853}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,p^{2}a,pb-\omega baaa,pc-baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1854}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,p^{2}a,pb-\omega ^{2}baaa,pc-baaa,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1855}
\end{equation}

\subsection{Descendants of 6.189}

$(5p+7)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,p^{2}a,pb-baa-xbaaa,pc,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.1856}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,p^2a-baaa,pb-baa,pc,\,\text{class }4\rangle 
\tag{7.1857}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,p^2a-xbaaa,pb-baa,pc,\,\text{class }%
4\rangle \;(\text{all }x)  \tag{7.1858}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,p^2a,pb-baa,pc-baaa,\,\text{class }4\rangle 
\tag{7.1859}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,p^2a-baaa,pb-baa,pc-baaa,\,\text{class }4\rangle
\tag{7.1860}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,p^2a-xbaaa,pb-baa,pc-baaa,\,\text{class }%
4\rangle \;(\text{all }x)  \tag{7.1861}
\end{equation}

\subsection{Descendants of 6.190}

2 algebras

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,p^2a,pb-baa,pc,\,\text{class }4\rangle 
\tag{7.1862}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,p^2a-baaa,pb-baa,pc,\,\text{class }4\rangle 
\tag{7.1863}
\end{equation}

\subsection{Descendants of 6.191}

$(5p+7)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,p^{2}a,pb-\omega baa-xbaaa,pc,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.1864}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,p^{2}a-baaa,pb-\omega baa,pc,\,\text{class }%
4\rangle  \tag{7.1865}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,p^{2}a-xbaaa,pb-\omega baa,pc,\,\text{class 
}4\rangle \;(\text{all }x)  \tag{7.1866}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,p^{2}a,pb-\omega baa,pc-baaa,\,\text{class }%
4\rangle  \tag{7.1867}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,p^{2}a-baaa,pb-\omega baa,pc-baaa,\,\text{class }%
4\rangle  \tag{7.1868}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,p^{2}a-xbaaa,pb-\omega baa,pc-baaa,\,\text{%
class }4\rangle \;(\text{all }x)  \tag{7.1869}
\end{equation}

\subsection{Descendants of 6.192}

2 algebras

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,p^{2}a,pb-\omega baa,pc,\,\text{class }%
4\rangle  \tag{7.1870}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,p^{2}a-baaa,pb-\omega baa,pc,\,\text{class }%
4\rangle  \tag{7.1871}
\end{equation}

\subsection{Descendants of 6.197}

$11+4\gcd (p-1,3)+2\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,ca,cb,p^2a-babb,pb,pc,\,\text{class }4\rangle 
\tag{7.1872}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca,cb,p^{2}a-\omega babb,pb,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.1873}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca,cb,p^{2}a-\omega ^{2}babb,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1874}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-babb,cb,p^2a-babb,pb,pc,\,\text{class }4\rangle 
\tag{7.1875}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-babb,cb,p^{2}a-\omega babb,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1876}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-babb,cb,p^{2}a-\omega ^{2}babb,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1877}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca,cb,p^2a-babb,pb,pc,\,\text{class }4\rangle 
\tag{7.1878}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca,cb,p^{2}a-\omega babb,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1879}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca,cb,p^{2}a-\omega ^{2}babb,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1880}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca-babb,cb,p^2a-babb,pb,pc,\,\text{class }4\rangle
\tag{7.1881}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca-babb,cb,p^{2}a-\omega babb,pb,pc,\,\text{class 
}4\rangle \;(p=1\func{mod}3)  \tag{7.1882}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca-babb,cb,p^{2}a-\omega ^{2}babb,pb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1883}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca,cb,p^2a,pb,pc,\,\text{class }4\rangle  \tag{7.1884}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca,cb,p^2a,pb,pc-babb,\,\text{class }4\rangle 
\tag{7.1885}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca,cb,p^2a,pb-babb,pc,\,\text{class }4\rangle 
\tag{7.1886}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-babb,cb,p^2a,pb,pc,\,\text{class }4\rangle 
\tag{7.1887}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-babb,cb,p^2a,pb,pc-babb,\,\text{class }4\rangle 
\tag{7.1888}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-babb,cb,p^2a,pb-babb,pc,\,\text{class }4\rangle 
\tag{7.1889}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca,cb,p^2a,pb,pc,\,\text{class }4\rangle 
\tag{7.1890}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca,cb,p^2a,pb,pc-babb,\,\text{class }4\rangle 
\tag{7.1891}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca,cb,p^2a,pb-babb,pc,\,\text{class }4\rangle 
\tag{7.1892}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca,cb,p^{2}a,pb-\omega babb,pc,\,\text{class }%
4\rangle  \tag{7.1893}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca,cb,p^{2}a,pb-\omega ^{2}babb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.1894}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca,cb,p^{2}a,pb-\omega ^{3}babb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.1895}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca-babb,cb,p^2a,pb,pc,\,\text{class }4\rangle 
\tag{7.1896}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca-babb,cb,p^2a,pb,pc-babb,\,\text{class }4\rangle
\tag{7.1897}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca-babb,cb,p^{2}a,pb,pc-\omega babb,\,\text{class 
}4\rangle  \tag{7.1898}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca-babb,cb,p^2a,pb-babb,pc,\,\text{class }4\rangle
\tag{7.1899}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca-babb,cb,p^{2}a,pb-\omega babb,pc,\,\text{class 
}4\rangle  \tag{7.1900}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca-babb,cb,p^{2}a,pb-\omega ^{2}babb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}4)  \tag{7.1901}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-babb,ca-babb,cb,p^{2}a,pb-\omega ^{3}babb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}4)  \tag{7.1902}
\end{equation}

\subsection{Descendants of 6.198}

$4+\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab,cb,p^2a-babb,pb,pc,\,\text{class }4\rangle 
\tag{7.1903}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab,cb,p^{2}a-\omega babb,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1904}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab,cb,p^{2}a-\omega ^{2}babb,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1905}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab,cb,p^2a,pb,pc,\,\text{class }4\rangle 
\tag{7.1906}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab,cb,p^2a,pb,pc-babb,\,\text{class }4\rangle 
\tag{7.1907}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab,cb,p^2a,pb-babb,pc,\,\text{class }4\rangle 
\tag{7.1908}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-bab,cb,p^2a,pb-babb,pc-babb,\,\text{class }4\rangle
\tag{7.1909}
\end{equation}

\subsection{Descendants of 6.207}

5 algebras

\begin{equation}
\langle a,b,c\,|\,baa,ca,cb,pb-ba,pc,\,\text{class }4\rangle  \tag{7.1910}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca,cb-p^3a,pb-ba,pc,\,\text{class }4\rangle 
\tag{7.1911}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-p^3a,ca,cb,pb-ba,pc,\,\text{class }4\rangle 
\tag{7.1912}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa-p^3a,ca,cb-p^3a,pb-ba,pc,\,\text{class }4\rangle 
\tag{7.1913}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,ca-p^3a,cb,pb-ba,pc,\,\text{class }4\rangle 
\tag{7.1914}
\end{equation}

\subsection{Descendants of 6.212}

4 algebras

\begin{equation}
\langle a,b,c\,|\,ca,cb,pb,pc-ba,\,\text{class }4\rangle  \tag{7.1915}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-p^3a,cb,pb,pc-ba,\,\text{class }4\rangle  \tag{7.1916}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-p^3a,pb,pc-ba,\,\text{class }4\rangle  \tag{7.1917}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca,cb-\omega p^{3}a,pb,pc-ba,\,\text{class }4\rangle 
\tag{7.1918}
\end{equation}

\subsection{Descendants of 6.215}

3 algebras

\begin{equation}
\langle a,b,c\,|\,ca-p^2a,cb,pb,pc-ba,\,\text{class }4\rangle  \tag{7.1919}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-p^2a,cb-p^3a,pb,pc-ba,\,\text{class }4\rangle 
\tag{7.1920}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-p^{2}a,cb-\omega p^{3}a,pb,pc-ba,\,\text{class }4\rangle
\tag{7.1921}
\end{equation}

\subsection{Descendants of 6.216}

4 algebras

\begin{equation}
\langle a,b,c\,|\,baa,bab,ca,cb,pa,pb,\,\text{class }4\rangle  \tag{7.1922}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,ca,cb-p^3c,pa,pb,\,\text{class }4\rangle 
\tag{7.1923}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab-p^3c,ca,cb,pa,pb,\,\text{class }4\rangle 
\tag{7.1924}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab-p^3c,ca-p^3c,cb,pa,pb,\,\text{class }4\rangle 
\tag{7.1925}
\end{equation}

\subsection{Descendants of 6.218}

$11+4\gcd (p-1,3)+2\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,pa,pb,p^2c,\,\text{class }4\rangle  \tag{7.1926}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,pa,pb,p^2c-baaa,\,\text{class }4\rangle 
\tag{7.1927}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,pa,pb,p^2c,\,\text{class }4\rangle 
\tag{7.1928}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,pa,pb,p^2c-baaa,\,\text{class }4\rangle 
\tag{7.1929}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baaa,pa,pb,p^2c,\,\text{class }4\rangle 
\tag{7.1930}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baaa,pa,pb,p^2c-baaa,\,\text{class }4\rangle 
\tag{7.1931}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baaa,pa,pb,p^2c,\,\text{class }4\rangle 
\tag{7.1932}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baaa,pa,pb,p^2c-baaa,\,\text{class }4\rangle
\tag{7.1933}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-\omega baaa,pa,pb,p^{2}c-baaa,\,\text{class 
}4\rangle  \tag{7.1934}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,pa,pb-baaa,p^2c,\,\text{class }4\rangle 
\tag{7.1935}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,pa,pb-baaa,p^2c,\,\text{class }4\rangle 
\tag{7.1936}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baaa,pa,pb-baaa,p^2c,\,\text{class }4\rangle 
\tag{7.1937}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baaa,pa,pb-baaa,p^2c,\,\text{class }4\rangle
\tag{7.1938}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,pa,pb-\omega baaa,p^{2}c,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.1939}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,pa,pb-\omega baaa,p^{2}c,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1940}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baaa,pa,pb-\omega baaa,p^{2}c,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1941}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baaa,pa,pb-\omega baaa,p^{2}c,\,\text{class 
}4\rangle \;(p=1\func{mod}3)  \tag{7.1942}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,pa,pb-\omega ^{2}baaa,p^{2}c,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1943}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,pa,pb-\omega ^{2}baaa,p^{2}c,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1944}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baaa,pa,pb-\omega ^{2}baaa,p^{2}c,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1945}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baaa,pa,pb-\omega ^{2}baaa,p^{2}c,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1946}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb,pa-baaa,pb,p^2c,\,\text{class }4\rangle 
\tag{7.1947}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baaa,pa-baaa,pb,p^2c,\,\text{class }4\rangle 
\tag{7.1948}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,pa-baaa,pb,p^2c,\,\text{class }4\rangle 
\tag{7.1949}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baaa,pa-baaa,pb,p^2c,\,\text{class }4\rangle
\tag{7.1950}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,pa-\omega baaa,pb,p^{2}c,\,\text{class }%
4\rangle  \tag{7.1951}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baaa,pa-\omega baaa,pb,p^{2}c,\,\text{class 
}4\rangle  \tag{7.1952}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,pa-\omega ^{2}baaa,pb,p^{2}c,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.1953}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baaa,pa-\omega ^{2}baaa,pb,p^{2}c,\,\text{%
class }4\rangle \;(p=1\func{mod}4)  \tag{7.1954}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb,pa-\omega ^{3}baaa,pb,p^{2}c,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.1955}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baaa,pa-\omega ^{3}baaa,pb,p^{2}c,\,\text{%
class }4\rangle \;(p=1\func{mod}4)  \tag{7.1956}
\end{equation}

\subsection{Descendants of 6.222}

$2p+5+3\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb,p^2c,\,\text{class }4\rangle 
\tag{7.1957}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-baaa,pa,pb,p^2c,\,\text{class }4\rangle 
\tag{7.1958}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb,p^2c-baaa,\,\text{class }4\rangle 
\tag{7.1959}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-baaa,pa,pb,p^2c-baaa,\,\text{class }4\rangle
\tag{7.1960}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baa,pa,pb,p^2c,\,\text{class }4\rangle 
\tag{7.1961}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baa,pa,pb,p^2c-baaa,\,\text{class }4\rangle
\tag{7.1962}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baa,pa,pb,p^{2}c-\omega baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1963}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baa,pa,pb,p^{2}c-\omega ^{2}baaa,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1964}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb-baaa,p^2c,\,\text{class }4\rangle 
\tag{7.1965}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb-\omega baaa,p^{2}c,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1966}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb-\omega ^{2}baaa,p^{2}c,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1967}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa-baaa,pb,p^2c,\,\text{class }4\rangle 
\tag{7.1968}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-baaa,pa,pb-xbaaa,p^2c,\,\text{class }%
4\rangle \;(x \neq 0)  \tag{7.1969}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-baaa,pa-baaa,pb,p^2c,\,\text{class }4\rangle
\tag{7.1970}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baa-baaa,pa-baaa,pb,p^2c,\,\text{class }%
4\rangle  \tag{7.1971}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baa,pa-\omega baaa,pb,p^{2}c,\,\text{class }%
4\rangle  \tag{7.1972}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baa,pa-\omega ^{2}baaa,pb,p^{2}c,\,\text{%
class }4\rangle \;(p=1\func{mod}4)  \tag{7.1973}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baa,pa-\omega ^{3}baaa,pb,p^{2}c,\,\text{%
class }4\rangle \;(p=1\func{mod}4)  \tag{7.1974}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baa,pa,pb-baaa,p^2c,\,\text{class }4\rangle
\tag{7.1975}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baa,pa,pb-\omega baaa,p^{2}c,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1976}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baa,pa,pb-\omega ^{2}baaa,p^{2}c,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1977}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baa,pa-xbaaa,pb-baaa,p^2c,\,\text{class }%
4\rangle \;(x \neq 0, x \sim x^{\prime }\text{ if }x^3=x^{\prime 3 }\func{mod%
} p)  \tag{7.1978}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baa,pa-xbaaa,pb-\omega baaa,p^{2}c,\,\text{%
class }4\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}%
\func{mod}p,\;p=1\func{mod}3)  \tag{7.1979}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaa,ca,cb-baa,pa-xbaaa,pb-\omega ^{2}baaa,p^{2}c,\,%
\text{class }4\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime
3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.1980}
\end{equation}

\subsection{Descendants of 6.228}

$p+1$ algebras

\begin{equation}
\langle a,b,c\,|\,ca,cb-xp^3c,pa-ba,pb,\,\text{class }4\rangle \;(\text{all }%
x)  \tag{7.1981}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ca-p^3c,cb,pa-ba,pb,\,\text{class }4\rangle  \tag{7.1982}
\end{equation}

\subsection{Descendants of 6.231}

$p^{2}+5p+14+(p+17)\gcd (p-1,3)+2\gcd (p-1,4)+\gcd (p-1,7)+\gcd (p-1,8)$
algebras

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.1983}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac,pa-babb,pb,pc,\,\text{class }4\rangle 
\tag{7.1984}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac,pa-\omega babb,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1985}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac,pa-\omega ^{2}babb,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.1986}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac,pa,pb-babb,pc,\,\text{class }4\rangle 
\tag{7.1987}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac,pa-babb,pb-babb,pc,\,\text{class }%
4\rangle  \tag{7.1988}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac,pa-\omega babb,pb-babb,pc,\,\text{class 
}4\rangle \;(p=1\func{mod}3)  \tag{7.1989}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac,pa-\omega ^{2}babb,pb-babb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1990}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac,pa,pb,pc-babb,\,\text{class }4\rangle 
\tag{7.1991}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa-babb,cac,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.1992}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa-babb,cac,pa-babb,pb,pc,\,\text{class }%
4\rangle  \tag{7.1993}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa-babb,cac,pa-\omega babb,pb,pc,\,\text{class 
}4\rangle \;(p=1\func{mod}3)  \tag{7.1994}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa-babb,cac,pa-\omega ^{2}babb,pb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1995}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa-babb,cac,pa,pb-babb,pc,\,\text{class }%
4\rangle  \tag{7.1996}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa-babb,cac,pa-babb,pb-babb,pc,\,\text{class }%
4\rangle  \tag{7.1997}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa-babb,cac,pa-\omega babb,pb-babb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.1998}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa-babb,cac,pa-\omega ^{2}babb,pb-babb,pc,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.1999}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa-babb,cac,pa,pb,pc-babb,\,\text{class }%
4\rangle  \tag{7.2000}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa-babb,cac,pa,pb-babb,pc-babb,\,\text{class }%
4\rangle  \tag{7.2001}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa-babb,cac,pa,pb-\omega babb,pc-babb,\,\text{%
class }4\rangle \;(p=1\func{mod}4)  \tag{7.2002}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa-babb,cac,pa,pb,pc-\omega babb,\,\text{class 
}4\rangle  \tag{7.2003}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa-babb,cac,pa,pb-babb,pc-\omega babb,\,\text{%
class }4\rangle  \tag{7.2004}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa-babb,cac,pa,pb-\omega babb,pc-\omega babb,\,%
\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.2005}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa,cac,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.2006}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa,cac,pa-babb,pb,pc,\,\text{class }%
4\rangle  \tag{7.2007}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa,cac,pa-\omega babb,pb,pc,\,\text{class 
}4\rangle \;(p=1\func{mod}3)  \tag{7.2008}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa,cac,pa-\omega ^{2}babb,pb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2009}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa,cac,pa,pb-babb,pc,\,\text{class }%
4\rangle  \tag{7.2010}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa,cac,pa-babb,pb-babb,pc,\,\text{class }%
4\rangle  \tag{7.2011}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa,cac,pa-\omega babb,pb-babb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2012}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa,cac,pa-\omega ^{2}babb,pb-babb,pc,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2013}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa,cac,pa,pb,pc-babb,\,\text{class }%
4\rangle  \tag{7.2014}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa,cac,pa-babb,pb,pc-babb,\,\text{class }%
4\rangle  \tag{7.2015}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa,cac,pa-\omega babb,pb,pc-babb,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2016}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa,cac,pa-\omega ^{2}babb,pb,pc-babb,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2017}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa-babb,cac,pa,pb,pc,\,\text{class }%
4\rangle  \tag{7.2018}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa-babb,cac,pa-babb,pb-xbabb,pc,\,\text{%
class }4\rangle \;(\text{all }x)  \tag{7.2019}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa-babb,cac,pa-\omega babb,pb-xbabb,pc,\,%
\text{class }4\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.2020}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa-babb,cac,pa-\omega
^{2}babb,pb-xbabb,pc,\,\text{class }4\rangle \;(\text{all }x,\;p=1\func{mod}%
3)  \tag{7.2021}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa-babb,cac,pa,pb-babb,pc,\,\text{class }%
4\rangle  \tag{7.2022}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa-babb,cac,pa,pb-\omega babb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2023}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa-babb,cac,pa,pb-\omega ^{2}babb,pc,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2024}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa-babb,cac,pa,pb-xbabb,pc-babb,\,\text{%
class }4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.2025}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac-babb,caa-babb,cac,pa,pb-xbabb,pc-\omega babb,\,%
\text{class }4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.2026}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.2027}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac,pa,pb-babb,pc,\,\text{class }%
4\rangle  \tag{7.2028}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac,pa,pb-\omega babb,pc,\,\text{class 
}4\rangle  \tag{7.2029}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac,pa,pb-\omega ^{2}babb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}4)  \tag{7.2030}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac,pa,pb-\omega ^{3}babb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}4)  \tag{7.2031}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac,pa-babb,pb,pc,\,\text{class }%
4\rangle  \tag{7.2032}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac,pa-\omega babb,pb,pc,\,\text{class 
}4\rangle \;(p=1\func{mod}3)  \tag{7.2033}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac,pa-\omega ^{2}babb,pb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2034}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac,pa-babb,pb-xbabb,pc,\,\text{class }%
4\rangle \;(x \neq 0, x \sim x^{\prime }\text{ if }x^3=x^{\prime 3 }\func{mod%
} p)  \tag{7.2035}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac,pa-\omega babb,pb-xbabb,pc,\,\text{%
class }4\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}%
\func{mod}p,\;p=1\func{mod}3)  \tag{7.2036}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac,pa-\omega ^{2}babb,pb-xbabb,pc,\,%
\text{class }4\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime
3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.2037}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac,pa,pb,pc-babb,\,\text{class }%
4\rangle  \tag{7.2038}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.2039}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa,pb,pc-babb,\,\text{class }%
4\rangle  \tag{7.2040}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa,pb-babb,pc,\,\text{class }%
4\rangle  \tag{7.2041}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa,pb-babb,pc-babb,\,\text{class }%
4\rangle  \tag{7.2042}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-babb,pb,pc,\,\text{class }%
4\rangle  \tag{7.2043}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega babb,pb,pc,\,\text{class 
}4\rangle  \tag{7.2044}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega ^{2}babb,pb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2045}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega ^{3}babb,pb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2046}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega ^{4}babb,pb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2047}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega ^{5}babb,pb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2048}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-babb,pb,pc-babb,\,\text{class }%
4\rangle  \tag{7.2049}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega babb,pb,pc-babb,\,\text{%
class }4\rangle  \tag{7.2050}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega ^{2}babb,pb,pc-babb,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2051}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega ^{3}babb,pb,pc-babb,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2052}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega ^{4}babb,pb,pc-babb,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2053}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega ^{5}babb,pb,pc-babb,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2054}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-babb,pb-babb,pc,\,\text{class }%
4\rangle  \tag{7.2055}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega babb,pb-babb,pc,\,\text{%
class }4\rangle  \tag{7.2056}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega ^{2}babb,pb-babb,pc,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2057}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega ^{3}babb,pb-babb,pc,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2058}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega ^{4}babb,pb-babb,pc,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2059}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega ^{5}babb,pb-babb,pc,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2060}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-babb,pb-babb,pc-xbabb,\,\text{%
class }4\rangle \;(x \neq 0, x \sim x^{\prime}\text{ if }x^6=x^{\prime 6 }%
\func{mod} p)  \tag{7.2061}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega babb,pb-babb,pc-xbabb,\,%
\text{class }4\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^6=x^{\prime
6}\func{mod}p)  \tag{7.2062}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega
^{2}babb,pb-babb,pc-xbabb,\,\text{class }4\rangle \;(x\neq 0,x\sim x^{\prime}%
\text{ if }x^6 =x^{\prime 6}\func{mod}p,\;p=1\func{mod}3)  \tag{7.2063}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega
^{3}babb,pb-babb,pc-xbabb,\,\text{class }4\rangle \;(x\neq 0,x\sim x^{\prime}%
\text{ if }x^6 =x^{\prime 6}\func{mod}p,\;p=1\func{mod}3)  \tag{7.2064}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega
^{4}babb,pb-babb,pc-xbabb,\,\text{class }4\rangle \;(x\neq 0,x\sim x^{\prime}%
\text{ if }x^6 =x^{\prime 6}\func{mod}p,\;p=1\func{mod}3)  \tag{7.2065}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,bac,caa,cac-babb,pa-\omega
^{5}babb,pb-babb,pc-xbabb,\,\text{class }4\rangle \;(x\neq 0,x\sim x^{\prime}%
\text{ if }x^6 =x^{\prime 6}\func{mod}p,\;p=1\func{mod}3)  \tag{7.2066}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac-babb,pa,pb,pc,\,\text{class }%
4\rangle  \tag{7.2067}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac-babb,pa,pb,pc-babb,\,\text{class }%
4\rangle  \tag{7.2068}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac-babb,pa,pb,pc-xbabb,\,\text{class }%
4\rangle \;(x=\omega ^{i},i=1,2,3,4,5,6,\;p=1\func{mod}7)  \tag{7.2069}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac-babb,pa,pb-xbabb,pc,\,\text{class }%
4\rangle \;(x=\omega ^{i},0\leq i<\gcd (p-1,8))  \tag{7.2070}
\end{equation}

In the following presentation we have $\;x=\omega ^{i},\;0\leq i<\gcd
(p-1,8),\;y\neq 0,\;y\sim y^{\prime }$ if $y^{8}=y^{\prime 8}\func{mod}p$:

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac-babb,pa,pb-xbabb,pc-ybabb,\,\text{%
class }4\rangle  \tag{7.2071}
\end{equation}

In the following presentation we have $\;x=\omega ^{i},\;0\leq i<2\gcd
(p-1,3)$:

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac-babb,pa-xbabb,pb,pc,\,\text{class }%
4\rangle  \tag{7.2072}
\end{equation}

In the following presentation we have $\;x=\omega ^{i},\;0\leq i<2\gcd
(p-1,3),\;y\neq 0,\;y\sim y^{\prime 6}=y^{\prime 6}\func{mod}p$:

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac-babb,pa-xbabb,pb,pc-ybabb,\,\text{%
class }4\rangle  \tag{7.2073}
\end{equation}

In the following presentation we have $\;x=\omega ^{i},\;0\leq i<2\gcd
(p-1,3),\;y\neq 0,\;y\sim y^{\prime 3}=y^{\prime 3}\func{mod}p,$ all $%
z,\;z\sim -z$:

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,bac,caa,cac-babb,pa-xbabb,pb-ybabb,pc-zbabb,\,%
\text{class }4\rangle  \tag{7.2074}
\end{equation}

\subsection{Descendants of 6.256}

$20+(p+11)\gcd (p-1,3)+4\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac,caa,cac,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.2075}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac,caa,cac,pa-baaa,pb,pc,\,\text{class }4\rangle 
\tag{7.2076}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac,caa,cac,pa,pb-baaa,pc,\,\text{class }4\rangle 
\tag{7.2077}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac,caa,cac,pa,pb-\omega baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2078}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac,caa,cac,pa,pb-\omega ^{2}baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2079}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac,caa,cac,pa,pb,pc-baaa,\,\text{class }4\rangle 
\tag{7.2080}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab,bac,caa,cac,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.2081}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab,bac,caa,cac,pa-baaa,pb,pc,\,\text{class }%
4\rangle  \tag{7.2082}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab,bac,caa,cac,pa,pb-baaa,pc,\,\text{class }%
4\rangle  \tag{7.2083}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab,bac,caa,cac,pa,pb-\omega baaa,pc,\,\text{class 
}4\rangle \;(p=1\func{mod}3)  \tag{7.2084}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab,bac,caa,cac,pa,pb-\omega ^{2}baaa,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2085}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab,bac,caa,cac,pa,pb,pc-baaa,\,\text{class }%
4\rangle  \tag{7.2086}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.2087}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac,pa-baaa,pb,pc,\,\text{class }%
4\rangle  \tag{7.2088}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac,pa-\omega baaa,pb,pc,\,\text{class 
}4\rangle  \tag{7.2089}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac,pa-\omega ^{2}baaa,pb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}4)  \tag{7.2090}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac,pa-\omega ^{3}baaa,pb,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}4)  \tag{7.2091}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac,pa,pb-baaa,pc,\,\text{class }%
4\rangle  \tag{7.2092}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac,pa,pb-\omega baaa,pc,\,\text{class 
}4\rangle \;(p=1\func{mod}3)  \tag{7.2093}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac,pa,pb-\omega ^{2}baaa,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2094}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac,pa,pb,pc-baaa,\,\text{class }%
4\rangle  \tag{7.2095}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab-baaa,bac,caa,cac,pa,pb,pc,\,\text{class }%
4\rangle  \tag{7.2096}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab-baaa,bac,caa,cac,pa-baaa,pb,pc,\,\text{class }%
4\rangle  \tag{7.2097}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab-baaa,bac,caa,cac,pa-\omega baaa,pb,pc,\,\text{%
class }4\rangle  \tag{7.2098}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab-baaa,bac,caa,cac,pa-\omega ^{2}baaa,pb,pc,\,%
\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.2099}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab-baaa,bac,caa,cac,pa-\omega ^{3}baaa,pb,pc,\,%
\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.2100}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab-baaa,bac,caa,cac,pa,pb-baaa,pc,\,\text{class }%
4\rangle  \tag{7.2101}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab-baaa,bac,caa,cac,pa,pb-\omega baaa,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2102}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab-baaa,bac,caa,cac,pa,pb-\omega ^{2}baaa,pc,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2103}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab-baaa,bac,caa,cac,pa,pb,pc-baaa,\,\text{class }%
4\rangle  \tag{7.2104}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab-baaa,bac,caa,cac,pa,pb,pc-\omega baaa,\,\text{%
class }4\rangle  \tag{7.2105}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac-baaa,caa,cac,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.2106}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac-baaa,caa,cac,pa-baaa,pb,pc,\,\text{class }%
4\rangle  \tag{7.2107}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac-baaa,caa,cac,pa,pb-baaa,pc,\,\text{class }%
4\rangle  \tag{7.2108}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac-baaa,caa,cac,pa,pb-\omega baaa,pc,\,\text{class 
}4\rangle \;(p=1\func{mod}3)  \tag{7.2109}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac-baaa,caa,cac,pa,pb-\omega ^{2}baaa,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2110}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac-baaa,caa,cac,pa,pb-baaa,pc-baaa,\,\text{class }%
4\rangle  \tag{7.2111}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac-baaa,caa,cac,pa,pb-\omega baaa,pc-baaa,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2112}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac-baaa,caa,cac,pa,pb-\omega ^{2}baaa,pc-baaa,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2113}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac-baaa,caa,cac,pa,pb,pc-baaa,\,\text{class }%
4\rangle  \tag{7.2114}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac,caa,cac-baaa,pa-baaa,pb,pc,\,\text{class }%
4\rangle  \tag{7.2115}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac,caa,cac-baaa,pa-\omega baaa,pb,pc,\,\text{class 
}4\rangle  \tag{7.2116}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac,caa,cac-baaa,pa,pb,pc,\,\text{class }4\rangle 
\tag{7.2117}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac,caa,cac-baaa,pa,pb-baaa,pc,\,\text{class }%
4\rangle  \tag{7.2118}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac,caa,cac-baaa,pa,pb-\omega baaa,pc,\,\text{class 
}4\rangle \;(p=1\func{mod}3)  \tag{7.2119}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac,caa,cac-baaa,pa,pb-\omega ^{2}baaa,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2120}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac,caa,cac-baaa,pa,pb,pc-baaa,\,\text{class }%
4\rangle  \tag{7.2121}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac,caa,cac-baaa,pa,pb-baaa,pc-baaa,\,\text{class }%
4\rangle  \tag{7.2122}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac,caa,cac-baaa,pa,pb-\omega baaa,pc-baaa,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2123}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,bac,caa,cac-baaa,pa,pb-\omega ^{2}baaa,pc-baaa,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2124}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac-baaa,pa,pb,pc,\,\text{class }%
4\rangle  \tag{7.2125}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac-baaa,pa-baaa,pb,pc,\,\text{class }%
4\rangle  \tag{7.2126}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac-baaa,pa-\omega baaa,pb,pc,\,\text{%
class }4\rangle  \tag{7.2127}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac-baaa,pa-\omega ^{2}baaa,pb,pc,\,%
\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.2128}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac-baaa,pa-\omega ^{3}baaa,pb,pc,\,%
\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.2129}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac-baaa,pa,pb,pc-baaa,\,\text{class }%
4\rangle  \tag{7.2130}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac-baaa,pa,pb,pc-\omega baaa,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2131}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac-baaa,pa,pb,pc-\omega ^{2}baaa,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2132}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac-baaa,pa,pb-baaa,pc-xbaaa,\,\text{%
class }4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.2133}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac-baaa,pa,pb-\omega baaa,pc-xbaaa,\,%
\text{class }4\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}3) 
\tag{7.2134}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,bac,caa,cac-baaa,pa,pb-\omega
^{2}baaa,pc-xbaaa,\,\text{class }4\rangle \;(\text{all }x,\,x\sim -x,\;p=1%
\func{mod}3)  \tag{7.2135}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-\omega baaa,bac,caa,cac-baaa,pa,pb,pc,\,\text{class 
}4\rangle  \tag{7.2136}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-\omega baaa,bac,caa,cac-baaa,pa-baaa,pb,pc,\,\text{%
class }4\rangle  \tag{7.2137}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-\omega baaa,bac,caa,cac-baaa,pa-\omega baaa,pb,pc,\,%
\text{class }4\rangle  \tag{7.2138}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-\omega baaa,bac,caa,cac-baaa,pa-\omega
^{2}baaa,pb,pc,\,\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.2139}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-\omega baaa,bac,caa,cac-baaa,pa-\omega
^{3}baaa,pb,pc,\,\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.2140}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-\omega baaa,bac,caa,cac-baaa,pa,pb,pc-baaa,\,\text{%
class }4\rangle  \tag{7.2141}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-\omega baaa,bac,caa,cac-baaa,pa,pb,pc-\omega baaa,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2142}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-\omega baaa,bac,caa,cac-baaa,pa,pb,pc-\omega
^{2}baaa,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2143}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-\omega baaa,bac,caa,cac-baaa,pa,pb-baaa,pc-xbaaa,\,%
\text{class }4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.2144}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-\omega baaa,bac,caa,cac-baaa,pa,pb-\omega
baaa,pc-xbaaa,\,\text{class }4\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{%
mod}3)  \tag{7.2145}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-\omega baaa,bac,caa,cac-baaa,pa,pb-\omega
^{2}baaa,pc-xbaaa,\,\text{class }4\rangle \;(\text{all }x,\,x\sim -x,\;p=1%
\func{mod}3)  \tag{7.2146}
\end{equation}

\subsection{Descendants of 6.261}

$4p+2+(p^{2}+3p+1)\gcd (p-1,3)+(p+1)\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac-xbaaa,caa,cac,pa,pb,pc,\,\text{class }%
4\rangle \;(x\neq -1,-\frac{1}{2})  \tag{7.2147}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac-xbaaa,caa,cac,pa-baaa,pb,pc,\,\text{class }%
4\rangle \;(x\neq -1,-\frac{1}{2})  \tag{7.2148}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac-xbaaa,caa,cac,pa,pb,pc-baaa,\,\text{class }%
4\rangle \;(x\neq -1,-\frac{1}{2})  \tag{7.2149}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac-xbaaa,caa,cac,pa,pb-baaa,pc,\,\text{class }%
4\rangle \;(x\neq -1,-\frac{1}{2})  \tag{7.2150}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac-xbaaa,caa,cac,pa,pb-\omega baaa,pc,\,\text{%
class }4\rangle \;(x\neq -1,-\frac{1}{2},\;p=1\func{mod}3)  \tag{7.2151}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac-xbaaa,caa,cac,pa,pb-\omega ^{2}baaa,pc,\,%
\text{class }4\rangle \;(x\neq -1,-\frac{1}{2},\;p=1\func{mod}3) 
\tag{7.2152}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac-xbaaa,caa,cac,pa,pb-baaa,pc-baaa,\,\text{%
class }4\rangle \;(x\neq -1,-\frac{1}{2})  \tag{7.2153}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac-xbaaa,caa,cac,pa,pb-\omega baaa,pc-baaa,\,%
\text{class }4\rangle \;(x\neq -1,-\frac{1}{2},\;p=1\func{mod}3) 
\tag{7.2154}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac-xbaaa,caa,cac,pa,pb-\omega
^{2}baaa,pc-baaa,\,\text{class }4\rangle \;(x\neq -1,-\frac{1}{2},\;p=1\func{%
mod}3)  \tag{7.2155}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac+baaa,caa,cac,pa,pb,pc,\,\text{class }%
4\rangle  \tag{7.2156}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac+baaa,caa,cac,pa-baaa,pb,pc,\,\text{class }%
4\rangle  \tag{7.2157}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac+baaa,caa,cac,pa,pb,pc-baaa,\,\text{class }%
4\rangle  \tag{7.2158}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac+baaa,caa,cac,pa-baaa,pb,pc-baaa,\,\text{%
class }4\rangle  \tag{7.2159}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac+baaa,caa,cac,pa,pb-baaa,pc,\,\text{class }%
4\rangle  \tag{7.2160}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac+baaa,caa,cac,pa,pb-\omega baaa,pc,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.2161}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac+baaa,caa,cac,pa,pb-\omega ^{2}baaa,pc,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2162}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac+baaa,caa,cac,pa,pb-baaa,pc-baaa,\,\text{%
class }4\rangle  \tag{7.2163}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac+baaa,caa,cac,pa,pb-\omega baaa,pc-baaa,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2164}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac+baaa,caa,cac,pa,pb-\omega
^{2}baaa,pc-baaa,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2165}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac+\frac{1}{2}baaa,caa,cac,pa,pb,pc,\,\text{%
class }4\rangle  \tag{7.2166}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac+\frac{1}{2}baaa,caa,cac,pa-baaa,pb,pc,\,%
\text{class }4\rangle  \tag{7.2167}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac+\frac{1}{2}baaa,caa,cac,pa,pb-baaa,pc,\,%
\text{class }4\rangle  \tag{7.2168}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac+\frac{1}{2}baaa,caa,cac,pa,pb-\omega
baaa,pc,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2169}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac+\frac{1}{2}baaa,caa,cac,pa,pb-\omega
^{2}baaa,pc,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2170}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,bac+\frac{1}{2}baaa,caa,cac,pa,pb,pc-baaa,\,%
\text{class }4\rangle  \tag{7.2171}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-baaa,bac+zbaaa,caa,cac,pa,pb,pc,\,\text{class }%
4\rangle  \tag{7.2172}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-baaa,bac+\frac{1}{2}baaa,caa,cac,pa-baaa,pb,pc,%
\,\text{class }4\rangle  \tag{7.2173}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-baaa,bac+\frac{1}{2}baaa,caa,cac,pa-\omega
baaa,pb,pc,\,\text{class }4\rangle  \tag{7.2174}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-baaa,bac+\frac{1}{2}baaa,caa,cac,pa-\omega
^{2}baaa,pb,pc,\,\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.2175}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-baaa,bac+\frac{1}{2}baaa,caa,cac,pa-\omega
^{3}baaa,pb,pc,\,\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.2176}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-baaa,bac+\frac{1}{2}baaa,caa,cac,pa,pb-baaa,pc,%
\,\text{class }4\rangle  \tag{7.2177}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-baaa,bac+zbaaa,caa,cac,pa,pb-\omega baaa,pc,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2178}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-baaa,bac+\frac{1}{2}baaa,caa,cac,pa,pb-\omega
^{2}baaa,pc,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2179}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-baaa,bac+\frac{1}{2}baaa,caa,cac,pa,pb,pc-baaa,%
\,\text{class }4\rangle  \tag{7.2180}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-baaa,bac+\frac{1}{2}baaa,caa,cac,pa,pb,pc-%
\omega baaa,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2181}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-baaa,bac+\frac{1}{2}baaa,caa,cac,pa,pb,pc-%
\omega ^{2}baaa,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.2182}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-xbaaa,bac,caa,cac,pa,pb,pc,\,\text{class }%
4\rangle \;(\text{all }x)  \tag{7.2183}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-xbaaa,bac,caa,cac,pa-baaa,pb,pc,\,\text{class }%
4\rangle \;(\text{all }x)  \tag{7.2184}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-xbaaa,bac,caa,cac,pa-\omega baaa,pb,pc,\,\text{%
class }4\rangle \;(\text{all }x)  \tag{7.2185}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-xbaaa,bac,caa,cac,pa-\omega ^{2}baaa,pb,pc,\,%
\text{class }4\rangle \;(\text{all }x,\;p=1\func{mod}4)  \tag{7.2186}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-xbaaa,bac,caa,cac,pa-\omega ^{3}baaa,pb,pc,\,%
\text{class }4\rangle \;(\text{all }x,\;p=1\func{mod}4)  \tag{7.2187}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-xbaaa,bac,caa,cac,pa,pb-baaa,pc,\,\text{class }%
4\rangle \;(\text{all }x)  \tag{7.2188}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-xbaaa,bac,caa,cac,pa,pb-\omega baaa,pc,\,\text{%
class }4\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.2189}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-xbaaa,bac,caa,cac,pa,pb-\omega ^{2}baaa,pc,\,%
\text{class }4\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.2190}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-xbaaa,bac,caa,cac,pa,pb-ybaaa,pc-baaa,\,\text{%
class }4\rangle \;(\text{all }x,\,y)  \tag{7.2191}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-xbaaa,bac,caa,cac,pa,pb-ybaaa,pc-\omega baaa,\,%
\text{class }4\rangle \;(\text{all }x,\,y,\;p=1\func{mod}3)  \tag{7.2192}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab-xbaaa,bac,caa,cac,pa,pb-ybaaa,pc-\omega
^{2}baaa,\,\text{class }4\rangle \;(\text{all }x,\,y,\;p=1\func{mod}3) 
\tag{7.2193}
\end{equation}

\subsection{Descendants of 6.267}

$p+3+3\gcd (p-1,3)+\gcd (p-1,4)+\gcd (p-1,5)$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-ba,pb,pc,\,\text{class }4\rangle  \tag{7.2194}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-ba-cacc,pb,pc,\,\text{class }4\rangle 
\tag{7.2195}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-ba-\omega cacc,pb,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.2196}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-ba-\omega ^{2}cacc,pb,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.2197}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-ba,pb,pc-cacc,\,\text{class }4\rangle 
\tag{7.2198}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-ba-cacc,pb,pc-cacc,\,\text{class }4\rangle 
\tag{7.2199}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-ba-\omega cacc,pb,pc-cacc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2200}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-ba-\omega ^{2}cacc,pb,pc-cacc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2201}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-ba,pb-cacc,pc,\,\text{class }4\rangle 
\tag{7.2202}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-cacc,pa-ba,pb-cacc,pc,\,\text{class }4\rangle 
\tag{7.2203}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-cacc,pa-ba,pb-xcacc,pc,\,\text{class }4\rangle
\;(x=\omega ^{i},\,i=1,2,3,4,\;p=1\func{mod}5)  \tag{7.2204}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-cacc,pa-ba,pb,pc,\,\text{class }4\rangle 
\tag{7.2205}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-cacc,pa-ba,pb,pc-cacc,\,\text{class }4\rangle 
\tag{7.2206}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-cacc,pa-ba,pb,pc-\omega cacc,\,\text{class }4\rangle
\tag{7.2207}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-cacc,pa-ba,pb,pc-\omega ^{2}cacc,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.2208}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-cacc,pa-ba,pb,pc-\omega ^{3}cacc,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.2209}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-cacc,pa-ba-cacc,pb,pc,\,\text{class }4\rangle 
\tag{7.2210}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-cacc,pa-ba-\omega cacc,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2211}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-cacc,pa-ba-\omega cacc,pb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2212}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-cacc,pa-ba-cacc,pb,pc-xcacc,\,\text{class }4\rangle
\;(x\neq 0,\,x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}\func{mod}p) 
\tag{7.2213}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-cacc,pa-ba-\omega cacc,pb,pc-xcacc,\,\text{class }%
4\rangle \;(x\neq 0,\,x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}\func{mod%
}p,\;p=1\func{mod}3)  \tag{7.2214}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-cacc,pa-ba-\omega cacc,pb,pc-xcacc,\,\text{class }%
4\rangle \;(x\neq 0,\,x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}\func{mod%
}p,\;p=1\func{mod}3)  \tag{7.2215}
\end{equation}

\subsection{Descendants of 6.269}

$(p+5)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-ba-cac,pb,pc,\,\text{class }4\rangle 
\tag{7.2216}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-ba-cac,pb-cacc,pc,\,\text{class }4\rangle 
\tag{7.2217}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-ba-cac,pb-xcacc,pc-cacc,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.2218}
\end{equation}

\subsection{Descendants of 6.271}

$(p+5)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-ba-\omega cac,pb,pc,\,\text{class }4\rangle 
\tag{7.2219}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-ba-\omega cac,pb-cacc,pc,\,\text{class }4\rangle
\tag{7.2220}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-ba-\omega cac,pb-xcacc,pc-cacc,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.2221}
\end{equation}

\subsection{Descendants of 6.273}

$1+\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,cac,pa-ba,pb-caa,pc,\,\text{class }4\rangle 
\tag{7.2222}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,cac-caaa,pa-ba,pb-caa,pc,\,\text{class }4\rangle 
\tag{7.2223}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,cac-\omega caaa,pa-ba,pb-caa,pc,\,\text{class }4\rangle
\tag{7.2224}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,cac-\omega ^{2}caaa,pa-ba,pb-caa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.2225}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,cac-\omega ^{3}caaa,pa-ba,pb-caa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.2226}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caaa,cac,pa-ba,pb-caa,pc,\,\text{class }4\rangle 
\tag{7.2227}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caaa,cac,pa-ba,pb-caa,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.2228}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega ^{2}caaa,cac,pa-ba,pb-caa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2229}
\end{equation}

\subsection{Descendants of 6.274}

$p+2+(2p+3)\gcd (p-1,3)+\gcd (p-1,5)$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,cac,pa-ba,pb,pc,\,\text{class }4\rangle  \tag{7.2230}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,cac,pa-ba,pb,pc-caaa,\,\text{class }4\rangle 
\tag{7.2231}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,cac,pa-ba,pb,pc-\omega caaa,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.2232}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,cac,pa-ba,pb,pc-\omega ^{2}caaa,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.2233}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,cac-caaa,pa-ba,pb,pc,\,\text{class }4\rangle 
\tag{7.2234}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,cac-caaa,pa-ba,pb,pc-caaa,\,\text{class }4\rangle 
\tag{7.2235}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,cac-caaa,pa-ba,pb,pc-\omega caaa,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2236}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,cac-caaa,pa-ba,pb,pc-\omega ^{2}caaa,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2237}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caaa,cac,pa-ba,pb,pc-xcaaa,\,\text{class }4\rangle \;(%
\text{all }x)  \tag{7.2238}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caaa,cac,pa-ba,pb,pc-xcaaa,\,\text{class }%
4\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.2239}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega ^{2}caaa,cac,pa-ba,pb,pc-xcaaa,\,\text{class }%
4\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.2240}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caaa,cac-caaa,pa-ba,pb,pc-xcaaa,\,\text{class }4\rangle
\;(\text{all }x)  \tag{7.2241}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caaa,cac-caaa,pa-ba,pb,pc-xcaaa,\,\text{class }%
4\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.2242}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega ^{2}caaa,cac-caaa,pa-ba,pb,pc-xcaaa,\,\text{%
class }4\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.2243}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,cac,pa-ba,pb-caaa,pc,\,\text{class }4\rangle 
\tag{7.2244}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caaa,cac,pa-ba,pb-caaa,pc,\,\text{class }4\rangle 
\tag{7.2245}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caaa,cac,pa-ba,pb-caaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2246}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega ^{2}caaa,cac,pa-ba,pb-caaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2247}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,cac-caaa,pa-ba,pb-caaa,pc,\,\text{class }4\rangle 
\tag{7.2248}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,cac-caaa,pa-ba,pb-xcaaa,pc,\,\text{class }4\rangle
\;(x=\omega ^{i},i=1,2,3,4,\;p=1\func{mod}5)  \tag{7.2249}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caaa,cac-caaa,pa-ba,pb-xcaaa,pc,\,\text{class }4\rangle
\;(x\neq 0,\,x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}\func{mod}p) 
\tag{7.2250}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caaa,cac-caaa,pa-ba,pb-xcaaa,pc,\,\text{class }%
4\rangle \;(x\neq 0,\,x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}\func{mod%
}p,\;p=1\func{mod}3)  \tag{7.2251}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega ^{2}caaa,cac-caaa,pa-ba,pb-xcaaa,pc,\,\text{%
class }4\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}%
\func{mod}p,\;p=1\func{mod}3)  \tag{7.2252}
\end{equation}

\subsection{Descendants of 6.275}

1 algebra

\begin{equation}
\langle a,b,c\,|\,cb,cac,pa-ba,pb,pc-caa,\,\text{class }4\rangle 
\tag{7.2253}
\end{equation}

\subsection{Descendants of 6.276}

1 algebra

\begin{equation}
\langle a,b,c\,|\,cb,cac,pa-ba,pb,pc-\omega caa,\,\text{class }4\rangle 
\tag{7.2254}
\end{equation}

\subsection{Descendants of 6.277}

$p$ algebras

\begin{equation}
\langle a,b,c\,|\,cb-caa,cac-xcaaa,pa-ba,pb-caa,pc,\,\text{class }4\rangle
\;(\text{all }x)  \tag{7.2255}
\end{equation}

\subsection{Descendants of 6.278}

$(5p+3)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,cb-caa,cac,pa-ba,pb,pc-xcaa,\,\text{class }4\rangle
\;(x\neq 0,-1,2,\frac{1}{2})  \tag{7.2256}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,cac,pa-ba,pb,pc-xcaaa,\,\text{class }4\rangle \;(%
\text{all }x,\,x\sim -x)  \tag{7.2257}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,cac,pa-ba,pb-caaa,pc-xcaaa,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.2258}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,cac,pa-ba,pb,pc+caa,\,\text{class }4\rangle 
\tag{7.2259}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,cac,pa-ba,pb-caaa,pc+caa,\,\text{class }4\rangle 
\tag{7.2260}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,cac,pa-ba,pb,pc-2caa-xcaaa,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.2261}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,cac,pa-ba,pb,pc-\frac{1}{2}caa,\,\text{class }%
4\rangle  \tag{7.2262}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,cac-caaa,pa-ba,pb,pc-\frac{1}{2}caa,\,\text{class }%
4\rangle  \tag{7.2263}
\end{equation}

\subsection{Descendants of 6.279}

$p$ algebras

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,cac-xcaaa,pa-ba,pb-caa,pc,\,\text{class }%
4\rangle \;(\text{all }x)  \tag{7.2264}
\end{equation}

\subsection{Descendants of 6.280}

$(5p+3)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,cac,pa-ba,pb,pc-xcaa,\,\text{class }4\rangle
\;(x\neq 0,-\omega ,2\omega ,\frac{\omega }{2})  \tag{7.2265}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,cac,pa-ba,pb,pc-xcaaa,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.2266}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,cac,pa-ba,pb-caaa,pc-xcaaa,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.2267}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,cac,pa-ba,pb,pc+\omega caa,\,\text{class }%
4\rangle  \tag{7.2268}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,cac,pa-ba,pb-caaa,pc+\omega caa,\,\text{%
class }4\rangle  \tag{7.2269}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,cac,pa-ba,pb,pc-2\omega caa-xcaaa,\,\text{%
class }4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.2270}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,cac,pa-ba,pb,pc-\frac{\omega }{2}caa,\,\text{%
class }4\rangle  \tag{7.2271}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega caa,cac-caaa,pa-ba,pb,pc-\frac{\omega }{2}caa,\,%
\text{class }4\rangle  \tag{7.2272}
\end{equation}

\subsection{Descendants of 6.281}

$2p^{2}+4p+4+2\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa,pb-ba,pc,\,\text{class }4\rangle  \tag{7.2273}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-cacc,pb-ba,pc,\,\text{class }4\rangle 
\tag{7.2274}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-\omega cacc,pb-ba,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.2275}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-\omega ^{2}cacc,pb-ba,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.2276}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-xcacc,pb-ba,pc-cacc,\,\text{class }4\rangle \;(%
\text{all }x,\,x\sim -x)  \tag{7.2277}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-xcacc,pb-ba,pc-\omega cacc,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.2278}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-cacc,pa-xcacc,pb-ba,pc-ycacc,\,\text{class }%
4\rangle \;(\text{all }x,\,y,x\sim -x)  \tag{7.2279}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-\omega cacc,pa-xcacc,pb-ba,pc-ycacc,\,\text{class }%
4\rangle \;(\text{all }x,\,y,\;x\sim -x)  \tag{7.2280}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa,pb-ba-cacc,pc,\,\text{class }4\rangle 
\tag{7.2281}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-cacc,pb-ba-cacc,pc,\,\text{class }4\rangle 
\tag{7.2282}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-\omega cacc,pb-ba-cacc,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2283}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-\omega ^{2}cacc,pb-ba-cacc,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2284}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-xcacc,pb-ba-cacc,pc-cacc,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.2285}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-xcacc,pb-ba-cacc,pc-\omega cacc,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.2286}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-cacc,pa-xcacc,pb-ba-cacc,pc-ycacc,\,\text{class }%
4\rangle \;(\text{all }x,\,y,x\sim -x)  \tag{7.2287}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-\omega cacc,pa-xcacc,pb-ba-cacc,pc-ycacc,\,\text{%
class }4\rangle \;(\text{all }x,\,y,\;x\sim -x)  \tag{7.2288}
\end{equation}

\subsection{Descendants of 6.282}

$p^{3}+p^{2}+2p+2+\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa,pb-ba-cac,pc,\,\text{class }4\rangle 
\tag{7.2289}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-cacc,pb-ba-cac,pc,\,\text{class }4\rangle 
\tag{7.2290}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-\omega cacc,pb-ba-cac,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.2291}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-\omega ^{2}cacc,pb-ba-cac,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2292}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-xcacc,pb-ba-cac,pc-cacc,\,\text{class }4\rangle
\;(\text{all }x,\,x\sim -x)  \tag{7.2293}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa,pa-xcacc,pb-ba-cac,pc-\omega cacc,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.2294}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-cacc,pa-xcacc,pb-ba-cac,pc-ycacc,\,\text{class }%
4\rangle \;(\text{all }x,\,y,\;x\sim -x)  \tag{7.2295}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-\omega cacc,pa-xcacc,pb-ba-cac,pc-ycacc,\,\text{%
class }4\rangle \;(\text{all }x,\,y,\;x\sim -x)  \tag{7.2296}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,caa-xcacc,pa-ycacc,pb-ba-cac-cacc,pc-zcacc,\,\text{%
class }4\rangle \;(\text{all }x,\,y,z)  \tag{7.2297}
\end{equation}

\subsection{Descendants of 6.289}

$6p$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,cac,pa,pb-ba,pc-xcaaa,\,\text{class }4\rangle \;(\text{%
all }x)  \tag{7.2298}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,cac-caaa,pa,pb-ba,pc-xcaaa,\,\text{class }4\rangle \;(%
\text{all }x)  \tag{7.2299}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caaa,cac,pa,pb-ba,pc-xcaaa,\,\text{class }4\rangle \;(%
\text{all }x)  \tag{7.2300}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caaa,cac-caaa,pa,pb-ba,pc-xcaaa,\,\text{class }4\rangle
\;(\text{all }x)  \tag{7.2301}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,cac,pa-caaa,pb-ba,pc,\,\text{class }4\rangle 
\tag{7.2302}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caaa,cac,pa-caaa,pb-ba,pc,\,\text{class }4\rangle 
\tag{7.2303}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,cac-caaa,pa-xcaaa,pb-ba,pc,\,\text{class }4\rangle \;(x
\neq 0)  \tag{7.2304}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caaa,cac-caaa,pa-xcaaa,pb-ba,pc,\,\text{class }4\rangle
\;(x \neq 0)  \tag{7.2305}
\end{equation}

\subsection{Descendants of 6.290}

$2p^{2}+p$ algebras

\begin{equation}
\langle a,b,c\,|\,cb-caa-xcaaa,cac,pa,pb-ba,pc-ycaaa,\,\text{class }4\rangle
\;(\text{all }x,\,y)  \tag{7.2306}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa-xcaaa,cac,pa-caaa,pb-ba,pc,\,\text{class }4\rangle
\;(\text{all }x)  \tag{7.2307}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-caa,cac-caaa,pa-xcaaa,pb-ba,pc-ycaaa,\,\text{class }%
4\rangle \;(\text{all }x,\,y)  \tag{7.2308}
\end{equation}

\subsection{Descendants of 6.294}

$p+4+5\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa,pb-ca,pc-babb,\,\text{class }4\rangle 
\tag{7.2309}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa,pb-ca,pc-\omega babb,\,\text{class }4\rangle 
\tag{7.2310}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa,pb-ca,pc,\,\text{class }4\rangle  \tag{7.2311}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-babb,pb-ca,pc,\,\text{class }4\rangle 
\tag{7.2312}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-\omega babb,pb-ca,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.2313}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-\omega ^{2}babb,pb-ca,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.2314}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa,pb-ca-babb,pc,\,\text{class }4\rangle 
\tag{7.2315}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-babb,pb-ca-babb,pc,\,\text{class }4\rangle 
\tag{7.2316}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-\omega babb,pb-ca-babb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2317}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-\omega ^{2}babb,pb-ca-babb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2318}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,pa,pb-ca,pc-babb,\,\text{class }4\rangle 
\tag{7.2319}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,pa,pb-ca,pc-\omega babb,\,\text{class }4\rangle
\tag{7.2320}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,pa,pb-ca,pc-xbabb,\,\text{class }4\rangle
\;(x=\omega ^{i},\;i=2,3,4,5,\;p=1\func{mod}3)  \tag{7.2321}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,pa,pb-ca,pc,\,\text{class }4\rangle 
\tag{7.2322}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,pa,pb-ca-babb,pc,\,\text{class }4\rangle 
\tag{7.2323}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,pa,pb-ca-\omega babb,pc,\,\text{class }4\rangle
\tag{7.2324}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,pa,pb-ca-\omega ^{2}babb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.2325}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,pa,pb-ca-\omega ^{3}babb,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.2326}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,pa-babb,pb-ca,pc,\,\text{class }4\rangle 
\tag{7.2327}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,pa-\omega babb,pb-ca,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2328}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,pa-\omega ^{2}babb,pb-ca,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2329}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,pa-babb,pb-ca-xbabb,pc,\,\text{class }4\rangle
\;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}\func{mod}p) 
\tag{7.2330}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,pa-\omega babb,pb-ca-xbabb,pc,\,\text{class }%
4\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}\func{mod}%
p,\;p=1\func{mod}3)  \tag{7.2331}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa-babb,pa-\omega ^{2}babb,pb-ca-xbabb,pc,\,\text{%
class }4\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}%
\func{mod}p,\;p=1\func{mod}3)  \tag{7.2332}
\end{equation}

\subsection{Descendants of 6.295}

$(p+5)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-bab-xbabb,pb-ca,pc,\,\text{class }4\rangle \;(%
\text{all }x,\,x\sim -x)  \tag{7.2333}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-bab,pb-ca,pc-babb,\,\text{class }4\rangle 
\tag{7.2334}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-bab,pb-ca,pc-\omega babb,\,\text{class }4\rangle
\tag{7.2335}
\end{equation}

\subsection{Descendants of 6.296}

$(p+5)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-\omega bab-xbabb,pb-ca,pc,\,\text{class }%
4\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.2336}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-\omega bab,pb-ca,pc-babb,\,\text{class }4\rangle
\tag{7.2337}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-\omega bab,pb-ca,pc-\omega babb,\,\text{class }%
4\rangle  \tag{7.2338}
\end{equation}

\subsection{Descendants of 6.297}

$2p+p\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa,pb-ca-bab,pc-xbabb,\,\text{class }4\rangle \;(%
\text{all }x)  \tag{7.2339}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-babb,pb-ca-bab,pc-xbabb,\,\text{class }4\rangle
\;(\text{all }x)  \tag{7.2340}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-\omega babb,pb-ca-bab,pc-xbabb,\,\text{class }%
4\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.2341}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-\omega ^{2}babb,pb-ca-bab,pc-xbabb,\,\text{class 
}4\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.2342}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-xbabb,pb-ca-bab-babb,pc+2babb,\,\text{class }%
4\rangle \;(\text{all }x)  \tag{7.2343}
\end{equation}

\subsection{Descendants of 6.298}

$(3p-1)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-bab,pb-ca-bab,pc-xbabb,\,\text{class }4\rangle
\;(\text{all }x)  \tag{7.2344}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-bab-xbabb,pb-ca-bab,pc-2babb,\,\text{class }%
4\rangle \;(x \neq 0, x \sim -x)  \tag{7.2345}
\end{equation}

\subsection{Descendants of 6.299}

$(3p-1)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-\omega bab,pb-ca-bab,pc-xbabb,\,\text{class }%
4\rangle \;(\text{all }x)  \tag{7.2346}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,baa,pa-\omega bab-xbabb,pb-ca-bab,pc-2babb,\,\text{%
class }4\rangle \;(x\neq 0,x\sim -x)  \tag{7.2347}
\end{equation}

\subsection{Descendants of 6.303}

$p^{2}+p+(p+1)\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,bab-baa,pa,pb-ca,pc,\,\text{class }4\rangle 
\tag{7.2348}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baa,pa,pb-ca-baaa,pc,\,\text{class }4\rangle 
\tag{7.2349}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baa,pa,pb-ca-\omega baaa,pc,\,\text{class }4\rangle
\;(p=1\func{mod}3)  \tag{7.2350}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baa,pa,pb-ca-\omega ^{2}baaa,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}3)  \tag{7.2351}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baa,pa-baaa,pb-ca-xbaaa,pc,\,\text{class }4\rangle
\;(\text{all }x)  \tag{7.2352}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baa,pa-\omega baaa,pb-ca-xbaaa,pc,\,\text{class }%
4\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.2353}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baa,pa-\omega ^{2}baaa,pb-ca-xbaaa,pc,\,\text{class 
}4\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.2354}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baa-baaa,pa-xbaaa,pb-ca-ybaaa,pc,\,\text{class }%
4\rangle \;(\text{all }x,\,y)  \tag{7.2355}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baa,pa,pb-ca,pc-baaa,\,\text{class }4\rangle 
\tag{7.2356}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baa,pa,pb-ca,pc-\omega baaa,\,\text{class }4\rangle
\tag{7.2357}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baa,pa,pb-ca,pc-\omega ^{2}baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.2358}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baa,pa,pb-ca,pc-\omega ^{3}baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.2359}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baa-baaa,pa,pb-ca,pc-xbaaa,\,\text{class }4\rangle
\;(x \neq 0)  \tag{7.2360}
\end{equation}

\subsection{Descendants of 6.304}

$p(p+1)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,bab-baa,pa,pb-ca-baa-ybaaa,pc-xbaaa,\,\text{class }%
4\rangle \;(x\neq 0,\text{ all }y,\;y\sim -y)  \tag{7.2361}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baa,pa-ybaaa,pb-ca-baa,pc,\,\text{class }4\rangle
\;(\text{all }y,\;y\sim -y)  \tag{7.2362}
\end{equation}

\subsection{Descendants of 6.305}

$p(p+1)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,bab-baa,pa,pb-ca-\omega baa-ybaaa,pc-xbaaa,\,\text{%
class }4\rangle \;(x\neq 0,\text{ all }y,y\sim -y)  \tag{7.2363}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baa,pa-ybaaa,pb-ca-\omega baa,pc,\,\text{class }%
4\rangle \;(\text{all }y,\;y\sim -y)  \tag{7.2364}
\end{equation}

\subsection{Descendants of 6.312}

$4p+2+2\gcd (p-1,3)+4\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,cb,bab,pa,pb-ca,pc,\,\text{class }4\rangle  \tag{7.2365}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,pa-baaa,pb-ca,pc,\,\text{class }4\rangle 
\tag{7.2366}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab,pa,pb-ca,pc,\,\text{class }4\rangle 
\tag{7.2367}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab,pa-baaa,pb-ca,pc,\,\text{class }4\rangle 
\tag{7.2368}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab,pa-\omega baaa,pb-ca,pc,\,\text{class }4\rangle
\tag{7.2369}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab,pa-xbaaa,pb-ca,pc,\,\text{class }4\rangle
\;(x=\omega ^{i},\;i=2,3,4,5,\;p=1\func{mod}3)  \tag{7.2370}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,pa,pb-ca,pc,\,\text{class }4\rangle 
\tag{7.2371}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,pa-baaa,pb-ca,pc,\,\text{class }4\rangle 
\tag{7.2372}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,pa-\omega baaa,pb-ca,pc,\,\text{class }4\rangle
\tag{7.2373}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,pa-\omega ^{2}baaa,pb-ca,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.2374}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,pa-\omega ^{3}baaa,pb-ca,pc,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.2375}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab-baaa,pa-xbaaa,pb-ca,pc,\,\text{class }4\rangle
\;(\text{all }x)  \tag{7.2376}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega baaa,bab-baaa,pa-xbaaa,pb-ca,pc,\,\text{class }%
4\rangle \;(\text{all }x)  \tag{7.2377}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,pa,pb-ca,pc-baaa,\,\text{class }4\rangle 
\tag{7.2378}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,pa,pb-ca,pc-\omega baaa,\,\text{class }4\rangle 
\tag{7.2379}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,pa,pb-ca,pc-\omega ^{2}baaa,\,\text{class }4\rangle
\;(p=1\func{mod}4)  \tag{7.2380}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab,pa,pb-ca,pc-\omega ^{3}baaa,\,\text{class }4\rangle
\;(p=1\func{mod}4)  \tag{7.2381}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab,pa,pb-ca,pc-baaa,\,\text{class }4\rangle 
\tag{7.2382}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab,pa,pb-ca,pc-\omega baaa,\,\text{class }4\rangle
\tag{7.2383}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab,pa,pb-ca,pc-\omega ^{2}baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.2384}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab,pa,pb-ca,pc-\omega ^{3}baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.2385}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,pa,pb-ca,pc-baaa,\,\text{class }4\rangle 
\tag{7.2386}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,pa,pb-ca,pc-\omega baaa,\,\text{class }4\rangle
\tag{7.2387}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,pa,pb-ca,pc-\omega ^{2}baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.2388}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb,bab-baaa,pa,pb-ca,pc-\omega ^{3}baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.2389}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baaa,bab-baaa,pa,pb-ca,pc-xbaaa,\,\text{class }4\rangle
\;(x \neq 0)  \tag{7.2390}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-\omega baaa,bab-baaa,pa,pb-ca,pc-xbaaa,\,\text{class }%
4\rangle \;(x\neq 0)  \tag{7.2391}
\end{equation}

\subsection{Descendants of 6.313}

$p+\gcd (p-1,4)+\gcd (p-1,5)$ algebras

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,pa,pb-ca,pc,\,\text{class }4\rangle 
\tag{7.2392}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,pa,pb-ca,pc-baaa,\,\text{class }4\rangle 
\tag{7.2393}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,pa,pb-ca,pc-\omega baaa,\,\text{class }4\rangle
\tag{7.2394}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,pa,pb-ca,pc-\omega ^{2}baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.2395}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,pa,pb-ca,pc-\omega ^{3}baaa,\,\text{class }%
4\rangle \;(p=1\func{mod}4)  \tag{7.2396}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,pa-baaa,pb-ca,pc,\,\text{class }4\rangle 
\tag{7.2397}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,pa-ybaaa,pb-ca,pc,\,\text{class }4\rangle
\;(y=\omega ^{i},\;i=1,2,3,4,\;p=1\func{mod}5)  \tag{7.2398}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,pa-baaa,pb-ca,pc-xbaaa,\,\text{class }4\rangle
\;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod}p) 
\tag{7.2399}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cb-baa,bab,pa-ybaaa,pb-ca,pc-xbaaa,\,\text{class }4\rangle
\;(y=\omega ^{i},i=1,2,3,4,\;x\neq 0,\;x\sim x^{\prime }\text{ if }%
x^{5}=x^{\prime 5}\func{mod}p,\;p=1\func{mod}5)  \tag{7.2400}
\end{equation}

\subsection{Descendants of 6.322}

3 algebras

\begin{equation}
\langle a,b,c\,|\,ba,ca,cb,pb,pc,\,\text{class }5\rangle  \tag{7.2401}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba-p^4a,ca,cb,pb,pc,\,\text{class }5\rangle  \tag{7.2402}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba,ca,cb-p^4a,pb,pc,\,\text{class }5\rangle  \tag{7.2403}
\end{equation}

\subsection{Descendants of 6.325}

$19+5\gcd (p-1,3)+6\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb,pa,pb,pc,\,\text{class }5\rangle 
\tag{7.2404}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb,pa-baaab,pb,pc,\,\text{class }5\rangle 
\tag{7.2405}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb,pa-\omega baaab,pb,pc,\,\text{class }%
5\rangle  \tag{7.2406}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb,pa,pb-baaab,pc,\,\text{class }5\rangle 
\tag{7.2407}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb,pa-baaab,pb-baaab,pc,\,\text{class }%
5\rangle  \tag{7.2408}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb,pa-\omega baaab,pb-baaab,pc,\,\text{class }%
5\rangle  \tag{7.2409}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb,pa-\omega ^{2}baaab,pb-baaab,pc,\,\text{%
class }5\rangle \;(p=1\func{mod}4)  \tag{7.2410}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb,pa-\omega ^{3}baaab,pb-baaab,pc,\,\text{%
class }5\rangle \;(p=1\func{mod}4)  \tag{7.2411}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb,pa,pb,pc-baaab,\,\text{class }5\rangle 
\tag{7.2412}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb,pa,pb,pc,\,\text{class }5\rangle 
\tag{7.2413}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb,pa-baaab,pb,pc,\,\text{class }%
5\rangle  \tag{7.2414}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb,pa-\omega baaab,pb,pc,\,\text{class }%
5\rangle  \tag{7.2415}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb,pa,pb-baaab,pc,\,\text{class }%
5\rangle  \tag{7.2416}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb,pa-baaab,pb-baaab,pc,\,\text{class }%
5\rangle  \tag{7.2417}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb,pa-\omega baaab,pb-baaab,pc,\,\text{%
class }5\rangle  \tag{7.2418}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb,pa-\omega ^{2}baaab,pb-baaab,pc,\,%
\text{class }5\rangle \;(p=1\func{mod}4)  \tag{7.2419}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb,pa-\omega ^{3}baaab,pb-baaab,pc,\,%
\text{class }5\rangle \;(p=1\func{mod}4)  \tag{7.2420}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb,pa,pb,pc-baaab,\,\text{class }%
5\rangle  \tag{7.2421}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb,pa,pb,pc,\,\text{class }5\rangle 
\tag{7.2422}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb,pa-baaaa,pb,pc,\,\text{class }5\rangle 
\tag{7.2423}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb,pa,pb-baaaa,pc,\,\text{class }5\rangle 
\tag{7.2424}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb,pa,pb-\omega baaaa,pc,\,\text{class }%
5\rangle  \tag{7.2425}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb,pa,pb-\omega ^{2}baaaa,pc,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.2426}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb,pa,pb-\omega ^{3}baaaa,pc,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.2427}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb,pa,pb,pc-baaaa,\,\text{class }5\rangle 
\tag{7.2428}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaaa,pa,pb,pc,\,\text{class }5\rangle 
\tag{7.2429}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaaa,pa-baaaa,pb,pc,\,\text{class }%
5\rangle  \tag{7.2430}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaaa,pa,pb-baaaa,pc,\,\text{class }%
5\rangle  \tag{7.2431}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaaa,pa,pb-\omega baaaa,pc,\,\text{class }%
5\rangle  \tag{7.2432}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaaa,pa,pb-\omega ^{2}baaaa,pc,\,\text{%
class }5\rangle \;(p=1\func{mod}4)  \tag{7.2433}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaaa,pa,pb-\omega ^{3}baaaa,pc,\,\text{%
class }5\rangle \;(p=1\func{mod}4)  \tag{7.2434}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaaa,pa,pb,pc-baaaa,\,\text{class }%
5\rangle  \tag{7.2435}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb,pa,pb,pc,\,\text{class }5\rangle 
\tag{7.2436}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb,pa-baaaa,pb,pc,\,\text{class }%
5\rangle  \tag{7.2437}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb,pa-\omega baaaa,pb,pc,\,\text{class }%
5\rangle  \tag{7.2438}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb,pa-xbaaaa,pb,pc,\,\text{class }%
5\rangle \;(x=\omega ^{i},\;i=2,3,4,5,\;p=1\func{mod}3)  \tag{7.2439}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb,pa,pb-baaaa,pc,\,\text{class }%
5\rangle  \tag{7.2440}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb,pa,pb-\omega baaaa,pc,\,\text{class }%
5\rangle  \tag{7.2441}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb,pa,pb-\omega ^{2}baaaa,pc,\,\text{%
class }5\rangle \;(p=1\func{mod}4)  \tag{7.2442}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb,pa,pb-\omega ^{3}baaaa,pc,\,\text{%
class }5\rangle \;(p=1\func{mod}4)  \tag{7.2443}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb,pa,pb,pc-baaaa,\,\text{class }%
5\rangle  \tag{7.2444}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb-baaaa,pa,pb,pc,\,\text{class }%
5\rangle  \tag{7.2445}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb-baaaa,pa-baaaa,pb,pc,\,\text{class }%
5\rangle  \tag{7.2446}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb-baaaa,pa-\omega baaaa,pb,pc,\,\text{%
class }5\rangle  \tag{7.2447}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb-baaaa,pa-xbaaaa,pb,pc,\,\text{class }%
5\rangle \;(x=\omega ^{i},\;i=2,3,4,5,\;p=1\func{mod}3)  \tag{7.2448}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb-baaaa,pa,pb-baaaa,pc,\,\text{class }%
5\rangle  \tag{7.2449}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb-baaaa,pa,pb-\omega baaaa,pc,\,\text{%
class }5\rangle  \tag{7.2450}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb-baaaa,pa,pb-\omega ^{2}baaaa,pc,\,%
\text{class }5\rangle \;(p=1\func{mod}4)  \tag{7.2451}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb-baaaa,pa,pb-\omega ^{3}baaaa,pc,\,%
\text{class }5\rangle \;(p=1\func{mod}4)  \tag{7.2452}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb-baaaa,pa,pb,pc-baaaa,\,\text{class }%
5\rangle  \tag{7.2453}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb-baaaa,pa,pb,pc-\omega baaaa,\,\text{%
class }5\rangle \;(p=1\func{mod}3)  \tag{7.2454}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaaa,ca,cb-baaaa,pa,pb,pc-\omega ^{2}baaaa,\,%
\text{class }5\rangle \;(p=1\func{mod}3)  \tag{7.2455}
\end{equation}

\subsection{Descendants of 6.326}

$4p+5+4\gcd (p-1,3)+2\gcd (p-1,4)+4\gcd (p-1,5)$ algebras

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb,pa,pb,pc,\,\text{class }5\rangle 
\tag{7.2456}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb,pa-baaab,pb,pc,\,\text{class }5\rangle
\tag{7.2457}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb,pa-\omega baaab,pb,pc,\,\text{class }%
5\rangle  \tag{7.2458}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb,pa-xbaaab,pb,pc,\,\text{class }%
5\rangle \;(x=\omega ^{i},\;i=2,3,4,5,\;p=1\func{mod}3)  \tag{7.2459}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb,pa,pb-baaab,pc,\,\text{class }5\rangle
\tag{7.2460}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb,pa,pb-ybaaab,pc,\,\text{class }%
5\rangle \;(y=\omega ^{i},\;i=1,2,3,4,\;p=1\func{mod}5)  \tag{7.2461}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb,pa-xbaaab,pb-baaab,pc,\,\text{class }%
5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}\func{mod%
}p)  \tag{7.2462}
\end{equation}

In the following presentation we have $\;y=\omega ^{i},\;i=1,2,3,4,\;x\neq
0,\;x\sim x^{\prime }$ if $x^{5}=x^{\prime 5}\func{mod}p,\;p=1\func{mod}5$:

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb,pa-xbaaab,pb-ybaaab,pc,\,\text{class }%
5\rangle  \tag{7.2463}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb,pa,pb,pc-baaab,\,\text{class }5\rangle
\tag{7.2464}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-baaab,cb,pa,pb,pc,\,\text{class }5\rangle
\tag{7.2465}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-baaab,cb,pa-baaab,pb,pc,\,\text{class }%
5\rangle  \tag{7.2466}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-baaab,cb,pa-\omega baaab,pb,pc,\,\text{%
class }5\rangle  \tag{7.2467}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-baaab,cb,pa-xbaaab,pb,pc,\,\text{class }%
5\rangle \;(x=\omega ^{i},\;i=2,3,4,5,\;p=1\func{mod}3)  \tag{7.2468}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-baaab,cb,pa,pb-baaab,pc,\,\text{class }%
5\rangle  \tag{7.2469}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-baaab,cb,pa,pb-xbaaab,pc,\,\text{class }%
5\rangle \;(x=\omega ^{i},\;i=1,2,3,4,\;p=1\func{mod}5)  \tag{7.2470}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-baaab,cb,pa-xbaaab,pb-baaab,pc,\,\text{%
class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}%
\func{mod}p)  \tag{7.2471}
\end{equation}

In the following presentation we have $\;y=\omega ^{i},\;i=1,2,3,4,\;x\neq
0,\;x\sim x^{\prime }$ if $x^{5}=x^{\prime 5}\func{mod}p,\;p=1\func{mod}5$:

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-baaab,cb,pa-xbaaab,pb-ybaaab,pc,\,\text{%
class }5\rangle  \tag{7.2472}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-baaab,cb,pa,pb,pc-baaab,\,\text{class }%
5\rangle  \tag{7.2473}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb,pa,pb,pc,\,\text{class }5\rangle 
\tag{7.2474}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb,pa-baaaa,pb,pc,\,\text{class }5\rangle
\tag{7.2475}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb,pa-xbaaaa,pb,pc,\,\text{class }%
5\rangle \;(x=\omega ^{i},\;i=1,2,3,4,\;p=1\func{mod}5)  \tag{7.2476}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb,pa,pb-baaaa,pc,\,\text{class }5\rangle
\tag{7.2477}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb,pa,pb-\omega baaaa,pc,\,\text{class }%
5\rangle  \tag{7.2478}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb,pa,pb-\omega ^{2}baaaa,pc,\,\text{%
class }5\rangle \;(p=1\func{mod}4)  \tag{7.2479}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb,pa,pb-\omega ^{3}baaaa,pc,\,\text{%
class }5\rangle \;(p=1\func{mod}4)  \tag{7.2480}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb,pa-xbaaaa,pb-baaaa,pc,\,\text{class }%
5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{4}=x^{\prime 4}\func{mod%
}p)  \tag{7.2481}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb,pa-xbaaaa,pb-\omega baaaa,pc,\,\text{%
class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{4}=x^{\prime 4}%
\func{mod}p)  \tag{7.2482}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb,pa-xbaaaa,pb-\omega ^{2}baaaa,pc,\,%
\text{class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }%
x^{4}=x^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.2483}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb,pa-xbaaaa,pb-\omega ^{3}baaaa,pc,\,%
\text{class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }%
x^{4}=x^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.2484}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb,pa,pb,pc-baaaa,\,\text{class }5\rangle
\tag{7.2485}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaaa,pa,pb,pc,\,\text{class }5\rangle
\tag{7.2486}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaaa,pa-baaaa,pb,pc,\,\text{class }%
5\rangle  \tag{7.2487}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaaa,pa-xbaaaa,pb,pc,\,\text{class }%
5\rangle \;(x=\omega ^{i},\;i=1,2,3,4,\;p=1\func{mod}5)  \tag{7.2488}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaaa,pa,pb-baaaa,pc,\,\text{class }%
5\rangle  \tag{7.2489}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaaa,pa,pb-\omega baaaa,pc,\,\text{%
class }5\rangle  \tag{7.2490}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaaa,pa,pb-\omega ^{2}baaaa,pc,\,%
\text{class }5\rangle \;(p=1\func{mod}4)  \tag{7.2491}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaaa,pa,pb-\omega ^{3}baaaa,pc,\,%
\text{class }5\rangle \;(p=1\func{mod}4)  \tag{7.2492}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaaa,pa-xbaaaa,pb-baaaa,pc,\,\text{%
class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{4}=x^{\prime 4}%
\func{mod}p)  \tag{7.2493}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaaa,pa-xbaaaa,pb-\omega baaaa,pc,\,%
\text{class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }%
x^{4}=x^{\prime 4}\func{mod}p)  \tag{7.2494}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaaa,pa-xbaaaa,pb-\omega
^{2}baaaa,pc,\,\text{class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if 
}x^{4}=x^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.2495}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaaa,pa-xbaaaa,pb-\omega
^{3}baaaa,pc,\,\text{class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if 
}x^{4}=x^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.2496}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaaa,pa,pb,pc-baaaa,\,\text{class }%
5\rangle  \tag{7.2497}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaaa,pa,pb,pc-\omega baaaa,\,\text{%
class }5\rangle  \tag{7.2498}
\end{equation}

\subsection{Descendants of 6.327}

$3p+12+2\gcd (p-1,3)+3\gcd (p-1,4)+\gcd (p-1,7)$ algebras

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb-baaa,pa,pb,pc,\,\text{class }5\rangle 
\tag{7.2499}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb-baaa,pa-baaab,pb,pc,\,\text{class }5\rangle
\tag{7.2500}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb-baaa,pa-\omega baaab,pb,pc,\,\text{class }%
5\rangle  \tag{7.2501}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb-baaa,pa,pb-baaab,pc,\,\text{class }5\rangle
\tag{7.2502}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb-baaa,pa-baaab,pb-baaab,pc,\,\text{class }%
5\rangle  \tag{7.2503}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb-baaa,pa-\omega baaab,pb-baaab,pc,\,\text{%
class }5\rangle  \tag{7.2504}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb-baaa,pa-\omega ^{2}baaab,pb-baaab,pc,\,%
\text{class }5\rangle \;(p=1\func{mod}4)  \tag{7.2505}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb-baaa,pa-\omega ^{3}baaab,pb-baaab,pc,\,%
\text{class }5\rangle \;(p=1\func{mod}4)  \tag{7.2506}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb-baaa,pa,pb,pc-baaab,\,\text{class }5\rangle
\tag{7.2507}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca,cb-baaa,pa,pb,pc-\omega baaab,\,\text{class }%
5\rangle  \tag{7.2508}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb-baaa,pa,pb,pc,\,\text{class }5\rangle
\tag{7.2509}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb-baaa,pa-baaab,pb,pc,\,\text{class }%
5\rangle  \tag{7.2510}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb-baaa,pa-\omega baaab,pb,pc,\,\text{%
class }5\rangle  \tag{7.2511}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb-baaa,pa-xbaaab,pb,pc,\,\text{class }%
5\rangle \;(x=\omega ^{i},\;i=2,3,4,5,\;p=1\func{mod}3)  \tag{7.2512}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb-baaa,pa,pb-baaab,pc,\,\text{class }%
5\rangle  \tag{7.2513}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb-baaa,pa,pb-xbaaab,pc,\,\text{class }%
5\rangle \;(x=\omega ^{i},\;i=1,2,3,4,5,6,\;p=1\func{mod}7)  \tag{7.2514}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb-baaa,pa-xbaaab,pb-baaab,pc,\,\text{%
class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{7}=x^{\prime 7}%
\func{mod}p)  \tag{7.2515}
\end{equation}

In the following presentation we have $\;y=\omega
^{i},\;i=1,2,3,4,5,6,\;x\neq 0,\;x\sim x^{\prime }$ if $x^{7}=x^{\prime 7}%
\func{mod}p,\;p=1\func{mod}7$:

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb-baaa,pa-xbaaab,pb-ybaaab,pc,\,\text{%
class }5\rangle  \tag{7.2516}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb-baaa,pa,pb,pc-baaab,\,\text{class }%
5\rangle  \tag{7.2517}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab,ca-baaab,cb-baaa,pa,pb,pc-\omega baaab,\,\text{%
class }5\rangle  \tag{7.2518}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaa,pa,pb,pc,\,\text{class }5\rangle 
\tag{7.2519}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaa,pa,pb-baaaa,pc,\,\text{class }5\rangle
\tag{7.2520}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaa,pa,pb-\omega baaaa,pc,\,\text{class }%
5\rangle  \tag{7.2521}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaa,pa,pb-\omega ^{2}baaaa,pc,\,\text{%
class }5\rangle \;(p=1\func{mod}4)  \tag{7.2522}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaa,pa,pb-\omega ^{3}baaaa,pc,\,\text{%
class }5\rangle \;(p=1\func{mod}4)  \tag{7.2523}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaa,pa,pb,pc-baaaa,\,\text{class }5\rangle
\tag{7.2524}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaa,pa,pb-baaaa,pc-baaaa,\,\text{class }%
5\rangle  \tag{7.2525}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaa,pa,pb-\omega baaaa,pc-baaaa,\,\text{%
class }5\rangle  \tag{7.2526}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaa,pa,pb-\omega ^{2}baaaa,pc-baaaa,\,%
\text{class }5\rangle \;(p=1\func{mod}4)  \tag{7.2527}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaa,pa,pb-\omega ^{3}baaaa,pc-baaaa,\,%
\text{class }5\rangle \;(p=1\func{mod}4)  \tag{7.2528}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaa,pa-baaaa,pb,pc,\,\text{class }5\rangle
\tag{7.2529}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaa-baaaa,pa,pb-xbaaaa,pc,\,\text{class }%
5\rangle \;(\text{all }x)  \tag{7.2530}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaa-baaaa,pa,pb-xbaaaa,pc-baaaa,\,\text{%
class }5\rangle \;(\text{all }x)  \tag{7.2531}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab,ca,cb-baaa-baaaa,pa-baaaa,pb,pc,\,\text{class }%
5\rangle  \tag{7.2532}
\end{equation}

\subsection{Descendants of 6.328}

$p^{2}+3p-2+(p+3)\gcd (p-1,3)+2\gcd (p-1,4)+2\gcd (p-1,5)$ algebras

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb-baaa,pa,pb,pc,\,\text{class }5\rangle
\tag{7.2533}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb-baaa,pa-baaab,pb,pc,\,\text{class }%
5\rangle  \tag{7.2534}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb-baaa,pa-\omega baaab,pb,pc,\,\text{%
class }5\rangle  \tag{7.2535}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb-baaa,pa-xbaaab,pb,pc,\,\text{class }%
5\rangle \;(x=\omega ^{i},\;i=2,3,4,5,\;p=1\func{mod}3)  \tag{7.2536}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb-baaa,pa,pb-baaab,pc,\,\text{class }%
5\rangle  \tag{7.2537}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb-baaa,pa,pb-xbaaab,pc,\,\text{class }%
5\rangle \;(x=\omega ^{i},\;i=1,2,3,4,\;p=1\func{mod}5)  \tag{7.2538}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb-baaa,pa-xbaaab,pb-baaab,pc,\,\text{%
class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}%
\func{mod}p)  \tag{7.2539}
\end{equation}

In the following presentation we have $\;y=\omega ^{i},\;i=1,2,3,4,\;x\neq
0,\;x\sim x^{\prime }$ if $x^{5}=x^{\prime 5}\func{mod}p,\;p=1\func{mod}5$:

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb-baaa,pa-xbaaab,pb-ybaaab,pc,\,\text{%
class }5\rangle  \tag{7.2540}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb-baaa,pa,pb,pc-baaab,\,\text{class }%
5\rangle  \tag{7.2541}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb-baaa,pa,pb,pc-\omega baaab,\,\text{%
class }5\rangle  \tag{7.2542}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb-baaa,pa,pb,pc-\omega ^{2}baaab,\,%
\text{class }5\rangle \;(p=1\func{mod}4)  \tag{7.2543}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca,cb-baaa,pa,pb,pc-\omega ^{3}baaab,\,%
\text{class }5\rangle \;(p=1\func{mod}4)  \tag{7.2544}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-baaab,cb-baaa,pa-xbaaab,pb,pc,\,\text{%
class }5\rangle \;(\text{all }x)  \tag{7.2545}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-baaab,cb-baaa,pa-xbaaab,pb-ybaaab,pc,\,%
\text{class }5\rangle \;(\text{all }x,\,y\neq 0,\;y\sim y^{\prime }\text{ if 
}y^{3}=y^{\prime 3}\func{mod}p)  \tag{7.2546}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-baaab,cb-baaa,pa,pb,pc-ybaaab,\,\text{%
class }5\rangle \;(y\neq 0,\;y\sim y^{\prime }\text{ if }y^{3}=y^{\prime 3}%
\func{mod}p)  \tag{7.2547}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-\omega baaab,cb-baaa,pa-xbaaab,pb,pc,\,%
\text{class }5\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.2548}
\end{equation}

In the next presentation we have $\;$all $x,\,y\neq 0,\;y\sim y^{\prime }$
if $y^{3}=y^{\prime 3}\func{mod}p,\;p=1\func{mod}3$:

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-\omega
baaab,cb-baaa,pa-xbaaab,pb-ybaaab,pc,\,\text{class }5\rangle \;(\text{all }%
x,\,y\neq 0,\;y\sim y^{\prime }\text{ if }y^{3}=y^{\prime 3}\func{mod}p,\;p=1%
\func{mod}3)  \tag{7.2549}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-\omega baaab,cb-baaa,pa,pb,pc-ybaaab,\,%
\text{class }5\rangle \;(y\neq 0,\;y\sim y^{\prime }\text{ if }%
y^{3}=y^{\prime 3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.2550}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-\omega
^{2}baaab,cb-baaa,pa-xbaaab,pb,pc,\,\text{class }5\rangle \;(\text{all }%
x,\;p=1\func{mod}3)  \tag{7.2551}
\end{equation}

In the next presentation we have $\;$all $x,\,y\neq 0,\;y\sim y^{\prime }$
if $y^{3}=y^{\prime 3}\func{mod}p,\;p=1\func{mod}3$:

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-\omega
^{2}baaab,cb-baaa,pa-xbaaab,pb-ybaaab,pc,\,\text{class }5\rangle 
\tag{7.2552}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaaa,bab-baaa,ca-\omega
^{2}baaab,cb-baaa,pa,pb,pc-ybaaab,\,\text{class }5\rangle \;(y\neq 0,y\sim
y^{\prime}\text{ if }y^3 =y^{\prime 3}\func{mod}p,\;p=1\func{mod}3) 
\tag{7.2553}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaa,pa,pb,pc,\,\text{class }5\rangle
\tag{7.2554}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaa,pa,pb-baaaa,pc,\,\text{class }%
5\rangle  \tag{7.2555}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaa,pa,pb-\omega baaaa,pc,\,\text{%
class }5\rangle  \tag{7.2556}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaa,pa,pb-\omega ^{2}baaaa,pc,\,%
\text{class }5\rangle \;(p=1\func{mod}4)  \tag{7.2557}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaa,pa,pb-\omega ^{3}baaaa,pc,\,%
\text{class }5\rangle \;(p=1\func{mod}4)  \tag{7.2558}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaa,pa-baaaa,pb,pc,\,\text{class }%
5\rangle  \tag{7.2559}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaa,pa-ybaaaa,pb,pc,\,\text{class }%
5\rangle \;(y=\omega ^{i},\;i=1,2,3,4,\;p=1\func{mod}5)  \tag{7.2560}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaa,pa-baaaa,pb-xbaaaa,pc,\,\text{%
class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{5}=x^{\prime 5}%
\func{mod}p)  \tag{7.2561}
\end{equation}

In the next presentation we have $\;y=\omega ^{i},\;i=1,2,3,4,\;x\neq
0,\;x\sim x^{\prime }$ if $x^{5}=x^{\prime 5}\func{mod}p,\;p=1\func{mod}5$:

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaa,pa-ybaaaa,pb-xbaaaa,pc,\,\text{%
class }5\rangle  \tag{7.2562}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaa,pa,pb,pc-baaaa,\,\text{class }%
5\rangle  \tag{7.2563}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaa,pa,pb,pc-\omega baaaa,\,\text{%
class }5\rangle \;(p=1\func{mod}3)  \tag{7.2564}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaa,pa,pb,pc-\omega ^{2}baaaa,\,%
\text{class }5\rangle \;(p=1\func{mod}3)  \tag{7.2565}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaa,pa,pb-xbaaaa,pc-baaaa,\,\text{%
class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}%
\func{mod}p)  \tag{7.2566}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaa,pa,pb-xbaaaa,pc-\omega baaaa,\,%
\text{class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }%
x^{3}=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.2567}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baaab,bab-baaa,ca,cb-baaa,pa,pb-xbaaaa,pc-\omega
^{2}baaaa,\,\text{class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }%
x^{3}=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.2568}
\end{equation}

\subsection{Descendants of 6.362}

$p^{2}+7p+3+2\gcd (p-1,3)+3\gcd (p-1,4)+\gcd (p-1,5)$ algebras

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb,pc,\,\text{class }5\rangle 
\tag{7.2569}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb-baaaa,pc,\,\text{class }5\rangle 
\tag{7.2570}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb-\omega baaaa,pc,\,\text{class }5\rangle
\tag{7.2571}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb-\omega ^{2}baaaa,pc,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.2572}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb-\omega ^{3}baaaa,pc,\,\text{class }%
5\rangle \;(p=1\func{mod}4)  \tag{7.2573}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb,pc-baaaa,\,\text{class }5\rangle 
\tag{7.2574}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb-baaaa,pc-baaaa,\,\text{class }5\rangle
\tag{7.2575}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb-\omega baaaa,pc-baaaa,\,\text{class }%
5\rangle  \tag{7.2576}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb-\omega ^{2}baaaa,pc-baaaa,\,\text{%
class }5\rangle \;(p=1\func{mod}4)  \tag{7.2577}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa,pb-\omega ^{3}baaaa,pc-baaaa,\,\text{%
class }5\rangle \;(p=1\func{mod}4)  \tag{7.2578}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa-baaaa,pb,pc,\,\text{class }5\rangle 
\tag{7.2579}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa,pa-baaaa,pb,pc-baaaa,\,\text{class }5\rangle
\tag{7.2580}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-baaaa,pa,pb-xbaaaa,pc,\,\text{class }%
5\rangle \;(\text{all }x)  \tag{7.2581}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-baaaa,pa,pb-xbaaaa,pc-baaaa,\,\text{class }%
5\rangle \;(\text{all }x)  \tag{7.2582}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-baaaa,pa-baaaa,pb,pc,\,\text{class }5\rangle
\tag{7.2583}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-baaaa,pa-xbaaaa,pb,pc-baaaa,\,\text{class }%
5\rangle \;(x\neq 0,\;x\sim -x)  \tag{7.2584}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-\omega baaaa,pa,pb-xbaaaa,pc,\,\text{class }%
5\rangle \;(\text{all }x)  \tag{7.2585}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-\omega baaaa,pa,pb-xbaaaa,pc-baaaa,\,\text{%
class }5\rangle \;(\text{all }x)  \tag{7.2586}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-\omega baaaa,pa-baaaa,pb,pc,\,\text{class }%
5\rangle  \tag{7.2587}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,ca,cb-baa-\omega baaaa,pa-xbaaaa,pb,pc-baaaa,\,\text{%
class }5\rangle \;(x\neq 0,\;x\sim -x)  \tag{7.2588}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa,pb,pc,\,\text{class }5\rangle 
\tag{7.2589}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa,pb,pc-baaaa,\,\text{class }5\rangle
\tag{7.2590}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa,pb,pc-ybaaaa,\,\text{class }%
5\rangle \;(y=\omega ^{i},\;i=1,2,3,4,\;p=1\func{mod}5)  \tag{7.2591}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa,pb-baaaa,pc,\,\text{class }5\rangle
\tag{7.2592}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa,pb-\omega baaaa,pc,\,\text{class }%
5\rangle  \tag{7.2593}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa,pb-\omega ^{2}baaaa,pc,\,\text{%
class }5\rangle \;(p=1\func{mod}4)  \tag{7.2594}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa,pb-\omega ^{3}baaaa,pc,\,\text{%
class }5\rangle \;(p=1\func{mod}4)  \tag{7.2595}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa,pb-baaaa,pc-xbaaaa,\,\text{class }%
5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{4}=x^{\prime 4}\func{mod%
}p)  \tag{7.2596}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa,pb-\omega baaaa,pc-xbaaaa,\,\text{%
class }5\rangle \;(x\neq 0,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}\func{%
mod}p)  \tag{7.2597}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa,pb-\omega ^{2}baaaa,pc-xbaaaa,\,%
\text{class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }%
x^{4}=x^{\prime 4}\func{mod}p,\;p=1\func{mod}4)  \tag{7.2598}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa,pb-\omega ^{3}baaaa,pc-xbaaaa,\,%
\text{class }5\rangle \;(x\neq 0,x\sim x^{\prime}\text{ if }x^4=x^{\prime 4}%
\func{mod}p,\;p=1\func{mod}4)  \tag{7.2599}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa-baaaa,pb,pc,\,\text{class }5\rangle
\tag{7.2600}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa-\omega baaaa,pb,pc,\,\text{class }%
5\rangle  \tag{7.2601}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa-ybaaaa,pb,pc,\,\text{class }%
5\rangle \;(y=\omega ^{i},\;i=2,3,4,5,\;p=1\func{mod}3)  \tag{7.2602}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa-baaaa,pb-xbaaaa,pc,\,\text{class }%
5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}\func{mod%
}p)  \tag{7.2603}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa-\omega baaaa,pb-xbaaaa,pc,\,\text{%
class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}%
\func{mod}p)  \tag{7.2604}
\end{equation}

In the next presentation we have $\;y=\omega ^{i},\;i=2,3,4,5,\;x\neq
0,\;x\sim x^{\prime }$ if $x^{3}=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3$:

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa-ybaaaa,pb-xbaaaa,pc,\,\text{class }%
5\rangle  \tag{7.2605}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa-baaaa,pb-zbaaaa,pc-xbaaaa,\,\text{%
class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{6}=x^{\prime 6}%
\func{mod}p,\text{ all }z)  \tag{7.2606}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa-\omega baaaa,pb-zbaaaa,pc-xbaaaa,\,%
\text{class }5\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }%
x^{6}=x^{\prime 6}\func{mod}p,\text{ all }z)  \tag{7.2607}
\end{equation}

In the next presentation we have $\;y=\omega ^{i},\;i=2,3,4,5,\;x\neq
0,\;x\sim x^{\prime }$ if $x^{6}=x^{\prime 6}\func{mod}p,$ all $z,\;p=1\func{%
mod}3$:

\begin{equation}
\langle a,b,c\,|\,bab-baaaa,ca,cb-baa,pa-ybaaaa,pb-zbaaaa,pc-xbaaaa,\,\text{%
class }5\rangle  \tag{7.2608}
\end{equation}

\subsection{Descendants of 6.85}

3 algebras

\begin{equation}
\langle a,b,c\,|\,ba,ca,cb,p^2b,p^2c,\,\text{class }3\rangle  \tag{7.2609}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba,ca-p^2a,cb,p^2b,p^2c,\,\text{class }3\rangle 
\tag{7.2610}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,ba,ca,cb-p^2a,p^2b,p^2c,\,\text{class }3\rangle 
\tag{7.2611}
\end{equation}

\subsection{Descendants of 6.86}

$p+17$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,bab,p^2a,p^2b,ca,cb,pc,\,\text{class }3\rangle 
\tag{7.2612}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,pba,p^2a,ca,cb,pc,\,\text{class }3\rangle 
\tag{7.2613}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,pba,p^2a,ca,cb-p^2b,pc,\,\text{class }3\rangle 
\tag{7.2614}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,pba,p^2a,ca-p^2b,cb,pc,\,\text{class }3\rangle 
\tag{7.2615}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,p^2a,p^2b-pba,ca,cb,pc,\,\text{class }3\rangle 
\tag{7.2616}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,pba,p^2a,p^2b,ca,cb,pc,\,\text{class }3\rangle 
\tag{7.2617}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,pba,p^2a,p^2b,ca,cb,pc-bab,\,\text{class }3\rangle 
\tag{7.2618}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,pba,p^2a,p^2b,ca-bab,cb,pc,\,\text{class }3\rangle 
\tag{7.2619}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,pba,p^2a,p^2b,ca-bab,cb,pc-bab,\,\text{class }3\rangle
\tag{7.2620}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,pba,p^2a,p^2b-bab,ca,cb,pc,\,\text{class }3\rangle 
\tag{7.2621}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,pba,p^2a,p^2b-bab,ca-bab,cb,pc,\,\text{class }3\rangle
\tag{7.2622}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,pba,p^2a-bab,p^2b,ca,cb,pc,\,\text{class }3\rangle 
\tag{7.2623}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,pba,p^2a-bab,p^2b,ca-bab,cb,pc,\,\text{class }3\rangle
\tag{7.2624}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,pba,p^{2}a-\omega bab,p^{2}b,ca,cb,pc,\,\text{class }%
3\rangle  \tag{7.2625}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,pba,p^{2}a-\omega bab,p^{2}b,ca-bab,cb,pc,\,\text{%
class }3\rangle  \tag{7.2626}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,pba-bab,p^2a-xbab,p^2b,ca,cb,pc,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.2627}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,pba-bab,p^2a+bab,p^2b,ca,cb,pc-bab,\,\text{class }%
3\rangle  \tag{7.2628}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,pba-bab,p^2a,p^2b-bab,ca,cb,pc,\,\text{class }3\rangle
\tag{7.2629}
\end{equation}

\subsection{Descendants of 6.87}

5 algebras

\begin{equation}
\langle a,b,c\,|\,p^2a,p^2b,ca,cb,pc-ba,\,\text{class }3\rangle  \tag{7.2630}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,pba,ca,cb,pc-ba,\,\text{class }3\rangle  \tag{7.2631}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,pba,ca,cb-p^2b,pc-ba,\,\text{class }3\rangle 
\tag{7.2632}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2a,pba,ca-p^2b,cb,pc-ba,\,\text{class }3\rangle 
\tag{7.2633}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^{2}a,pba,ca-\omega \ast p^{2}b,cb,pc-ba,\,\text{class }%
3\rangle  \tag{7.2634}
\end{equation}

\subsection{Descendants of 6.88}

26 algebras

\begin{equation}
\langle a,b,c\,|\,baa,bab,p^2b,ca,cb,pa,\,\text{class }3\rangle  \tag{7.2635}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,p^2b,ca,cb-p^2c,pa,\,\text{class }3\rangle 
\tag{7.2636}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,p^2b,ca-p^2c,cb,pa,\,\text{class }3\rangle 
\tag{7.2637}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,p^2c,ca,cb,pa,\,\text{class }3\rangle  \tag{7.2638}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,p^2c,ca,cb-p^2b,pa,\,\text{class }3\rangle 
\tag{7.2639}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,p^2c,ca-p^2b,cb,pa,\,\text{class }3\rangle 
\tag{7.2640}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^2b,p^2c,ca,cb,pa,\,\text{class }3\rangle 
\tag{7.2641}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^2b,p^2c,ca-bab,cb,pa,\,\text{class }3\rangle 
\tag{7.2642}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^2b,p^2c,ca,cb,pa-bab,\,\text{class }3\rangle 
\tag{7.2643}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^2b,p^2c,ca-bab,cb,pa-bab,\,\text{class }3\rangle 
\tag{7.2644}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^{2}b,p^{2}c,ca,cb,pa-\omega bab,\,\text{class }%
3\rangle  \tag{7.2645}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^{2}b,p^{2}c,ca-bab,cb,pa-\omega bab,\,\text{class }%
3\rangle  \tag{7.2646}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^2b,p^2c-bab,ca,cb,pa,\,\text{class }3\rangle 
\tag{7.2647}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^2b,p^2c-bab,ca-bab,cb,pa,\,\text{class }3\rangle 
\tag{7.2648}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^2b-bab,p^2c,ca,cb,pa,\,\text{class }3\rangle 
\tag{7.2649}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^2b-bab,p^2c,ca-bab,cb,pa,\,\text{class }3\rangle 
\tag{7.2650}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^2b,p^2c,ca,cb,pa,\,\text{class }3\rangle 
\tag{7.2651}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^2b,p^2c,ca,cb,pa-baa,\,\text{class }3\rangle 
\tag{7.2652}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^2b,p^2c,ca,cb-baa,pa,\,\text{class }3\rangle 
\tag{7.2653}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^2b,p^2c,ca,cb-baa,pa-baa,\,\text{class }3\rangle 
\tag{7.2654}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^2b,p^2c-baa,ca,cb,pa,\,\text{class }3\rangle 
\tag{7.2655}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^2b,p^2c-baa,ca,cb-baa,pa,\,\text{class }3\rangle 
\tag{7.2656}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^2b-baa,p^2c,ca,cb,pa,\,\text{class }3\rangle 
\tag{7.2657}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^2b-baa,p^2c,ca,cb-baa,pa,\,\text{class }3\rangle 
\tag{7.2658}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^{2}b-\omega baa,p^{2}c,ca,cb,pa,\,\text{class }%
3\rangle  \tag{7.2659}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^{2}b-\omega baa,p^{2}c,ca,cb-baa,pa,\,\text{class }%
3\rangle  \tag{7.2660}
\end{equation}

\subsection{Descendants of 6.89}

$p+6$ algebras

\begin{equation}
\langle a,b,c\,|\,p^2b,p^2c,ca,cb,pa-ba,\,\text{class }3\rangle  \tag{7.2661}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^2c,ca,cb,pa-ba,\,\text{class }3\rangle  \tag{7.2662}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^2c,ca,cb-p^2b,pa-ba,\,\text{class }3\rangle 
\tag{7.2663}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^2c,ca-p^2b,cb,pa-ba,\,\text{class }3\rangle 
\tag{7.2664}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^2b,ca,cb-xp^2c,pa-ba,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.2665}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^2b,ca-p^2c,cb,pa-ba,\,\text{class }3\rangle 
\tag{7.2666}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,p^2b,p^2c-bab,ca,cb,pa-ba,\,\text{class }3\rangle 
\tag{7.2667}
\end{equation}

\subsection{Descendants of 6.90}

30 algebras

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cac,cb,pb,pc,\,\text{class }3\rangle 
\tag{7.2668}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cac,cb-p^2a,pb,pc,\,\text{class }3\rangle 
\tag{7.2669}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,p^2a,cb,pb,pc,\,\text{class }3\rangle 
\tag{7.2670}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,p^2a,cb-caa,pb,pc,\,\text{class }3\rangle 
\tag{7.2671}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,p^2a,cb,pb,pc-caa,\,\text{class }3\rangle 
\tag{7.2672}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,p^2a,cb-caa,pb,pc-caa,\,\text{class }%
3\rangle  \tag{7.2673}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,p^{2}a,cb,pb,pc-\omega caa,\,\text{class }%
3\rangle  \tag{7.2674}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,p^{2}a,cb-caa,pb,pc-\omega caa,\,\text{%
class }3\rangle  \tag{7.2675}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,p^2a,cb,pb-caa,pc,\,\text{class }3\rangle 
\tag{7.2676}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,p^2a,cb-caa,pb-caa,pc,\,\text{class }%
3\rangle  \tag{7.2677}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,p^2a-caa,cb,pb,pc,\,\text{class }3\rangle 
\tag{7.2678}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,p^2a-caa,cb-caa,pb,pc,\,\text{class }%
3\rangle  \tag{7.2679}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cb,p^2a,pb,pc,\,\text{class }3\rangle 
\tag{7.2680}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cb,p^2a,pb,pc-cac,\,\text{class }3\rangle 
\tag{7.2681}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cb,p^2a,pb-cac,pc,\,\text{class }3\rangle 
\tag{7.2682}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cb,p^2a-cac,pb,pc,\,\text{class }3\rangle 
\tag{7.2683}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cb,p^{2}a-\omega cac,pb,pc,\,\text{class }%
3\rangle  \tag{7.2684}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,cb,p^2a,pb,pc,\,\text{class }3\rangle 
\tag{7.2685}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,cb,p^2a,pb,pc-baa,\,\text{class }%
3\rangle  \tag{7.2686}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,cb,p^2a,pb-baa,pc,\,\text{class }%
3\rangle  \tag{7.2687}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,cb,p^{2}a,pb-\omega baa,pc,\,\text{%
class }3\rangle  \tag{7.2688}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,cb,p^2a-baa,pb,pc,\,\text{class }%
3\rangle  \tag{7.2689}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,cb,p^{2}a-\omega baa,pb,pc,\,\text{%
class }3\rangle  \tag{7.2690}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+bab,cb,p^2a,pb,pc,\,\text{class }3\rangle 
\tag{7.2691}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+bab,cb,p^2a,pb,pc-bab,\,\text{class }%
3\rangle  \tag{7.2692}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+bab,cb,p^2a,pb-bab,pc-bab,\,\text{class }%
3\rangle  \tag{7.2693}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+bab,cb,p^2a-bab,pb,pc,\,\text{class }%
3\rangle  \tag{7.2694}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+\omega bab,cb,p^{2}a,pb,pc,\,\text{class }%
3\rangle  \tag{7.2695}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+\omega bab,cb,p^{2}a,pb,pc-bab,\,\text{%
class }3\rangle  \tag{7.2696}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+\omega bab,cb,p^{2}a-bab,pb,pc,\,\text{%
class }3\rangle  \tag{7.2697}
\end{equation}

\subsection{Descendants of 6.91}

$2p+88+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cac,cb,pa,pc,\,\text{class }3\rangle 
\tag{7.2698}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cac,cb-p^2b,pa,pc,\,\text{class }3\rangle 
\tag{7.2699}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cb,p^2b,pa,pc,\,\text{class }3\rangle 
\tag{7.2700}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cb,p^2b,pa-cac,pc,\,\text{class }3\rangle 
\tag{7.2701}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cb,p^{2}b,pa-\omega cac,pc,\,\text{class }%
3\rangle  \tag{7.2702}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cb,p^2b,pa,pc-cac,\,\text{class }3\rangle 
\tag{7.2703}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cb,p^2b,pa-cac,pc-cac,\,\text{class }%
3\rangle  \tag{7.2704}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cb,p^{2}b,pa-\omega cac,pc-cac,\,\text{%
class }3\rangle  \tag{7.2705}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,caa,cb,p^2b-cac,pa,pc,\,\text{class }3\rangle 
\tag{7.2706}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+bab,cb,p^2b,pa,pc,\,\text{class }3\rangle 
\tag{7.2707}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+bab,cb,p^2b,pa-bab,pc,\,\text{class }%
3\rangle  \tag{7.2708}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+bab,cb,p^{2}b,pa-\omega bab,pc,\,\text{%
class }3\rangle  \tag{7.2709}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+bab,cb,p^2b,pa,pc-bab,\,\text{class }%
3\rangle  \tag{7.2710}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+bab,cb,p^2b,pa-bab,pc-bab,\,\text{class }%
3\rangle  \tag{7.2711}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+bab,cb,p^{2}b,pa-\omega bab,pc-bab,\,\text{%
class }3\rangle  \tag{7.2712}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+bab,cb,p^2b-bab,pa,pc,\,\text{class }%
3\rangle  \tag{7.2713}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+\omega bab,cb,p^{2}b,pa,pc,\,\text{class }%
3\rangle  \tag{7.2714}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+\omega bab,cb,p^{2}b,pa-bab,pc,\,\text{%
class }3\rangle  \tag{7.2715}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+\omega bab,cb,p^{2}b,pa-\omega bab,pc,\,%
\text{class }3\rangle  \tag{7.2716}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+\omega bab,cb,p^{2}b,pa,pc-bab,\,\text{%
class }3\rangle  \tag{7.2717}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+\omega bab,cb,p^{2}b,pa-bab,pc-bab,\,\text{%
class }3\rangle  \tag{7.2718}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+\omega bab,cb,p^{2}b,pa-\omega
bab,pc-bab,\,\text{class }3\rangle  \tag{7.2719}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac+\omega bab,cb,p^{2}b-bab,pa,pc,\,\text{%
class }3\rangle  \tag{7.2720}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,cb,p^2b,pa,pc,\,\text{class }3\rangle 
\tag{7.2721}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,cb,p^2b,pa-baa,pc,\,\text{class }%
3\rangle  \tag{7.2722}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,cb,p^{2}b,pa-\omega baa,pc,\,\text{%
class }3\rangle  \tag{7.2723}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,cb,p^2b,pa,pc-baa,\,\text{class }%
3\rangle  \tag{7.2724}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,cb,p^2b,pa-baa,pc-baa,\,\text{class }%
3\rangle  \tag{7.2725}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,cb,p^{2}b,pa-\omega baa,pc-baa,\,\text{%
class }3\rangle  \tag{7.2726}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,cb,p^2b-baa,pa,pc,\,\text{class }%
3\rangle  \tag{7.2727}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac-baa,cb,p^{2}b-\omega baa,pa,pc,\,\text{%
class }3\rangle  \tag{7.2728}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac+baa,cb,p^2b,pa,pc,\,\text{class }%
3\rangle  \tag{7.2729}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac+baa,cb,p^2b,pa,pc-baa,\,\text{class }%
3\rangle  \tag{7.2730}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac+baa,cb,p^{2}b,pa,pc-\omega baa,\,\text{%
class }3\rangle \;(p=1\func{mod}4)  \tag{7.2731}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac+baa,cb,p^{2}b,pa-baa,pc-xbaa,\,\text{%
class }3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.2732}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac+baa,cb,p^{2}b,pa-\omega baa,pc-xbaa,\,%
\text{class }3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.2733}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac+baa,cb,p^2b-baa,pa,pc,\,\text{class }%
3\rangle  \tag{7.2734}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac+baa,cb,p^{2}b-\omega baa,pa,pc,\,\text{%
class }3\rangle  \tag{7.2735}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac+\omega baa,cb,p^{2}b,pa,pc,\,\text{%
class }3\rangle  \tag{7.2736}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac+\omega baa,cb,p^{2}b,pa,pc-baa,\,\text{%
class }3\rangle  \tag{7.2737}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac+\omega baa,cb,p^{2}b,pa,pc-\omega
baa,\,\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.2738}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac+\omega baa,cb,p^{2}b,pa-baa,pc-xbaa,\,%
\text{class }3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.2739}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac+\omega baa,cb,p^{2}b,pa-\omega
baa,pc-xbaa,\,\text{class }3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.2740}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac+\omega baa,cb,p^{2}b-baa,pa,pc,\,\text{%
class }3\rangle  \tag{7.2741}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac+\omega baa,cb,p^{2}b-\omega
baa,pa,pc,\,\text{class }3\rangle  \tag{7.2742}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,caa,cac,cb,p^2b,pa,pc,\,\text{class }3\rangle 
\tag{7.2743}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,caa,cac,cb,p^2b,pa,pc-bac,\,\text{class }3\rangle 
\tag{7.2744}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,caa,cac,cb,p^2b,pa-bac,pc,\,\text{class }3\rangle 
\tag{7.2745}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,caa,cac,cb,p^2b,pa-bac,pc-bac,\,\text{class }%
3\rangle  \tag{7.2746}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,caa,cac,cb,p^2b-bac,pa,pc,\,\text{class }3\rangle 
\tag{7.2747}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,caa-bac,cac,cb,p^2b,pa,pc,\,\text{class }3\rangle 
\tag{7.2748}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,caa-bac,cac,cb,p^2b,pa,pc-bac,\,\text{class }%
3\rangle  \tag{7.2749}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,caa-bac,cac,cb,p^{2}b,pa,pc-\omega bac,\,\text{%
class }3\rangle  \tag{7.2750}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,caa-bac,cac,cb,p^2b,pa-bac,pc,\,\text{class }%
3\rangle  \tag{7.2751}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,caa-bac,cac,cb,p^2b,pa-bac,pc-bac,\,\text{class }%
3\rangle  \tag{7.2752}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,caa-bac,cac,cb,p^{2}b,pa-bac,pc-\omega bac,\,\text{%
class }3\rangle  \tag{7.2753}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,caa-bac,cac,cb,p^2b-bac,pa,pc,\,\text{class }%
3\rangle  \tag{7.2754}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,cb,p^2b,pa,pc,\,\text{class }3\rangle 
\tag{7.2755}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,cb,p^2b,pa,pc-caa,\,\text{class }3\rangle 
\tag{7.2756}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,cb,p^{2}b,pa,pc-\omega caa,\,\text{class }%
3\rangle  \tag{7.2757}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,cb-caa,p^2b,pa,pc,\,\text{class }3\rangle 
\tag{7.2758}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,cb-caa,p^2b,pa,pc-caa,\,\text{class }%
3\rangle  \tag{7.2759}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,cb-caa,p^{2}b,pa,pc-\omega caa,\,\text{%
class }3\rangle  \tag{7.2760}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,cb,p^2b,pa-caa,pc,\,\text{class }3\rangle 
\tag{7.2761}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,cb-caa,p^2b,pa-caa,pc,\,\text{class }%
3\rangle  \tag{7.2762}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,cb,p^2b-caa,pa,pc,\,\text{class }3\rangle 
\tag{7.2763}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cac,cb-caa,p^2b-caa,pa,pc,\,\text{class }%
3\rangle  \tag{7.2764}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa-bab,cac,cb,p^2b,pa,pc,\,\text{class }3\rangle 
\tag{7.2765}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa-bab,cac,cb,p^2b,pa,pc-bab,\,\text{class }%
3\rangle  \tag{7.2766}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa-bab,cac,cb,p^{2}b,pa,pc-\omega bab,\,\text{%
class }3\rangle  \tag{7.2767}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa-bab,cac,cb,p^2b,pa-bab,pc,\,\text{class }%
3\rangle  \tag{7.2768}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa-bab,cac,cb,p^{2}b,pa-\omega bab,pc,\,\text{%
class }3\rangle  \tag{7.2769}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa-bab,cac,cb,p^2b-bab,pa,pc,\,\text{class }%
3\rangle  \tag{7.2770}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac,cb,p^2b,pa,pc,\,\text{class }3\rangle 
\tag{7.2771}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac,cb,p^2b,pa,pc-bab,\,\text{class }3\rangle 
\tag{7.2772}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac,cb,p^2b,pa-bab,pc,\,\text{class }3\rangle 
\tag{7.2773}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac,cb,p^{2}b,pa-\omega bab,pc,\,\text{class }%
3\rangle  \tag{7.2774}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,caa,cac,cb,p^2b-bab,pa,pc,\,\text{class }3\rangle 
\tag{7.2775}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac,cb,p^2b,pa,pc,\,\text{class }3\rangle 
\tag{7.2776}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac,cb,p^2b,pa,pc-baa,\,\text{class }3\rangle 
\tag{7.2777}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac,cb,p^2b,pa-baa,pc,\,\text{class }3\rangle 
\tag{7.2778}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac,cb,p^2b-baa,pa,pc,\,\text{class }3\rangle 
\tag{7.2779}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac,cb,p^{2}b-\omega baa,pa,pc,\,\text{class }%
3\rangle  \tag{7.2780}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac,cb-baa,p^2b,pa,pc,\,\text{class }3\rangle 
\tag{7.2781}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac,cb-baa,p^2b,pa,pc-baa,\,\text{class }%
3\rangle  \tag{7.2782}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac,cb-baa,p^2b,pa-baa,pc,\,\text{class }%
3\rangle  \tag{7.2783}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac,cb-baa,p^2b-baa,pa,pc,\,\text{class }%
3\rangle  \tag{7.2784}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,caa,cac,cb-baa,p^{2}b-\omega baa,pa,pc,\,\text{%
class }3\rangle  \tag{7.2785}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac,cb,p^2b,pa,pc,\,\text{class }3\rangle 
\tag{7.2786}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac,cb,p^2b,pa-baa,pc,\,\text{class }%
3\rangle  \tag{7.2787}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac,cb,p^{2}b,pa-\omega baa,pc,\,\text{%
class }3\rangle  \tag{7.2788}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac,cb,p^2b,pa,pc-baa,\,\text{class }%
3\rangle  \tag{7.2789}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac,cb,p^2b-baa,pa,pc,\,\text{class }%
3\rangle  \tag{7.2790}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,caa,cac,cb,p^{2}b-\omega baa,pa,pc,\,\text{%
class }3\rangle  \tag{7.2791}
\end{equation}

\subsection{Descendants of 6.92}

$2p+13$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,caa,cac,cb,pb-ba,pc,\,\text{class }3\rangle 
\tag{7.2792}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,cac,cb-p^2a,pb-ba,pc,\,\text{class }3\rangle 
\tag{7.2793}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,p^2a,cb,pb-ba,pc,\,\text{class }3\rangle 
\tag{7.2794}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,p^2a,cb,pb-ba,pc-cac,\,\text{class }3\rangle 
\tag{7.2795}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,p^2a,cb,pb-ba-cac,pc,\,\text{class }3\rangle 
\tag{7.2796}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,p^2a,cb,pb-ba-cac,pc-cac,\,\text{class }3\rangle 
\tag{7.2797}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,p^2a-cac,cb,pb-ba,pc,\,\text{class }3\rangle 
\tag{7.2798}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,p^{2}a-\omega cac,cb,pb-ba,pc,\,\text{class }%
3\rangle  \tag{7.2799}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,cac,p^2a,cb,pb-ba,pc-xcaa,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.2800}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,cac,p^2a,cb-caa,pb-ba,pc-xcaa,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.2801}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,cac,p^2a-caa,cb,pb-ba,pc,\,\text{class }3\rangle 
\tag{7.2802}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,cac,p^2a-caa,cb-caa,pb-ba,pc,\,\text{class }3\rangle 
\tag{7.2803}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac,p^2a,cb,pb-ba,pc,\,\text{class }3\rangle 
\tag{7.2804}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac-baa,p^2a,cb,pb-ba,pc,\,\text{class }3\rangle 
\tag{7.2805}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa-baa,cac,p^2a,cb,pb-ba,pc,\,\text{class }3\rangle 
\tag{7.2806}
\end{equation}

\subsection{Descendants of 6.93}

$p+15+2\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,bab,caa,cb,pb-ca,pc,\,\text{class }3\rangle 
\tag{7.2807}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,caa,cb-p^2a,pb-ca,pc,\,\text{class }3\rangle 
\tag{7.2808}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,caa,cb-\omega p^{2}a,pb-ca,pc,\,\text{class }%
3\rangle  \tag{7.2809}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,p^2a,cb,pb-ca,pc,\,\text{class }3\rangle 
\tag{7.2810}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,p^2a-caa,cb,pb-ca,pc,\,\text{class }3\rangle 
\tag{7.2811}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,p^2a,cb,pb-ca,pc,\,\text{class }3\rangle 
\tag{7.2812}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,p^2a,cb,pb-ca,pc-bab,\,\text{class }3\rangle 
\tag{7.2813}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,p^2a,cb,pb-ca-bab,pc,\,\text{class }3\rangle 
\tag{7.2814}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,p^2a-bab,cb,pb-ca,pc,\,\text{class }3\rangle 
\tag{7.2815}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,p^{2}a-\omega bab,cb,pb-ca,pc,\,\text{class }%
3\rangle  \tag{7.2816}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa-bab,p^2a-xbab,cb,pb-ca,pc,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.2817}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa-bab,p^2a+bab,cb,pb-ca,pc-bab,\,\text{class }%
3\rangle  \tag{7.2818}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa-bab,p^{2}a+bab,cb,pb-ca,pc-\omega bab,\,\text{%
class }3\rangle  \tag{7.2819}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^2a,cb,pb-ca,pc,\,\text{class }3\rangle 
\tag{7.2820}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^2a,cb,pb-ca,pc-baa,\,\text{class }3\rangle 
\tag{7.2821}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^{2}a,cb,pb-ca,pc-\omega baa,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.2822}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^{2}a,cb,pb-ca,pc-\omega ^{2}baa,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.2823}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^2a,cb-baa,pb-ca,pc,\,\text{class }3\rangle 
\tag{7.2824}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^2a,cb-baa,pb-ca,pc-baa,\,\text{class }3\rangle 
\tag{7.2825}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^{2}a,cb-baa,pb-ca,pc-\omega baa,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.2826}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^{2}a,cb-baa,pb-ca,pc-\omega ^{2}baa,\,\text{%
class }3\rangle \;(p=1\func{mod}3)  \tag{7.2827}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^2a-baa,cb,pb-ca,pc,\,\text{class }3\rangle 
\tag{7.2828}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^2a-baa,cb-baa,pb-ca,pc,\,\text{class }3\rangle 
\tag{7.2829}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^{2}a-baa,cb-\omega baa,pb-ca,pc,\,\text{class }%
3\rangle  \tag{7.2830}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^{2}a-baa,cb-\omega ^{2}baa,pb-ca,pc,\,\text{%
class }3\rangle \;(p=1\func{mod}4)  \tag{7.2831}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^{2}a-baa,cb-\omega ^{3}baa,pb-ca,pc,\,\text{%
class }3\rangle \;(p=1\func{mod}4)  \tag{7.2832}
\end{equation}

\subsection{Descendants of 6.94}

$2p+15+3\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c\,|\,caa,cac,cb,pa,pc-ba,\,\text{class }3\rangle  \tag{7.2833}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac,cb-p^2b,pa,pc-ba,\,\text{class }3\rangle 
\tag{7.2834}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,p^2b,cb,pa,pc-ba,\,\text{class }3\rangle  \tag{7.2835}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,p^2b,cb,pa-cac,pc-ba,\,\text{class }3\rangle 
\tag{7.2836}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,p^{2}b,cb,pa-\omega cac,pc-ba,\,\text{class }3\rangle 
\tag{7.2837}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,p^2b,cb,pa,pc-ba-cac,\,\text{class }3\rangle 
\tag{7.2838}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,p^2b,cb,pa-cac,pc-ba-cac,\,\text{class }3\rangle 
\tag{7.2839}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,p^{2}b,cb,pa-\omega cac,pc-ba-cac,\,\text{class }%
3\rangle  \tag{7.2840}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,p^2b-cac,cb,pa,pc-ba,\,\text{class }3\rangle 
\tag{7.2841}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,p^2b,cb,pa,pc-ba,\,\text{class }3\rangle  \tag{7.2842}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,p^2b,cb,pa-caa,pc-ba,\,\text{class }3\rangle 
\tag{7.2843}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,p^2b,cb-caa,pa,pc-ba,\,\text{class }3\rangle 
\tag{7.2844}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,p^2b,cb-caa,pa-caa,pc-ba,\,\text{class }3\rangle 
\tag{7.2845}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,p^{2}b,cb-caa,pa-\omega caa,pc-ba,\,\text{class }%
3\rangle  \tag{7.2846}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,p^{2}b,cb-caa,pa-\omega ^{2}caa,pc-ba,\,\text{class }%
3\rangle \;(p=1\func{mod}4)  \tag{7.2847}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,p^{2}b,cb-caa,pa-\omega ^{3}caa,pc-ba,\,\text{class }%
3\rangle \;(p=1\func{mod}4)  \tag{7.2848}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,p^2b-caa,cb,pa,pc-ba,\,\text{class }3\rangle 
\tag{7.2849}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,p^{2}b-\omega caa,cb,pa,pc-ba,\,\text{class }3\rangle
\;(p=1\func{mod}3)  \tag{7.2850}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,p^{2}b-\omega ^{2}caa,cb,pa,pc-ba,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.2851}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,p^2b-caa,cb-caa,pa,pc-ba,\,\text{class }3\rangle 
\tag{7.2852}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,p^{2}b-\omega caa,cb-caa,pa,pc-ba,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.2853}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,p^{2}b-\omega ^{2}caa,cb-caa,pa,pc-ba,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.2854}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac-caa,p^2b,cb,pa,pc-ba,\,\text{class }3\rangle 
\tag{7.2855}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac-caa,p^2b,cb,pa,pc-ba-caa,\,\text{class }3\rangle 
\tag{7.2856}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac-caa,p^{2}b,cb,pa,pc-ba-\omega caa,\,\text{class }%
3\rangle  \tag{7.2857}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac-caa,p^2b,cb,pa-caa,pc-ba-xcaa,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.2858}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac-caa,p^{2}b,cb,pa-\omega caa,pc-ba-xcaa,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.2859}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac-caa,p^2b-caa,cb,pa,pc-ba,\,\text{class }3\rangle 
\tag{7.2860}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac-caa,p^{2}b-\omega caa,cb,pa,pc-ba,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.2861}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac-caa,p^{2}b-\omega ^{2}caa,cb,pa,pc-ba,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.2862}
\end{equation}

\subsection{Descendants of 6.95}

$5p+10$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,bab,cb,pa,pc-ca,\,\text{class }3\rangle  \tag{7.2863}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,cb-p^2b,pa,pc-ca,\,\text{class }3\rangle 
\tag{7.2864}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^2b,cb,pa,pc-ca,\,\text{class }3\rangle  \tag{7.2865}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^2b,cb,pa-bab,pc-ca,\,\text{class }3\rangle 
\tag{7.2866}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^{2}b,cb,pa-\omega bab,pc-ca,\,\text{class }3\rangle 
\tag{7.2867}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^2b,cb,pa,pc-ca-bab,\,\text{class }3\rangle 
\tag{7.2868}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^2b,cb,pa-bab,pc-ca-bab,\,\text{class }3\rangle 
\tag{7.2869}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^{2}b,cb,pa-\omega bab,pc-ca-bab,\,\text{class }%
3\rangle  \tag{7.2870}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^2b-bab,cb,pa,pc-ca,\,\text{class }3\rangle 
\tag{7.2871}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^2b-xbaa,cb,pa,pc-ca,\,\text{class }3\rangle \;(\text{%
all }x)  \tag{7.2872}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^2b-xbaa,cb-baa,pa,pc-ca,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.2873}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^2b,cb,pa-baa,pc-ca,\,\text{class }3\rangle 
\tag{7.2874}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,p^2b,cb-baa,pa-baa,pc-ca,\,\text{class }3\rangle 
\tag{7.2875}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,p^2b-xbaa,cb,pa,pc-ca,\,\text{class }3\rangle \;(x
\neq 0)  \tag{7.2876}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,p^2b,cb,pa-xbaa,pc-ca,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.2877}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,p^2b,cb,pa-xbaa,pc-ca-baa,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.2878}
\end{equation}

\subsection{Descendants of 6.96}

$2p+26$ algebras

\begin{equation}
\langle a,b,c\,|\,bab,caa,cac,cb,pa-ba,pc,\,\text{class }3\rangle 
\tag{7.2879}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,cac,cb-p^2b,pa-ba,pc,\,\text{class }3\rangle 
\tag{7.2880}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^2b,cb,pa-ba,pc,\,\text{class }3\rangle 
\tag{7.2881}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^2b,cb,pa-ba-cac,pc,\,\text{class }3\rangle 
\tag{7.2882}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^{2}b,cb,pa-ba-\omega cac,pc,\,\text{class }%
3\rangle  \tag{7.2883}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^2b,cb,pa-ba,pc-cac,\,\text{class }3\rangle 
\tag{7.2884}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^2b,cb,pa-ba-cac,pc-cac,\,\text{class }3\rangle 
\tag{7.2885}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^{2}b,cb,pa-ba-\omega cac,pc-cac,\,\text{class }%
3\rangle  \tag{7.2886}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,caa,p^2b-cac,cb,pa-ba,pc,\,\text{class }3\rangle 
\tag{7.2887}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cac,p^2b,cb,pa-ba,pc,\,\text{class }3\rangle 
\tag{7.2888}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cac,p^2b,cb,pa-ba,pc-caa,\,\text{class }3\rangle 
\tag{7.2889}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cac,p^{2}b,cb,pa-ba,pc-\omega caa,\,\text{class }%
3\rangle  \tag{7.2890}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cac,p^2b,cb-caa,pa-ba,pc-xcaa,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.2891}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cac,p^{2}b,cb-\omega caa,pa-ba,pc-xcaa,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.2892}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cac,p^2b-caa,cb,pa-ba,pc,\,\text{class }3\rangle 
\tag{7.2893}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cac,p^2b-caa,cb-caa,pa-ba,pc,\,\text{class }3\rangle 
\tag{7.2894}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,cac,p^{2}b-caa,cb-\omega caa,pa-ba,pc,\,\text{class }%
3\rangle  \tag{7.2895}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac,p^2b,cb,pa-ba,pc,\,\text{class }3\rangle 
\tag{7.2896}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac,p^2b,cb,pa-ba,pc-bab,\,\text{class }3\rangle 
\tag{7.2897}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac,p^2b-bab,cb,pa-ba,pc,\,\text{class }3\rangle 
\tag{7.2898}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac+bab,p^2b,cb,pa-ba,pc,\,\text{class }3\rangle 
\tag{7.2899}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac+bab,p^2b-bab,cb,pa-ba,pc,\,\text{class }3\rangle 
\tag{7.2900}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac+\omega bab,p^{2}b,cb,pa-ba,pc,\,\text{class }%
3\rangle  \tag{7.2901}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,cac+\omega bab,p^{2}b-bab,cb,pa-ba,pc,\,\text{class }%
3\rangle  \tag{7.2902}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,caa-bab,p^2b,cb,pa-ba,pc,\,\text{class }3\rangle 
\tag{7.2903}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,caa-bab,p^2b,cb,pa-ba,pc-bab,\,\text{class }3\rangle 
\tag{7.2904}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,caa-bab,p^{2}b,cb,pa-ba,pc-\omega bab,\,\text{class }%
3\rangle  \tag{7.2905}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,cac,caa-bab,p^2b-bab,cb,pa-ba,pc,\,\text{class }3\rangle 
\tag{7.2906}
\end{equation}

\subsection{Descendants of 6.97}

$3p+18+\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,bab,bac,cb-xp^2b,pa-ca,pc,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.2907}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,p^2b,cb,pa-ca,pc,\,\text{class }3\rangle 
\tag{7.2908}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bab,p^2b-bac,cb,pa-ca,pc,\,\text{class }3\rangle 
\tag{7.2909}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,p^2b,cb,pa-ca,pc,\,\text{class }3\rangle 
\tag{7.2910}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,p^2b,cb,pa-ca-bab,pc,\,\text{class }3\rangle 
\tag{7.2911}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,p^{2}b,cb,pa-ca-\omega bab,pc,\,\text{class }%
3\rangle  \tag{7.2912}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,p^2b,cb,pa-ca,pc-bab,\,\text{class }3\rangle 
\tag{7.2913}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,bac,p^2b-bab,cb,pa-ca,pc,\,\text{class }3\rangle 
\tag{7.2914}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,p^2b,cb,pa-ca,pc,\,\text{class }3\rangle 
\tag{7.2915}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,p^2b,cb-baa,pa-ca,pc,\,\text{class }3\rangle 
\tag{7.2916}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,p^{2}b,cb-\omega baa,pa-ca,pc,\,\text{class }%
3\rangle  \tag{7.2917}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,p^2b,cb,pa-ca,pc-baa,\,\text{class }3\rangle 
\tag{7.2918}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,p^2b,cb-baa,pa-ca,pc-baa,\,\text{class }3\rangle 
\tag{7.2919}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,p^{2}b,cb-\omega baa,pa-ca,pc-baa,\,\text{class }%
3\rangle  \tag{7.2920}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,p^2b-baa,cb-xbaa,pa-ca,pc,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.2921}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab,bac,p^{2}b-\omega baa,cb-xbaa,pa-ca,pc,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.2922}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,p^2b,cb,pa-ca,pc,\,\text{class }3\rangle 
\tag{7.2923}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,p^2b,cb,pa-ca-baa,pc,\,\text{class }3\rangle 
\tag{7.2924}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,p^{2}b,cb,pa-ca-\omega baa,pc,\,\text{class }%
3\rangle  \tag{7.2925}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,p^2b,cb,pa-ca,pc-baa,\,\text{class }3\rangle 
\tag{7.2926}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,p^{2}b,cb,pa-ca,pc-\omega baa,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.2927}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,p^{2}b,cb,pa-ca,pc-\omega ^{2}baa,\,\text{%
class }3\rangle \;(p=1\func{mod}3)  \tag{7.2928}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,p^2b-baa,cb,pa-ca,pc,\,\text{class }3\rangle 
\tag{7.2929}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,bab-baa,bac,p^{2}b-\omega baa,cb,pa-ca,pc,\,\text{class }%
3\rangle  \tag{7.2930}
\end{equation}

\subsection{Descendants of 6.98}

$(5p+1)/2$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,caa,cb,pb-ba,pc-ca,\,\text{class }3\rangle 
\tag{7.2931}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,cb-p^2a,pb-ba,pc-ca,\,\text{class }3\rangle 
\tag{7.2932}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^2a,cb,pb-ba,pc-ca,\,\text{class }3\rangle 
\tag{7.2933}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^2a,cb,pb-ba-caa,pc-ca,\,\text{class }3\rangle 
\tag{7.2934}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,cb,pb-ba,pc-xca,\,\text{class }3\rangle \;(x\neq
0,1,-1,\;x\sim x^{-1})  \tag{7.2935}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,cb-p^{2}a,pb-ba,pc-xca,\,\text{class }3\rangle
\;(x\neq 0,1,-1,\;x\sim x^{-1})  \tag{7.2936}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^{2}a,cb,pb-ba,pc-xca,\,\text{class }3\rangle
\;(x\neq 0,1,-1,\;x\sim x^{-1})  \tag{7.2937}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,p^{2}a,cb,pb-ba,pc-xca,\,\text{class }3\rangle
\;(x\neq 0,1,-1,\;x\sim x^{-1})  \tag{7.2938}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa-baa,p^{2}a,cb,pb-ba,pc-xca,\,\text{class }3\rangle
\;(x\neq 0,1,-1,\;x\sim x^{-1})  \tag{7.2939}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,cb,pb-ba,pc+ca,\,\text{class }3\rangle 
\tag{7.2940}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,cb-p^2a,pb-ba,pc+ca,\,\text{class }3\rangle 
\tag{7.2941}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^2a,cb,pb-ba,pc+ca,\,\text{class }3\rangle 
\tag{7.2942}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa-baa,p^2a,cb,pb-ba,pc+ca,\,\text{class }3\rangle 
\tag{7.2943}
\end{equation}

\subsection{Descendants of 6.99}

$p+4$ algebras

\begin{equation}
\langle a,b,c\,|\,baa,caa,cb,pb-ba-ca,pc-ca,\,\text{class }3\rangle 
\tag{7.2944}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,cb-p^2a,pb-ba-ca,pc-ca,\,\text{class }3\rangle 
\tag{7.2945}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,caa,cb-\omega p^{2}a,pb-ba-ca,pc-ca,\,\text{class }%
3\rangle  \tag{7.2946}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,baa,p^2a,cb,pb-ba-ca,pc-ca,\,\text{class }3\rangle 
\tag{7.2947}
\end{equation}

\begin{equation}
\langle a,b,c\,|\,caa,p^2a,cb,pb-ba-ca,pc-ca-xbaa,\,\text{class }3\rangle \;(%
\text{all }x)  \tag{7.2948}
\end{equation}

\subsection{Descendants of 4.1}

1361 algebras when $p=3,\;p^{5}+2p^{4}+7p^{3}+25p^{2}+88p+270+(p+4)\gcd
(p-1,3)+\gcd (p-1,4)$ algebras when $p>3$

\begin{equation}
\langle a,b,c,d\,|\,ba,ca,da,cb,db,dc,pd,\,\text{class }2\rangle 
\tag{7.2949}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,dc,pa,pb,\,\text{class }2\rangle 
\tag{7.2950}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,dc,pa-ba,pb,\,\text{class }2\rangle 
\tag{7.2951}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,dc,pb,pc-ba,\,\text{class }2\rangle 
\tag{7.2952}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,dc,pb,pc,\,\text{class }2\rangle 
\tag{7.2953}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,dc,pb-ba,pc,\,\text{class }2\rangle 
\tag{7.2954}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,dc,pc,pd,\,\text{class }2\rangle 
\tag{7.2955}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,dc,pc-ba,pd,\,\text{class }2\rangle 
\tag{7.2956}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,dc-ba,pa,pc,\,\text{class }2\rangle 
\tag{7.2957}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,dc-ba,pa-ba,pc,\,\text{class }2\rangle 
\tag{7.2958}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,dc-ba,pa,pb,\,\text{class }2\rangle 
\tag{7.2959}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,dc-ba,pa-ba,pb,\,\text{class }2\rangle 
\tag{7.2960}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pb,pc,pd,\,\text{class }2\rangle 
\tag{7.2961}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa,pc,pd,\,\text{class }2\rangle 
\tag{7.2962}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pb-ba,pc,pd,\,\text{class }2\rangle 
\tag{7.2963}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pb-ca,pc,pd,\,\text{class }2\rangle 
\tag{7.2964}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa,pc-ba,pd,\,\text{class }2\rangle 
\tag{7.2965}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa,pc-ca,pd,\,\text{class }2\rangle 
\tag{7.2966}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa-ba,pc,pd,\,\text{class }2\rangle 
\tag{7.2967}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa-ca,pc,pd,\,\text{class }2\rangle 
\tag{7.2968}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pb-ba,pc-xca,pd,\,\text{class }2\rangle \;(x
\neq 0, x \sim x^{-1})  \tag{7.2969}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pb-ba-ca,pc-ca,pd,\,\text{class }2\rangle 
\tag{7.2970}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pb-\omega ca,pc-ba,pd,\,\text{class }2\rangle
\tag{7.2971}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pb-xca,pc-ba-ca,pd,\,\text{class }2\rangle
\;(\text{all }x\text{ with }1+4x\text{ not a square }\func{mod}p) 
\tag{7.2972}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa-ca,pc-ba,pd,\,\text{class }2\rangle 
\tag{7.2973}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa-ba,pc-ca,pd,\,\text{class }2\rangle 
\tag{7.2974}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa,pb,pc,\,\text{class }2\rangle 
\tag{7.2975}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa-ba,pb,pc,\,\text{class }2\rangle 
\tag{7.2976}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa,pb-ba,pc,\,\text{class }2\rangle 
\tag{7.2977}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa-ca,pb-ba,pc,\,\text{class }2\rangle 
\tag{7.2978}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa,pb-ca,pc,\,\text{class }2\rangle 
\tag{7.2979}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa-ba,pb-ca,pc,\,\text{class }2\rangle 
\tag{7.2980}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa,pb-ba,pc-xca,\,\text{class }2\rangle
\;(x\neq 0,\;x\sim x^{-1})  \tag{7.2981}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa,pb-ba-ca,pc-ca,\,\text{class }2\rangle 
\tag{7.2982}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa,pb-\omega ca,pc-ba,\,\text{class }2\rangle
\tag{7.2983}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa,pb-xca,pc-ba-ca,\,\text{class }2\rangle
\;(\text{all }x\text{ with }1+4x\text{ not a square }\func{mod}p) 
\tag{7.2984}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa,pb,pd-ca,\,\text{class }2\rangle 
\tag{7.2985}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa,pb-ba,pd-ca,\,\text{class }2\rangle 
\tag{7.2986}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa-ba,pb,pd-ca,\,\text{class }2\rangle 
\tag{7.2987}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa,pc,pd-ca,\,\text{class }2\rangle 
\tag{7.2988}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa,pc-ba,pd-ca,\,\text{class }2\rangle 
\tag{7.2989}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa-ba,pc,pd-ca,\,\text{class }2\rangle 
\tag{7.2990}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pb,pc,pd-ca,\,\text{class }2\rangle 
\tag{7.2991}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pb,pc-ba,pd-ca,\,\text{class }2\rangle 
\tag{7.2992}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pb-ba,pc,pd-ca,\,\text{class }2\rangle 
\tag{7.2993}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pb,pc,\,\text{class }2\rangle 
\tag{7.2994}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pb,pc-ba,\,\text{class }2\rangle 
\tag{7.2995}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pb,pc-dc,\,\text{class }2\rangle 
\tag{7.2996}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pb,pc-ba-dc,\,\text{class }2\rangle 
\tag{7.2997}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pb-ba,pc,\,\text{class }2\rangle 
\tag{7.2998}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pb-ba,pc-ba,\,\text{class }2\rangle 
\tag{7.2999}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pb-ba,pc-dc,\,\text{class }2\rangle 
\tag{7.3000}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pb-ba,pc-ba-dc,\,\text{class }2\rangle 
\tag{7.3001}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pb-dc,pc,\,\text{class }2\rangle 
\tag{7.3002}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pb-dc,pc-ba,\,\text{class }2\rangle 
\tag{7.3003}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pb-dc,pc-dc,\,\text{class }2\rangle 
\tag{7.3004}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pb-dc,pc-ba-dc,\,\text{class }2\rangle 
\tag{7.3005}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pb-ba-dc,pc,\,\text{class }2\rangle 
\tag{7.3006}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pb-ba-dc,pc-ba,\,\text{class }2\rangle 
\tag{7.3007}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pb-ba-dc,pc-xba-dc,\,\text{class }%
2\rangle \;(\text{all }x)  \tag{7.3008}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa-ba,pb-dc,pc,\,\text{class }2\rangle 
\tag{7.3009}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa-ba,pb-dc,pc-ba,\,\text{class }2\rangle 
\tag{7.3010}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa-ba,pb-dc,pc-dc,\,\text{class }2\rangle 
\tag{7.3011}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa-ba,pb-dc,pc-ba-dc,\,\text{class }2\rangle
\tag{7.3012}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pc,pd-pb,\,\text{class }2\rangle 
\tag{7.3013}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pc,pd+ba-pb,\,\text{class }2\rangle 
\tag{7.3014}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pc,pd+ba+dc-pb,\,\text{class }2\rangle 
\tag{7.3015}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pc-ba,pd-pb,\,\text{class }2\rangle 
\tag{7.3016}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pc-ba,pd+dc-pb,\,\text{class }2\rangle 
\tag{7.3017}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pc-dc,pd-pb,\,\text{class }2\rangle 
\tag{7.3018}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pc-dc,pd+ba-pb,\,\text{class }2\rangle 
\tag{7.3019}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pc-ba-dc,pd-pb,\,\text{class }2\rangle 
\tag{7.3020}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa,pc-ba-dc,pd+ba-pb,\,\text{class }2\rangle
\tag{7.3021}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa-ba,pc-ba,pd-pb,\,\text{class }2\rangle 
\tag{7.3022}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa-ba,pc-ba,pd+dc-pb,\,\text{class }2\rangle
\tag{7.3023}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa-ba,pc-ba-xdc,pd-pb,\,\text{class }%
2\rangle \;(x \neq 0)  \tag{7.3024}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa-ba,pc-xdc,pd-pb,\,\text{class }2\rangle
\;(x\neq 0,\,x\sim x^{-1})  \tag{7.3025}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa-dc,pc-ba,pd-pb,\,\text{class }2\rangle 
\tag{7.3026}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa-dc,pc-\omega ba,pd-pb,\,\text{class }%
2\rangle  \tag{7.3027}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa-dc,pc-xba-dc,pd-pb,\,\text{class }%
2\rangle \;(x \neq 0)  \tag{7.3028}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa-ba-dc,pc-xba-ydc,pd-pb,\,\text{class }%
2\rangle \;(x\neq 0,\;y\neq 0,\;y\sim y^{-1})  \tag{7.3029}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,cb,db,pa-ba-dc,pc-xba-xdc,pd+ba-pb,\,\text{class }%
2\rangle \;(x\neq 0,\;x\sim x^{-1})  \tag{7.3030}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa,pb,pc,\,\text{class }2\rangle 
\tag{7.3031}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa-ba,pb,pc,\,\text{class }2\rangle 
\tag{7.3032}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa-ca,pb,pc,\,\text{class }2\rangle 
\tag{7.3033}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa,pb-ba,pc,\,\text{class }2\rangle 
\tag{7.3034}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa,pb-ca,pc,\,\text{class }2\rangle 
\tag{7.3035}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa-ba,pb-ca,pc,\,\text{class }2\rangle 
\tag{7.3036}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa-ca,pb-ba,pc,\,\text{class }2\rangle 
\tag{7.3037}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa,pb,pc-ba,\,\text{class }2\rangle 
\tag{7.3038}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa-ca,pb,pc-ba,\,\text{class }2\rangle 
\tag{7.3039}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa,pb-ba,pc-ba,\,\text{class }2\rangle 
\tag{7.3040}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa-ca,pb-ba,pc-ba,\,\text{class }2\rangle
\tag{7.3041}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa,pb-ba-xca,pc-ba,\,\text{class }%
2\rangle \;(x \neq 0)  \tag{7.3042}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa,pb-ca,pc-ba,\,\text{class }2\rangle 
\tag{7.3043}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa,pb-\omega ca,pc-ba,\,\text{class }%
2\rangle  \tag{7.3044}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa,pb,pc-ca,\,\text{class }2\rangle 
\tag{7.3045}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa-ba,pb,pc-ca,\,\text{class }2\rangle 
\tag{7.3046}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa,pb-xba,pc-ca,\,\text{class }2\rangle
\;(x \neq 0)  \tag{7.3047}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pa,pb-ba-ca,pc-ca,\,\text{class }2\rangle
\tag{7.3048}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pb,pc,pd,\,\text{class }2\rangle 
\tag{7.3049}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pb-ba,pc,pd,\,\text{class }2\rangle 
\tag{7.3050}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pb-ca,pc,pd,\,\text{class }2\rangle 
\tag{7.3051}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pb,pc-ba,pd,\,\text{class }2\rangle 
\tag{7.3052}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pb-ca,pc-ba,pd,\,\text{class }2\rangle 
\tag{7.3053}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pb-\omega ca,pc-ba,pd,\,\text{class }%
2\rangle  \tag{7.3054}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pb-xba,pc-ca,pd,\,\text{class }2\rangle
\;(\text{all }x)  \tag{7.3055}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pb,pc,pd-ba,\,\text{class }2\rangle 
\tag{7.3056}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pb-ca,pc,pd-ba,\,\text{class }2\rangle 
\tag{7.3057}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pb,pc,pd-ca,\,\text{class }2\rangle 
\tag{7.3058}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pb-ba,pc,pd-ca,\,\text{class }2\rangle 
\tag{7.3059}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pb,pc-ba,pd-ca,\,\text{class }2\rangle 
\tag{7.3060}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc,pb,pc-ca,pd-ba,\,\text{class }2\rangle 
\tag{7.3061}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc-\omega ba,pa,pb,pc,\,\text{class }2\rangle
\tag{7.3062}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc-\omega ba,pa-ba,pb,pc,\,\text{class }%
2\rangle  \tag{7.3063}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc-\omega ba,pa,pb-ba-xca,pc,\,\text{class }%
2\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.3064}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc-\omega ba,pa-ca,pb-ba-xca,pc,\,\text{%
class }2\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.3065}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc-\omega ba,pa,pb-ca,pc,\,\text{class }%
2\rangle  \tag{7.3066}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc-\omega ba,pa-ba,pb-ca,pc,\,\text{class }%
2\rangle  \tag{7.3067}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,cb,db-ca,dc-\omega ba,pa,pb-xba-yca,pc-zba-tca,\,%
\text{class }2\rangle  \tag{7.3068}
\end{equation}%
Here, in presentation 7.3068, we put the parameters $x,y,z,t$ in a matrix $%
A=\left( 
\begin{array}{ll}
x & y \\ 
z & t%
\end{array}%
\right) $. (We require $A$ to be non-singular.) We have an action on the
matrices $A$ defined by%
\[
A\rightarrow \alpha ^{-1}\left( 
\begin{array}{ll}
\mu & \nu \\ 
\pm \omega \nu & \pm \mu%
\end{array}%
\right) A\left( 
\begin{array}{ll}
\mu & \nu \\ 
\pm \omega \nu & \pm \mu%
\end{array}%
\right) ^{-1}. 
\]%
Two matrices give isomorphic algebras if they lie in the same orbit under
this action. There are $(p+1)^{2}/2$ orbits.

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa,pb,pc,pd,\,\text{class }2\rangle 
\tag{7.3069}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa-cb,pb,pc,pd,\,\text{class }2\rangle 
\tag{7.3070}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa-ba,pb,pc,pd,\,\text{class }2\rangle 
\tag{7.3071}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa-ca,pb-xcb,pc,pd,\,\text{class }2\rangle \;(x
\neq 0, x \sim x^{-1})  \tag{7.3072}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa-ca-cb,pb-cb,pc,pd,\,\text{class }2\rangle 
\tag{7.3073}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa-\omega cb,pb-ca,pc,pd,\,\text{class }2\rangle
\tag{7.3074}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa-xcb,pb-ca-cb,pc,pd,\,\text{class }2\rangle
\;(\text{all }x\text{ with }1+4x\text{ not a square }\func{mod}p) 
\tag{7.3075}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa-ba,pb-ca,pc,pd,\,\text{class }2\rangle 
\tag{7.3076}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa-ba,pb-cb,pc,pd,\,\text{class }2\rangle 
\tag{7.3077}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa-ba,pb-cb,pc-xba-ca,pd,\,\text{class }%
2\rangle \;(\text{all }x)  \tag{7.3078}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa-ba,pb-ca,pc-cb,pd,\,\text{class }2\rangle 
\tag{7.3079}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa-ba,pb-ca,pc+cb,pd,\,\text{class }2\rangle 
\tag{7.3080}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa-ba,pb-ca,pc-\omega ba+cb,pd,\,\text{class }%
2\rangle  \tag{7.3081}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa,pb,pc,pd-cb,\,\text{class }2\rangle 
\tag{7.3082}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa-ba,pb,pc,pd-cb,\,\text{class }2\rangle 
\tag{7.3083}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa,pb-ba,pc,pd-cb,\,\text{class }2\rangle 
\tag{7.3084}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa-ca,pb-ba,pc,pd-cb,\,\text{class }2\rangle 
\tag{7.3085}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa,pb-ca,pc,pd-cb,\,\text{class }2\rangle 
\tag{7.3086}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa-ba,pb-ca,pc,pd-cb,\,\text{class }2\rangle 
\tag{7.3087}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa,pb-ba,pc-xca,pd-cb,\,\text{class }2\rangle
\;(x \neq 0, x \sim x^{-1})  \tag{7.3088}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa,pb-ba-ca,pc-ca,pd-cb,\,\text{class }2\rangle
\tag{7.3089}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa,pb-\omega ca,pc-ba,pd-cb,\,\text{class }%
2\rangle  \tag{7.3090}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,da,db,dc,pa,pb-xca,pc-ba-ca,pd-cb,\,\text{class }%
2\rangle \;(\text{all }x\text{ with }1+4x\text{ not a square }\func{mod}p) 
\tag{7.3091}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb,pc-cb,pd,\,\text{class }2\rangle 
\tag{7.3092}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-dc,pc-cb,pd,\,\text{class }2\rangle 
\tag{7.3093}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-ba,pc-cb,pd,\,\text{class }2\rangle 
\tag{7.3094}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-ba-dc,pc-cb,pd,\,\text{class }2\rangle 
\tag{7.3095}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-cb,pc-cb,pd,\,\text{class }2\rangle 
\tag{7.3096}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-cb,pc-cb-dc,pd,\,\text{class }2\rangle 
\tag{7.3097}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-cb,pc-ba-cb-dc,pd,\,\text{class }2\rangle
\tag{7.3098}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb,pc,pd,\,\text{class }2\rangle 
\tag{7.3099}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb,pc-dc,pd,\,\text{class }2\rangle 
\tag{7.3100}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb,pc-ba,pd,\,\text{class }2\rangle 
\tag{7.3101}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb,pc-ba-dc,pd,\,\text{class }2\rangle 
\tag{7.3102}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-dc,pc-dc,pd,\,\text{class }2\rangle 
\tag{7.3103}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-dc,pc-ba,pd,\,\text{class }2\rangle 
\tag{7.3104}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-dc,pc-ba-dc,pd,\,\text{class }2\rangle 
\tag{7.3105}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-ba,pc-dc,pd,\,\text{class }2\rangle 
\tag{7.3106}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-ba,pc-ba-dc,pd,\,\text{class }2\rangle 
\tag{7.3107}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-ba-dc,pc-ba-xdc,pd,\,\text{class }%
2\rangle \;(x \neq 0)  \tag{7.3108}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-zcb,pc-xba-ycb,pd-dc,\,\text{class }%
2\rangle \;(x=0,1,\;y=0,1,\;z\neq 0,-1)  \tag{7.3109}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb,pc,pd-dc,\,\text{class }2\rangle 
\tag{7.3110}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb,pc-ba,pd-dc,\,\text{class }2\rangle 
\tag{7.3111}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-ba,pc,pd-dc,\,\text{class }2\rangle 
\tag{7.3112}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-ba,pc-ba,pd-dc,\,\text{class }2\rangle 
\tag{7.3113}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb,pc-cb,pd-dc,\,\text{class }2\rangle 
\tag{7.3114}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-ba,pc-cb,pd-dc,\,\text{class }2\rangle 
\tag{7.3115}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb+cb,pc,pd-dc,\,\text{class }2\rangle 
\tag{7.3116}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb+cb,pc-cb,pd-dc,\,\text{class }2\rangle 
\tag{7.3117}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb+cb,pc-ba,pd-dc,\,\text{class }2\rangle 
\tag{7.3118}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb+cb,pc-ba-cb,pd-dc,\,\text{class }2\rangle
\tag{7.3119}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb+cb-dc,pc,pd-dc,\,\text{class }2\rangle 
\tag{7.3120}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb+cb-dc,pc-cb,pd-dc,\,\text{class }2\rangle
\tag{7.3121}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb+cb-dc,pc-ba,pd-dc,\,\text{class }2\rangle
\tag{7.3122}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb+cb-dc,pc-ba-cb,pd-dc,\,\text{class }%
2\rangle  \tag{7.3123}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-ba-cb-tdc,pc-ba-ydc,pd-cb,\,\text{class }%
2\rangle \;(\text{all }t,y)  \tag{7.3124}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-cb-tdc,pc-xba-ydc,pd-cb,\,\text{class }%
2\rangle \;(\text{all }t,\;x=0,1,\;y=0,1)  \tag{7.3125}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-tdc,pc-xba-ydc,pd-cb,\,\text{class }%
2\rangle \;(t=0,1,\omega ,\;x=0,1,\;y=0,1)  \tag{7.3126}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-ba-tdc,pc-ydc,pd-cb,\,\text{class }%
2\rangle \;(t=0,1,\omega ,\;y=0,1)  \tag{7.3127}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-ba-tdc,pc-ba-ydc,pd-cb,\,\text{class }%
2\rangle \;(t=1,\omega ,\text{ all }y)  \tag{7.3128}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-ba,pc-ba-ydc,pd-cb,\,\text{class }%
2\rangle \;(y=0,1,\omega )  \tag{7.3129}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-ba-cb-tdc,pc-ydc,pd-cb,\,\text{class }%
2\rangle \;(\text{all }t,\;y=0,1)  \tag{7.3130}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-cb,pc,pd-ba,\,\text{class }2\rangle 
\tag{7.3131}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-cb,pc-dc,pd-ba,\,\text{class }2\rangle 
\tag{7.3132}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-cb,pc-cb,pd-ba,\,\text{class }2\rangle 
\tag{7.3133}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-cb,pc-cb-dc,pd-ba,\,\text{class }2\rangle
\tag{7.3134}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-ydc,pc-zcb-tdc,pd-ba,\,\text{class }%
2\rangle \;(y=0,1,\;z=0,1,\;t=0,1)  \tag{7.3135}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-xcb,pc-tdc,pd-ba-dc,\,\text{class }%
2\rangle \;(\text{all }x,\;t=0,1)  \tag{7.3136}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-xcb-dc,pc-ydc,pd-ba-dc,\,\text{class }%
2\rangle \;(x=0,-1, y=0,1)  \tag{7.3137}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb-xcb-ydc,pc-cb,pd-ba-dc,\,\text{class }%
2\rangle \;(\text{all }x,\;\,y=0,1)  \tag{7.3138}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa,pb+cb-xdc,pc-cb-dc,pd-ba-dc,\,\text{class }%
2\rangle \;(\text{all }x)  \tag{7.3139}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-dc,pb-xba-ycb,pc-cb,pd-dc,\,\text{class }%
2\rangle \;(x=0,1,\text{ all }y)  \tag{7.3140}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-dc,pb-ycb,pc-xba,pd-dc,\,\text{class }%
2\rangle \;(x=0,1,\text{ all }y)  \tag{7.3141}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-dc,pb-ba,pc-zba,pd-dc,\,\text{class }%
2\rangle \;(z=0,1)  \tag{7.3142}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-dc,pb-xba-cb,pc-yba-dc,pd-cb,\,\text{class }%
2\rangle \;(\text{all }x,\,y)  \tag{7.3143}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-dc,pb-ba-cb,pc-yba,pd-cb,\,\text{class }%
2\rangle \;(\text{all }y)  \tag{7.3144}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-dc,pb-cb,pc-yba,pd-cb,\,\text{class }%
2\rangle \;(y=0,1,\omega )  \tag{7.3145}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-dc,pb-ba,pc-yba-dc,pd-cb,\,\text{class }%
2\rangle \;(\text{all }y)  \tag{7.3146}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-dc,pb,pc-yba-dc,pd-cb,\,\text{class }%
2\rangle \;(y=0,1)  \tag{7.3147}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-dc,pb-xba,pc-yba,pd-cb,\,\text{class }%
2\rangle \;(x=0,1,\;y=0,1)  \tag{7.3148}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-dc,pb-ba,pc-\omega ba,pd-cb,\,\text{class }%
2\rangle \;(p=1\func{mod}3)  \tag{7.3149}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-dc,pb-ba,pc-\omega ^{2}ba,pd-cb,\,\text{%
class }2\rangle \;(p=1\func{mod}3)  \tag{7.3150}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-dc,pb,pc,pd-ba,\,\text{class }2\rangle 
\tag{7.3151}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-dc,pb-cb,pc-cb-dc,pd-ba,\,\text{class }%
2\rangle  \tag{7.3152}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-dc,pb-cb,pc-ycb,pd-ba,\,\text{class }%
2\rangle \;(\text{all }y)  \tag{7.3153}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-dc,pb-xcb,pc-ycb,pd-ba-dc,\,\text{class }%
2\rangle \;(\text{all }x,\,y)  \tag{7.3154}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-dc,pb-xcb,pc-(1+x)^{-1}cb-dc,pd-ba-dc,\,%
\text{class }2\rangle \;(x\neq -1)  \tag{7.3155}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-ba-ydc,pc-zba-tcb,pd-dc,\,\text{class }%
2\rangle \;(y=0,1,\text{ all }z,t)  \tag{7.3156}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-ydc,pc-zba-cb,pd-dc,\,\text{class }%
2\rangle \;(y=0,1,\text{ all }z)  \tag{7.3157}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-ydc,pc-zba,pd-dc,\,\text{class }%
2\rangle \;(y=0,1,\;z=0,1,\omega )  \tag{7.3158}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb,pc,pd-cb,\,\text{class }2\rangle 
\tag{7.3159}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb,pc-ba,pd-cb,\,\text{class }2\rangle 
\tag{7.3160}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb,pc-\omega ba,pd-cb,\,\text{class }%
2\rangle  \tag{7.3161}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-dc,pc-ba,pd-cb,\,\text{class }2\rangle
\tag{7.3162}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-dc,pc-\omega ba,pd-cb,\,\text{class }%
2\rangle  \tag{7.3163}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-\omega dc,pc-\omega ba,pd-cb,\,\text{%
class }2\rangle  \tag{7.3164}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-ydc,pc-zba-dc,pd-cb,\,\text{class }%
2\rangle \;(y=1,\omega ,\text{ all }z)  \tag{7.3165}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb,pc-xba-dc,pd-cb,\,\text{class }%
2\rangle \;(x=0,1,\omega )  \tag{7.3166}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-ba-ydc,pc-zba-tdc,pd-cb,\,\text{class }%
2\rangle \;(y=1,\omega ,\text{ all }z,\;t\neq 0,1,-1,\;t\sim t^{-1}) 
\tag{7.3167}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-ba,pc-zba-tdc,pd-cb,\,\text{class }%
2\rangle \;(z=0,1,\omega ,\;t\neq 0,1,-1,\;t\sim t^{-1})  \tag{7.3168}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-ba-dc,pc-xba-tdc,pd-cb,\,\text{class }%
2\rangle \;(t=1,-1, x \neq 0)  \tag{7.3169}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-ba-\omega dc,pc-(\omega \ast
x^{2})ba-tdc,pd-cb,\,\text{class }2\rangle \;(t=1,-1,\;x\neq 0,\;x\sim -x) 
\tag{7.3170}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-ba,pc-zba-tdc,pd-cb,\,\text{class }%
2\rangle \;(t=1,-1,z=0,1,\omega )  \tag{7.3171}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-ba-ydc,pc-zba-tdc,pd-cb-dc,\,\text{%
class }2\rangle \;(\text{all }y,z,t)  \tag{7.3172}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-ydc,pc-zba-dc,pd-cb-dc,\,\text{class }%
2\rangle \;(\text{all }y,z)  \tag{7.3173}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-ydc,pc-zba,pd-cb-dc,\,\text{class }%
2\rangle \;(\text{all }y,\;z=0,1,\omega )  \tag{7.3174}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-ba-ydc,pc-zcb-tdc,pd-ba-dc,\,\text{%
class }2\rangle \;(\text{all }y,z,t)  \tag{7.3175}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-dc,pc-zcb-tdc,pd-ba-dc,\,\text{class }%
2\rangle \;(\text{all }z,t)  \tag{7.3176}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb,pc-cb-tdc,pd-ba-dc,\,\text{class }%
2\rangle \;(\text{all }t)  \tag{7.3177}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb,pc-tdc,pd-ba-dc,\,\text{class }%
2\rangle \;(t=0,1,\omega )  \tag{7.3178}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-cb,pb-xba-ydc,pc-zba-tdc,pd-ba-cb-dc,\,\text{%
class }2\rangle \;(\text{all }x,\,y,z,t,\;(x,\,y,z,t)\sim (t+1,z+1,y-1,x-1))
\tag{7.3179}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-ba,pb-xcb,pc-tcb,pd-dc,\,\text{class }%
2\rangle \;(\text{all }x,t,\;(x,t)\sim (t,x))  \tag{7.3180}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-ba,pb-xcb,pc-ba+cb,pd-dc,\,\text{class }%
2\rangle \;(\text{all }x)  \tag{7.3181}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-ba,pb+cb-dc,pc-ba+cb,pd-dc,\,\text{class }%
2\rangle  \tag{7.3182}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-ba,pb-xcb,pc-zcb,pd-ba-dc,\,\text{class }%
2\rangle \;(\text{all }x,z)  \tag{7.3183}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-ba,pb+cb-dc,pc-zcb,pd-ba-dc,\,\text{class }%
2\rangle \;(\text{all }z)  \tag{7.3184}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-ba,pb-xcb,pc+cb-dc,pd-ba-dc,\,\text{class }%
2\rangle \;(x \neq -1)  \tag{7.3185}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-ba-dc,pb-xcb,pc-zcb,pd-ba-tdc,\,\text{class }%
2\rangle \;(t\neq 0,\;\text{all }x,z,\;(x,z)\sim (tz,\frac{x}{t})) 
\tag{7.3186}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-ba-dc,pb-xcb,pc-zcb-dc,pd-ba-dc,\,\text{%
class }2\rangle \;(x,\;z\neq 0,\;(x,z)\sim (z,x))  \tag{7.3187}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-ba-dc,pb-dc,pc-zcb,pd-ba-dc,\,\text{class }%
2\rangle \;(z \neq 0)  \tag{7.3188}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-ba-dc,pb,pc-dc,pd-ba-dc,\,\text{class }%
2\rangle  \tag{7.3189}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-ba-dc,pb-dc,pc-xdc,pd-ba-dc,\,\text{class }%
2\rangle \;(x\neq 0,\;x\sim x^{-1})  \tag{7.3190}
\end{equation}

In the presentation below we have $\;t\neq 0,1,\;$all $x,z$ such that $%
(x+t)(1+z)=1\func{mod}p,\;(x,z)\sim (tz,\frac{x}{t})$:

\begin{equation}
\langle a,b,c,d\,|\,ca,da,db,pa-ba-dc,pb-xcb,pc-zcb-dc,pd-ba-tdc,\,\text{%
class }2\rangle  \tag{7.3191}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa,pb-ba,pc-ca,pd-xda,\,\text{class }2\rangle
\;(x \neq 0)  \tag{7.3192}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa,pb-ba-ca,pc-ca,pd-da,\,\text{class }2\rangle
\tag{7.3193}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa,pb-xda,pc-ba-yda,pd-ca-da,\,\text{class }%
2\rangle \;(x\neq 0,\text{ all }y)  \tag{7.3194}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa,pb-xda,pc-ba-yda,pd-ca,\,\text{class }%
2\rangle \;(x\neq 0,\;x\sim -x,\,y=1,\omega )  \tag{7.3195}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa,pb-da,pc-ba,pd-ca,\,\text{class }2\rangle 
\tag{7.3196}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa,pb-\omega da,pc-ba,pd-ca,\,\text{class }%
2\rangle \;(p=1\func{mod}3)  \tag{7.3197}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa,pb-\omega ^{2}da,pc-ba,pd-ca,\,\text{class }%
2\rangle \;(p=1\func{mod}3)  \tag{7.3198}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa,pb-ca,pc-da,pd,\,\text{class }2\rangle 
\tag{7.3199}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa-ba,pb-ca,pc-da,pd,\,\text{class }2\rangle 
\tag{7.3200}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa,pb-ba,pc-da,pd,\,\text{class }2\rangle 
\tag{7.3201}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa-ca,pb-ba,pc-da,pd,\,\text{class }2\rangle 
\tag{7.3202}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa,pb-ba,pc-ca,pd,\,\text{class }2\rangle 
\tag{7.3203}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa-da,pb-ba,pc-ca,pd,\,\text{class }2\rangle 
\tag{7.3204}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa,pb-xca,pc-ba-ca,pd,\,\text{class }2\rangle
\;(x \neq 0)  \tag{7.3205}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa-da,pb-xca,pc-ba-ca,pd,\,\text{class }%
2\rangle \;(x \neq 0)  \tag{7.3206}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa,pb-ca,pc-ba,pd,\,\text{class }2\rangle 
\tag{7.3207}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa-da,pb-ca,pc-ba,pd,\,\text{class }2\rangle 
\tag{7.3208}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa,pb-\omega ca,pc-ba,pd,\,\text{class }2\rangle
\tag{7.3209}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa-da,pb-\omega ca,pc-ba,pd,\,\text{class }%
2\rangle  \tag{7.3210}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa,pb-ba,pc,pd,\,\text{class }2\rangle 
\tag{7.3211}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa-ca,pb-ba,pc,pd,\,\text{class }2\rangle 
\tag{7.3212}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa,pb-ca,pc,pd,\,\text{class }2\rangle 
\tag{7.3213}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa-ba,pb-ca,pc,pd,\,\text{class }2\rangle 
\tag{7.3214}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa-da,pb-ca,pc,pd,\,\text{class }2\rangle 
\tag{7.3215}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa,pb,pc,pd,\,\text{class }2\rangle 
\tag{7.3216}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,db,dc,pa-ba,pb,pc,pd,\,\text{class }2\rangle 
\tag{7.3217}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb,pc,pd,\,\text{class }2\rangle 
\tag{7.3218}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb-ba,pc,pd,\,\text{class }2\rangle 
\tag{7.3219}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb-cb,pc,pd,\,\text{class }2\rangle 
\tag{7.3220}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb-cb,pc-ba,pd,\,\text{class }2\rangle 
\tag{7.3221}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb-xdb,pc-ba,pd,\,\text{class }2\rangle
\;(\text{all }x)  \tag{7.3222}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb,pc-cb,pd,\,\text{class }2\rangle 
\tag{7.3223}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb-ba,pc-cb,pd,\,\text{class }2\rangle 
\tag{7.3224}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb-db,pc-cb,pd,\,\text{class }2\rangle 
\tag{7.3225}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb,pc-db,pd,\,\text{class }2\rangle 
\tag{7.3226}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb-ba,pc-db,pd,\,\text{class }2\rangle 
\tag{7.3227}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb-\omega ba,pc-db,pd,\,\text{class }%
2\rangle  \tag{7.3228}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb-cb,pc-db,pd,\,\text{class }2\rangle 
\tag{7.3229}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb-xdb,pc-ba,pd-cb,\,\text{class }%
2\rangle \;(\text{all }x)  \tag{7.3230}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb,pc-ba,pd-db,\,\text{class }2\rangle 
\tag{7.3231}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb-cb,pc-ba,pd-db,\,\text{class }2\rangle
\tag{7.3232}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb,pc-cb,pd-db,\,\text{class }2\rangle 
\tag{7.3233}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb-ba,pc-cb,pd-db,\,\text{class }2\rangle
\tag{7.3234}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb,pc-cb-db,pd-db,\,\text{class }2\rangle
\tag{7.3235}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb-ba,pc-cb-db,pd-db,\,\text{class }%
2\rangle  \tag{7.3236}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb-\omega ba,pc-cb-db,pd-db,\,\text{class 
}2\rangle  \tag{7.3237}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb,pc-xdb,pd-cb-db,\,\text{class }%
2\rangle \;(x\neq 0,-\frac{1}{4})  \tag{7.3238}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb-ba,pc-xdb,pd-cb-db,\,\text{class }%
2\rangle \;(x\neq 0,-\frac{1}{4})  \tag{7.3239}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb,pc-db,pd-cb,\,\text{class }2\rangle 
\tag{7.3240}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb-ba,pc-db,pd-cb,\,\text{class }2\rangle
\tag{7.3241}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb,pc-\omega db,pd-cb,\,\text{class }%
2\rangle  \tag{7.3242}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa,pb-ba,pc-\omega db,pd-cb,\,\text{class }%
2\rangle  \tag{7.3243}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-ba,pb,pc,pd,\,\text{class }2\rangle 
\tag{7.3244}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-ba,pb-cb,pc,pd,\,\text{class }2\rangle 
\tag{7.3245}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-ba,pb,pc-xcb,pd,\,\text{class }2\rangle
\;(x \neq 0)  \tag{7.3246}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-ba,pb-db,pc-xcb,pd,\,\text{class }%
2\rangle \;(x \neq 0)  \tag{7.3247}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-ba,pb,pc,pd-cb,\,\text{class }2\rangle 
\tag{7.3248}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-ba,pb-db,pc,pd-cb,\,\text{class }2\rangle
\tag{7.3249}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-ba,pb,pc-xcb,pd-xdb,\,\text{class }%
2\rangle \;(x \neq 0)  \tag{7.3250}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-ba,pb,pc-ba+cb,pd+db,\,\text{class }%
2\rangle  \tag{7.3251}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-ba,pb,pc-xcb,pd-ydb,\,\text{class }%
2\rangle \;(x,\,y\neq 0,\;x\neq y,\;(x,\,y)\sim (y,x))  \tag{7.3252}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-ba,pb,pc-xcb,pd-ba+db,\,\text{class }%
2\rangle \;(x \neq 0,-1)  \tag{7.3253}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-ba,pb,pc-xcb,pd-cb-xdb,\,\text{class }%
2\rangle \;(x \neq 0)  \tag{7.3254}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-ba,pb,pc-ba+cb,pd-cb+db,\,\text{class }%
2\rangle  \tag{7.3255}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-ba,pb,pc-xdb,pd-cb-ydb,\,\text{class }%
2\rangle \;(x\neq 0,\text{ all }y\text{ with }4x+y^{2}\text{ not a square }%
\func{mod}p)  \tag{7.3256}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb,pc,pd,\,\text{class }2\rangle 
\tag{7.3257}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-ba,pc,pd,\,\text{class }2\rangle 
\tag{7.3258}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-db,pc,pd,\,\text{class }2\rangle 
\tag{7.3259}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-\omega db,pc,pd,\,\text{class }%
2\rangle  \tag{7.3260}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb,pc,pd-ba,\,\text{class }2\rangle 
\tag{7.3261}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-ba,pc,pd-ba,\,\text{class }2\rangle
\tag{7.3262}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-db,pc,pd-ba,\,\text{class }2\rangle
\tag{7.3263}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-\omega db,pc,pd-ba,\,\text{class }%
2\rangle  \tag{7.3264}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-xba-ydb,pc,pd-db,\,\text{class }%
2\rangle \;(x=0,1,\text{ all }y)  \tag{7.3265}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-ydb,pc-ba,pd-xdb,\,\text{class }%
2\rangle \;(\text{all }x,\,y,\;x\sim -x)  \tag{7.3266}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-ydb,pc-\omega ba,pd-xdb,\,\text{%
class }2\rangle \;(\text{all }x,\,y,\;x\sim -x)  \tag{7.3267}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-ba-(x^{2}+1)db,pc-ba,pd-xdb,\,\text{%
class }2\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.3268}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-ba-(x^{2}+\omega )db,pc-\omega
ba,pd-xdb,\,\text{class }2\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.3269}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-xba-zdb,pc-db,pd,\,\text{class }%
2\rangle \;(\text{all }x,\;z=1,\omega )  \tag{7.3270}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb,pc-db,pd,\,\text{class }2\rangle 
\tag{7.3271}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-ba,pc-db,pd,\,\text{class }2\rangle
\tag{7.3272}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-\omega ba,pc-db,pd,\,\text{class }%
2\rangle  \tag{7.3273}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-\omega ^{2}ba,pc-db,pd,\,\text{%
class }2\rangle \;(p=1\func{mod}4)  \tag{7.3274}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-\omega ^{3}ba,pc-db,pd,\,\text{%
class }2\rangle \;(p=1\func{mod}4)  \tag{7.3275}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb,pc-db,pd-ba,\,\text{class }2\rangle
\tag{7.3276}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb,pc-db,pd-\omega ba,\,\text{class }%
2\rangle \;(p=1\func{mod}3)  \tag{7.3277}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb,pc-db,pd-\omega ^{2}ba,\,\text{%
class }2\rangle \;(p=1\func{mod}3)  \tag{7.3278}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-xdb,pc-db,pd-ba,\,\text{class }%
2\rangle \;(x \neq 0, x \sim x^{\prime }\text{ if }x^3=x^{\prime 3 }\func{mod%
} p)  \tag{7.3279}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-xdb,pc-db,pd-\omega ba,\,\text{%
class }2\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}%
\func{mod}p,\;p=1\func{mod}3)  \tag{7.3280}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-xdb,pc-db,pd-\omega ^{2}ba,\,\text{%
class }2\rangle \;(x\neq 0,x \sim x^{\prime }\text{ if }x^3=x^{\prime 3}%
\func{mod}p,\;p=1\func{mod}3)  \tag{7.3281}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-xba-ydb,pc-db,pd-ba,\,\text{class }%
2\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}\func{mod%
}p,\text{ all }y)  \tag{7.3282}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-xba-ydb,pc-db,pd-\omega ba,\,\text{%
class }2\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}%
\func{mod}p,\text{ all }y,\;p=1\func{mod}3)  \tag{7.3283}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,da,dc-ba,pa-cb,pb-xba-ydb,pc-db,pd-\omega ^{2}ba,\,%
\text{class }2\rangle \;(x\neq 0,\;x\sim x^{\prime }\text{ if }%
x^{3}=x^{\prime 3}\func{mod}p,\text{ all }y,\;p=1\func{mod}3)  \tag{7.3284}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|%
\,cb,da,db-ca,pa-jba-kca-mdc,pb-nba-rca-sdc,pc-tba-uca-vdc,pd-xba-yca-zdc,\,%
\text{class }2\rangle  \tag{7.3285}
\end{equation}%
In presentation 7.3285 we have 12 parameters giving the values of $%
pa,pb,pc,pd$. We write%
\[
\left( 
\begin{array}{c}
pa \\ 
pb \\ 
pc \\ 
pd%
\end{array}%
\right) =A\left( 
\begin{array}{c}
ba \\ 
ca \\ 
dc%
\end{array}%
\right) 
\]%
with $A$ a $4\times 3$ matrix. The isomorphism classes given by the matrices
correspond to their orbits under transformations of the form 
\[
A\longmapsto (\alpha \delta -\beta \gamma )^{-1}\left( 
\begin{array}{cccc}
\alpha \lambda & \beta \lambda & \beta \mu & -\alpha \mu \\ 
\gamma \lambda & \delta \lambda & \delta \mu & -\gamma \mu \\ 
\gamma \nu & \delta \nu & \delta \xi & -\gamma \xi \\ 
-\alpha \nu & -\beta \nu & -\beta \xi & \alpha \xi%
\end{array}%
\right) A\left( 
\begin{array}{ccc}
\lambda ^{2} & 2\lambda \mu & \mu ^{2} \\ 
\lambda \nu & \lambda \xi +\mu \nu & \mu \xi \\ 
\nu ^{2} & 2\nu \xi & \xi ^{2}%
\end{array}%
\right) ^{-1}. 
\]%
We note that if we multiply $\alpha ,\beta ,\gamma ,\delta $ through by a
factor $k$ (in the expression above), and multiply $\lambda ,\mu ,\nu ,\xi $
through by a factor $l$, then the image of $A$ is multiplied by a factor $%
k^{-1}l^{-1}$. So we can ignore the factor $(\alpha \delta -\beta \gamma
)^{-1}$ and still get the same orbits.

If we take $\mu =0$ in the matrices above, then we obtain a subgroup $H$ of
the automorphism group of index $p+1$. The subspace spanned by $a,b$ is
invariant under $H$, and it turns out that $pa,pb$ has 11 orbits under the
action of $H$. So the problem of computing orbits of 12 parameters reduces
to 11 problems with 6 parameters. There are 550 orbits when $p=3$ and 
\begin{eqnarray*}
\allowbreak p^{5}+p^{4}+4p^{3}+6p^{2}+18p+19\text{ if }p &=&1\func{mod}3, \\
\allowbreak p^{5}+p^{4}+4p^{3}+6p^{2}+16p+17\text{ if }p &=&2\func{mod}3
\end{eqnarray*}%
orbits when $p>3$. This implies that we need presentations with at least 5
parameters to describe this situation, so 6 parameters is not at all bad!
Nevertheless computing the orbits for a specific prime $p$ require serious
computational effort.

\begin{equation}
\langle a,b,c,d\,|\,da,db-\omega
ca,dc-ba,pa-jba-kca-mcb,pb-nba-rca-scb,pc-tba-uca-vcb,pd-xba-yca-zcb,\,\text{%
class }2\rangle  \tag{7.3286}
\end{equation}%
In presentation 7.3286 we again have 12 parameters giving the values of $%
pa,pb,pc,pd$. We let 
\[
\left( 
\begin{array}{c}
pa \\ 
pb \\ 
pc \\ 
pd%
\end{array}%
\right) =A\left( 
\begin{array}{c}
ba \\ 
ca \\ 
cb%
\end{array}%
\right) 
\]%
where $A$ is a $4\times 3$ matrix. The isomorphism classes given by the
matrices correspond to their orbits under transformations of the form%
\[
A\mapsto \left( 
\begin{array}{cccc}
\alpha & 0 & 0 & \delta \\ 
\pm \lambda & \pm \gamma & \pm \omega \beta & \pm \mu \\ 
\nu & \beta & \gamma & \xi \\ 
\pm \omega \delta & 0 & 0 & \pm \alpha%
\end{array}%
\right) AB^{-1}, 
\]%
where%
\[
B=\left( 
\begin{array}{ccc}
\pm (\alpha \gamma -\omega \beta \delta ) & \pm (\omega \alpha \beta -\omega
\gamma \delta ) & 0 \\ 
\alpha \beta -\gamma \delta & \alpha \gamma -\omega \beta \delta & 0 \\ 
\pm (\beta \lambda -\gamma \nu +\omega \beta \xi -\gamma \mu ) & \pm (\gamma
\lambda -\omega \beta \mu +\omega \gamma \xi -\omega \beta \nu ) & \pm
(\gamma ^{2}-\omega \beta ^{2})%
\end{array}%
\right) . 
\]

We note that $\langle a,d\rangle +L^{2}$ is a characteristic subalgebra, and
first investigate the orbits of $pa,pd$. We consider three separate cases: $%
pa=pd=0$, $pa$ and $pd$ span a one dimensional subspace, and $pa,pd$ are
linearly independent. It turns out that there are $p+4$ orbits of $pa,pd$.
We pick a set of orbit representatives for $pa,pd$, and work out a set of
orbit representatives for $pc,pd$ in each case. There are $\frac{9}{2}p+%
\frac{13}{2}+3p^{2}+\frac{1}{2}p^{4}+\frac{1}{2}p^{3}$ orbits of matrices $A$%
.

\subsection{Descendants of 5.3}

$p^{4}+5p^{3}+19p^{2}+64p+140+(p+6)\gcd (p-1,3)+(p+7)\gcd (p-1,4)+\gcd
(p-1,5)$ algebras

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc,pa,pb,pc-baa,pd-bab,\,\text{class }%
3\rangle  \tag{7.3287}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc,pa,pb,pc-bab,pd,\,\text{class }3\rangle 
\tag{7.3288}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc,pa-baa,pb,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.3289}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc,pa,pb-baa,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.3290}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc,pa,pb-\omega baa,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.3291}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc,pa,pb,pc,pd,\,\text{class }3\rangle 
\tag{7.3292}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc,pa-bab,pb,pc,pd,\,\text{class }3\rangle 
\tag{7.3293}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc,pa-\omega bab,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3294}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc,pa-baa,pb,pc,pd,\,\text{class }3\rangle 
\tag{7.3295}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc,pa-baa,pb-xbab,pc,pd,\,\text{class }%
3\rangle \;(x\neq 0,\,x\sim x^{-1})  \tag{7.3296}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc,pa-baa-bab,pb-bab,pc,pd,\,\text{class }%
3\rangle  \tag{7.3297}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc,pa-baa-\omega bab,pb-bab,pc,pd,\,\text{%
class }3\rangle  \tag{7.3298}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc,pa-\omega bab,pb-baa,pc,pd,\,\text{class }%
3\rangle  \tag{7.3299}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc,pa-xbab,pb-baa-bab,pc,pd,\,\text{class }%
3\rangle \;(\text{all }x\text{ with }1+4x\text{ not a square }\func{mod}p) 
\tag{7.3300}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa,pb,pc,pd,\,\text{class }3\rangle 
\tag{7.3301}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-bab,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3302}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-\omega bab,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3303}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-baa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3304}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-xbaa,pb-bab,pc,pd,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.3305}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-baa-bab,pb-bab,pc,pd,\,\text{class 
}3\rangle  \tag{7.3306}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-baa-\omega bab,pb-bab,pc,pd,\,%
\text{class }3\rangle  \tag{7.3307}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-baa-xbab,pb-baa,pc,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3308}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa,pb-baa,pc,pd,\,\text{class }%
3\rangle  \tag{7.3309}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-bab,pb-baa,pc,pd,\,\text{class }%
3\rangle  \tag{7.3310}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-\omega bab,pb-baa,pc,pd,\,\text{%
class }3\rangle  \tag{7.3311}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-baa-xbab,pb-\omega baa,pc,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.3312}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa,pb-\omega baa,pc,pd,\,\text{class }%
3\rangle  \tag{7.3313}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-bab,pb-\omega baa,pc,pd,\,\text{%
class }3\rangle  \tag{7.3314}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-\omega bab,pb-\omega baa,pc,pd,\,%
\text{class }3\rangle  \tag{7.3315}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa,pb-xbaa,pc-bab,pd,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.3316}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-baa,pb,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.3317}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa,pb,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.3318}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-bab,pb,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.3319}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-\omega bab,pb,pc-baa,pd,\,\text{%
class }3\rangle  \tag{7.3320}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-\omega ^{2}bab,pb,pc-baa,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.3321}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-\omega ^{3}bab,pb,pc-baa,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.3322}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-xbab,pb-bab,pc-baa,pd,\,\text{%
class }3\rangle \;(\text{all }x,\,x \sim x^{\prime }\text{ if }x^3=x^{\prime
3}\func{mod}p)  \tag{7.3323}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-xbab,pb-\omega bab,pc-baa,pd,\,%
\text{class }3\rangle \;(\text{all }x,\,x \sim x^{\prime }\text{ if }%
x^3=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.3324}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-xbab,pb-\omega ^{2}bab,pc-baa,pd,\,%
\text{class }3\rangle \;(\text{all }x,\,x \sim x^{\prime }\text{ if }%
x^3=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.3325}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa,pb,pc-bab,pd-baa,\,\text{class }%
3\rangle  \tag{7.3326}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa,pb,pc,pd-baa,\,\text{class }%
3\rangle  \tag{7.3327}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-bab,pb,pc,pd-baa,\,\text{class }%
3\rangle  \tag{7.3328}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-\omega bab,pb,pc,pd-baa,\,\text{%
class }3\rangle  \tag{7.3329}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa,pb-bab,pc,pd-baa,\,\text{class }%
3\rangle  \tag{7.3330}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-bab,pb-bab,pc,pd-baa,\,\text{class 
}3\rangle  \tag{7.3331}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-\omega bab,pb-bab,pc,pd-baa,\,%
\text{class }3\rangle  \tag{7.3332}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa,pb,pc-baa,pd-bab,\,\text{class }%
3\rangle  \tag{7.3333}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa,pb-baa,pc,pd-bab,\,\text{class }%
3\rangle  \tag{7.3334}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa,pb-\omega baa,pc,pd-bab,\,\text{%
class }3\rangle  \tag{7.3335}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa,pb,pc,pd-bab,\,\text{class }%
3\rangle  \tag{7.3336}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc,pa-baa,pb,pc,pd-bab,\,\text{class }%
3\rangle  \tag{7.3337}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa,pb,pc,pd,\,\text{class }3\rangle 
\tag{7.3338}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3339}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-bab,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3340}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa-bab,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3341}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-\omega bab,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3342}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa-\omega bab,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3343}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-bab,pb-baa,pc,pd,\,\text{class }%
3\rangle  \tag{7.3344}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-bab,pb-\omega baa,pc,pd,\,\text{%
class }3\rangle  \tag{7.3345}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-\omega bab,pb-\omega baa,pc,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.3346}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-\omega bab,pb-baa,pc,pd,\,\text{%
class }3\rangle \;(p=3\func{mod}4)  \tag{7.3347}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa-xbab,pb-baa,pc,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3348}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa-xbab,pb-\omega baa,pc,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.3349}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa,pb-xbab,pc,pd,\,\text{class }%
3\rangle \;(x\neq 0,\,x\sim x^{-1})  \tag{7.3350}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa-bab,pb-xbab,pc,pd,\,\text{%
class }3\rangle \;(x \neq 0)  \tag{7.3351}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa-\omega bab,pb-xbab,pc,pd,\,%
\text{class }3\rangle \;(x\neq 0)  \tag{7.3352}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa-bab,pb-xbaa-ybab,pc,pd,\,\text{%
class }3\rangle \;(x\neq 0,\,y\neq 0,1,-1,\,y\sim y^{-1})  \tag{7.3353}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa-\omega
bab,pb-xbaa-ybab,pc,pd,\,\text{class }3\rangle \;(x\neq 0,\,y\neq
0,1,-1,\,y\sim y^{-1})  \tag{7.3354}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa-bab,pb-x^{2}baa+bab,pc,pd,\,%
\text{class }3\rangle \;(x\neq 0,\,x\sim -x)  \tag{7.3355}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa-bab,pb-x^{2}baa-bab,pc,pd,\,%
\text{class }3\rangle \;(x\neq 0,\,x\sim -x)  \tag{7.3356}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa-bab,pb-(x^{2}\ast \omega
)baa+bab,pc,pd,\,\text{class }3\rangle \;(x\neq 0,\,x\sim -x)  \tag{7.3357}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa-bab,pb-(x^{2}\ast \omega
)baa-bab,pc,pd,\,\text{class }3\rangle \;(x\neq 0,\,x\sim -x)  \tag{7.3358}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa-\omega bab,pb-(x^{2}\ast
\omega )baa+bab,pc,pd,\,\text{class }3\rangle \;(x\neq 0,\,x\sim -x,\;p=1%
\func{mod}4)  \tag{7.3359}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa-\omega bab,pb-(x^{2}\ast
\omega )baa-bab,pc,pd,\,\text{class }3\rangle \;(x\neq 0,\,x\sim -x,\;p=1%
\func{mod}4)  \tag{7.3360}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa-\omega
bab,pb-x^{2}baa+bab,pc,pd,\,\text{class }3\rangle \;(x\neq 0,\,x\sim -x,\;p=3%
\func{mod}4)  \tag{7.3361}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa-\omega
bab,pb-x^{2}baa-bab,pc,pd,\,\text{class }3\rangle \;(x\neq 0,\,x\sim -x,\;p=3%
\func{mod}4)  \tag{7.3362}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa,pb,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.3363}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-bab,pb,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.3364}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-\omega bab,pb,pc-baa,pd,\,\text{%
class }3\rangle  \tag{7.3365}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-xbab,pb-bab,pc-baa,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3366}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-xbab,pb-ybab,pc-baa-bab,pd,\,\text{%
class }3\rangle \;(\text{all }x,\,y,\,(x,\,y)\sim (y,x))  \tag{7.3367}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa,pb,pc-bab,pd-baa,\,\text{class }%
3\rangle  \tag{7.3368}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa,pb,pc,pd-baa,\,\text{class }%
3\rangle  \tag{7.3369}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-bab,pb,pc,pd-baa,\,\text{class }%
3\rangle  \tag{7.3370}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-\omega bab,pb,pc,pd-baa,\,\text{%
class }3\rangle  \tag{7.3371}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa,pb-bab,pc,pd-baa,\,\text{class }%
3\rangle  \tag{7.3372}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-bab,pb-bab,pc,pd-baa,\,\text{class 
}3\rangle  \tag{7.3373}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-\omega bab,pb-bab,pc,pd-baa,\,%
\text{class }3\rangle  \tag{7.3374}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa,pb,pc-baa,pd-baa-bab,\,\text{class 
}3\rangle  \tag{7.3375}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa,pb,pc,pd-baa-bab,\,\text{class }%
3\rangle  \tag{7.3376}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa,pb,pc,pd-baa-bab,\,\text{class 
}3\rangle  \tag{7.3377}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-\omega baa,pb,pc,pd-baa-bab,\,%
\text{class }3\rangle  \tag{7.3378}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa,pb-(\omega \ast
x^{2})baa,pc,pd-baa-bab,\,\text{class }3\rangle \;(x\neq 0,\,x\sim -x) 
\tag{7.3379}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-baa,pb-x^{2}baa,pc,pd-baa-bab,\,%
\text{class }3\rangle \;(x\neq 0,\text{ equivalence classes }\{{x,-x,}\frac{{%
1}}{x}{,-}\frac{{1}}{x}\})  \tag{7.3380}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,pa-\omega baa,pb-(\omega \ast
x^{2})baa,pc,pd-baa-bab,\,\text{class }3\rangle \;(x\neq 0,\text{
equivalence classes }\{{x,-x,}\frac{{1}}{x}{,-}\frac{{1}}{x}\})  \tag{7.3381}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb-\omega baa,da,db,dc,pa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3382}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb-\omega baa,da,db,dc,pa-xbaa-bab,pb,pc,pd,\,%
\text{class }3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.3383}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb-\omega baa,da,db,dc,pa-xbaa-\omega
bab,pb,pc,pd,\,\text{class }3\rangle \;(\text{all }x,\,x\sim -x) 
\tag{7.3384}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb-\omega baa,da,db,dc,pa-baa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3385}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb-\omega baa,da,db,dc,pa-\omega baa,pb,pc,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.3386}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb-\omega
baa,da,db,dc,pa-xbaa-ybab,pb-zbaa-tbab,pc,pd,\,\text{class }3\rangle 
\tag{7.3387}
\end{equation}%
In presentation 7.3387, if we write the parameters $x,y,z,t$ in a matrix $%
A=\left( 
\begin{array}{cc}
x & y \\ 
z & t%
\end{array}%
\right) $ (which is assumed to be non-singular) then two matrices give
isomorphic algebras if and only if they are in the same orbit under the
action%
\[
A\rightarrow \frac{1}{\det P}PAP^{-1}, 
\]%
where $P$ lies in the group of non-singular matrices of the form%
\[
\left( 
\begin{array}{ll}
\alpha & \beta \\ 
\pm \omega \beta & \pm \alpha%
\end{array}%
\right) . 
\]

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb-\omega
baa,da,db,dc,pa-xbaa,pb-ybaa,pc-bab,pd,\,\text{class }3\rangle \;(\text{all }%
x,\,y,\,x\sim -x)  \tag{7.3388}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb-\omega baa,da,db,dc,pa,pb,pc-bab,pd-baa,\,%
\text{class }3\rangle  \tag{7.3389}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb-\omega baa,da,db,dc,pa-bab,pb-xbab,pc,pd-baa,\,%
\text{class }3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.3390}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb-\omega baa,da,db,dc,pa-\omega
bab,pb-xbab,pc,pd-baa,\,\text{class }3\rangle \;(\text{all }x,\,x\sim -x) 
\tag{7.3391}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb-\omega baa,da,db,dc,pa,pb,pc,pd-baa,\,\text{%
class }3\rangle  \tag{7.3392}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb-\omega baa,da,db,dc,pa,pb-bab,pc,pd-baa,\,%
\text{class }3\rangle  \tag{7.3393}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb-\omega baa,da,db,dc,pa,pb-\omega
bab,pc,pd-baa,\,\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.3394}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3395}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb-baa,pc,pd,\,\text{class }%
3\rangle  \tag{7.3396}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb-\omega baa,pc,pd,\,\text{%
class }3\rangle  \tag{7.3397}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb-bab,pc,pd,\,\text{class }%
3\rangle  \tag{7.3398}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-baa,pb-xbab,pc,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3399}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-baa,pb-baa-bab,pc,pd,\,\text{%
class }3\rangle  \tag{7.3400}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-baa,pb-\omega baa-bab,pc,pd,\,%
\text{class }3\rangle  \tag{7.3401}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-bab,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3402}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-bab,pb-baa,pc,pd,\,\text{class 
}3\rangle  \tag{7.3403}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-bab,pb-\omega baa,pc,pd,\,%
\text{class }3\rangle  \tag{7.3404}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-bab,pb-xbaa-bab,pc,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3405}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-\omega bab,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3406}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-\omega bab,pb-baa,pc,pd,\,%
\text{class }3\rangle  \tag{7.3407}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-\omega bab,pb-\omega
baa,pc,pd,\,\text{class }3\rangle  \tag{7.3408}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-\omega bab,pb-xbaa-bab,pc,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.3409}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb,pc,pd-baa,\,\text{class }%
3\rangle  \tag{7.3410}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb-bab,pc,pd-baa,\,\text{class 
}3\rangle  \tag{7.3411}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-bab,pb,pc,pd-baa,\,\text{class 
}3\rangle  \tag{7.3412}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-\omega bab,pb,pc,pd-baa,\,%
\text{class }3\rangle  \tag{7.3413}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb,pc,pd-bab,\,\text{class }%
3\rangle  \tag{7.3414}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb-baa,pc,pd-bab,\,\text{class 
}3\rangle  \tag{7.3415}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb-\omega baa,pc,pd-bab,\,%
\text{class }3\rangle  \tag{7.3416}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-baa,pb-xbaa,pc,pd-bab,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3417}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb,pc-baa,pd-xbab,\,\text{%
class }3\rangle \;(x \neq 0)  \tag{7.3418}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb,pc-baa,pd-baa-bab,\,\text{%
class }3\rangle  \tag{7.3419}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-xbab,pb,pc-baa,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3420}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-xbab,pb-bab,pc-baa,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3421}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb,pc-bab,pd-xbaa-bab,\,\text{%
class }3\rangle \;(x \neq 0)  \tag{7.3422}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb,pc-bab,pd-baa,\,\text{class 
}3\rangle  \tag{7.3423}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb,pc-bab,pd-\omega baa,\,%
\text{class }3\rangle  \tag{7.3424}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.3425}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-baa,pb,pc-bab,pd,\,\text{class 
}3\rangle  \tag{7.3426}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-\omega baa,pb,pc-bab,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3427}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-\omega ^{2}baa,pb,pc-bab,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3428}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb-baa,pc-bab,pd,\,\text{class 
}3\rangle  \tag{7.3429}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb-\omega baa,pc-bab,pd,\,%
\text{class }3\rangle  \tag{7.3430}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb-\omega ^{2}baa,pc-bab,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.3431}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa,pb-\omega ^{3}baa,pc-bab,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.3432}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-baa,pb-xbaa,pc-bab,pd,\,\text{%
class }3\rangle \;(x\neq 0,\,x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}%
\func{mod}p)  \tag{7.3433}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-\omega baa,pb-xbaa,pc-bab,pd,\,%
\text{class }3\rangle \;(x\neq 0,\,x\sim x^{\prime }\text{ if }%
x^{3}=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.3434}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-\omega
^{2}baa,pb-xbaa,pc-bab,pd,\,\text{class }3\rangle \;(x\neq 0,\,x\sim
x^{\prime }\text{ if }x^{3}=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3) 
\tag{7.3435}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,da,db-bab,dc,pa-xbaa,pb-ybaa,pc-bab,pd-bab,\,%
\text{class }3\rangle \;(\text{all }x,\,y)  \tag{7.3436}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|%
\,ca-bab,cb-baa,da,db-bab,dc,pa-rbaa-sbab,pb-tbaa-ubab,pc-vbaa-xbab,pd-ybaa-zbab,\,%
\text{class }3\rangle  \tag{7.3437}
\end{equation}%
In presentation 7.3437 we write%
\[
\left( 
\begin{array}{c}
pa \\ 
pb \\ 
pc \\ 
pd%
\end{array}%
\right) =A\left( 
\begin{array}{c}
baa \\ 
bab%
\end{array}%
\right) 
\]%
where $A$ is $4\times 2$ matrix. Two matrices $A$ give isomorphic algebras
if they lie in the same orbit under the action%
\[
A\rightarrow \frac{1}{\alpha ^{2}+\beta ^{2}}\left( 
\begin{array}{cccc}
\alpha & -\beta & \gamma & \delta \\ 
\pm \beta & \pm \alpha & \pm \lambda & \pm \mu \\ 
0 & 0 & \alpha ^{2}-\beta ^{2} & -4\alpha \beta \\ 
0 & 0 & \pm \alpha \beta & \pm (\alpha ^{2}-\beta ^{2})%
\end{array}%
\right) A\left( 
\begin{array}{cc}
\pm \alpha & \mp \beta \\ 
\beta & \alpha%
\end{array}%
\right) ^{-1}. 
\]

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb-\omega
baa,da,db-bab,dc,pa-rbaa-sbab,pb-tbaa-ubab,pc-vbaa-xbab,pd-ybaa-zbab,\,\text{%
class }3\rangle  \tag{7.3438}
\end{equation}%
In presentation 7.3438 we write%
\[
\left( 
\begin{array}{c}
pa \\ 
pb \\ 
pc \\ 
pd%
\end{array}%
\right) =A\left( 
\begin{array}{c}
baa \\ 
bab%
\end{array}%
\right) 
\]%
where $A$ is $4\times 2$ matrix. Two matrices $A$ give isomorphic algebras
if they lie in the same orbit under the action%
\[
A\rightarrow \frac{1}{\alpha ^{2}+\omega \beta ^{2}}\left( 
\begin{array}{cccc}
\alpha & \beta & \gamma & \delta \\ 
\mp \omega \beta & \pm \alpha & \pm \lambda & \pm \mu \\ 
0 & 0 & \alpha ^{2}-\omega \beta ^{2} & 4\omega \alpha \beta \\ 
0 & 0 & \mp \alpha \beta & \pm (\alpha ^{2}-\omega \beta ^{2})%
\end{array}%
\right) A\left( 
\begin{array}{cc}
\pm \alpha & \pm \beta \\ 
-\omega \beta & \alpha%
\end{array}%
\right) ^{-1}. 
\]

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb,pc,pd,\,\text{class }3\rangle 
\tag{7.3439}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb-baa,pc,pd,\,\text{class }%
3\rangle  \tag{7.3440}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb-\omega baa,pc,pd,\,\text{class }%
3\rangle  \tag{7.3441}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb-bab,pc,pd,\,\text{class }%
3\rangle  \tag{7.3442}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-baa,pb-xbab,pc,pd,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.3443}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-baa,pb-baa-bab,pc,pd,\,\text{class 
}3\rangle  \tag{7.3444}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-baa,pb-\omega baa-bab,pc,pd,\,%
\text{class }3\rangle  \tag{7.3445}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-bab,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3446}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-bab,pb-baa,pc,pd,\,\text{class }%
3\rangle  \tag{7.3447}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-bab,pb-\omega baa,pc,pd,\,\text{%
class }3\rangle  \tag{7.3448}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-bab,pb-xbaa-bab,pc,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3449}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-\omega bab,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3450}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-\omega bab,pb-baa,pc,pd,\,\text{%
class }3\rangle  \tag{7.3451}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-\omega bab,pb-\omega baa,pc,pd,\,%
\text{class }3\rangle  \tag{7.3452}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-\omega bab,pb-xbaa-bab,pc,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.3453}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.3454}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb-baa,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.3455}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb-\omega baa,pc-baa,pd,\,\text{%
class }3\rangle  \tag{7.3456}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb-bab,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.3457}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-baa,pb-xbab,pc-baa,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3458}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-baa,pb-baa-bab,pc-baa,pd,\,\text{%
class }3\rangle  \tag{7.3459}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-baa,pb-\omega baa-bab,pc-baa,pd,\,%
\text{class }3\rangle  \tag{7.3460}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-bab,pb,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.3461}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-bab,pb-baa,pc-baa,pd,\,\text{class 
}3\rangle  \tag{7.3462}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-bab,pb-\omega baa,pc-baa,pd,\,%
\text{class }3\rangle  \tag{7.3463}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-bab,pb-xbaa-bab,pc-baa,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3464}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-\omega bab,pb,pc-baa,pd,\,\text{%
class }3\rangle  \tag{7.3465}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-\omega bab,pb-baa,pc-baa,pd,\,%
\text{class }3\rangle  \tag{7.3466}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-\omega bab,pb-\omega
baa,pc-baa,pd,\,\text{class }3\rangle  \tag{7.3467}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-\omega bab,pb-xbaa-bab,pc-baa,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.3468}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-baa-xbab,pb-baa-ybab,pc-bab,pd,\,%
\text{class }3\rangle \;(\text{all }x,\,y)  \tag{7.3469}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-baa-xbab,pb-\omega
baa-ybab,pc-bab,pd,\,\text{class }3\rangle \;(\text{all }x,\,y)  \tag{7.3470}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-baa,pb-xbab,pc-bab,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3471}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-baa-bab,pb-xbab,pc-bab,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3472}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-baa-\omega bab,pb-xbab,pc-bab,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.3473}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-xbab,pb-baa-bab,pc-bab,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3474}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-xbab,pb-\omega baa-bab,pc-bab,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.3475}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb-baa,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.3476}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-bab,pb-baa,pc-bab,pd,\,\text{class 
}3\rangle  \tag{7.3477}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-\omega bab,pb-baa,pc-bab,pd,\,%
\text{class }3\rangle  \tag{7.3478}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb-\omega baa,pc-bab,pd,\,\text{%
class }3\rangle  \tag{7.3479}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-bab,pb-\omega baa,pc-bab,pd,\,%
\text{class }3\rangle  \tag{7.3480}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-\omega bab,pb-\omega
baa,pc-bab,pd,\,\text{class }3\rangle  \tag{7.3481}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.3482}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-bab,pb,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.3483}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-\omega bab,pb,pc-bab,pd,\,\text{%
class }3\rangle  \tag{7.3484}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb-bab,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.3485}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-bab,pb-bab,pc-bab,pd,\,\text{class 
}3\rangle  \tag{7.3486}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-\omega bab,pb-bab,pc-bab,pd,\,%
\text{class }3\rangle  \tag{7.3487}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb,pc-baa,pd-bab,\,\text{class }%
3\rangle  \tag{7.3488}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb-baa,pc-baa,pd-bab,\,\text{class 
}3\rangle  \tag{7.3489}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb-\omega baa,pc-baa,pd-bab,\,%
\text{class }3\rangle  \tag{7.3490}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb-\omega ^{2}baa,pc-baa,pd-bab,\,%
\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.3491}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb-\omega ^{3}baa,pc-baa,pd-bab,\,%
\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.3492}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa,pb-bab,pc-baa,pd-bab,\,\text{class 
}3\rangle  \tag{7.3493}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-baa,pb-xbab,pc-baa,pd-bab,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3494}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-baa,pb-xbaa-bab,pc-baa,pd-bab,\,%
\text{class }3\rangle \;(x \neq 0)  \tag{7.3495}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-bab,pb-xbaa-ybab,pc-baa,pd-bab,\,%
\text{class }3\rangle \;(\text{all }x,\,y,\,y\sim -y)  \tag{7.3496}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,pa-\omega
bab,pb-xbaa-ybab,pc-baa,pd-bab,\,\text{class }3\rangle \;(\text{all }%
x,\,y,\,y\sim -y)  \tag{7.3497}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3498}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-baa,pc,pd,\,\text{class }%
3\rangle  \tag{7.3499}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-\omega baa,pc,pd,\,\text{%
class }3\rangle  \tag{7.3500}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-bab,pc,pd,\,\text{class }%
3\rangle  \tag{7.3501}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-baa,pb-xbab,pc,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3502}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-baa,pb-baa-bab,pc,pd,\,\text{%
class }3\rangle  \tag{7.3503}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-baa,pb-\omega baa-bab,pc,pd,\,%
\text{class }3\rangle  \tag{7.3504}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-bab,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3505}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-bab,pb-baa,pc,pd,\,\text{class 
}3\rangle  \tag{7.3506}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-bab,pb-\omega baa,pc,pd,\,%
\text{class }3\rangle  \tag{7.3507}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-bab,pb-xbaa-bab,pc,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3508}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega bab,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3509}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega bab,pb-baa,pc,pd,\,%
\text{class }3\rangle  \tag{7.3510}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega bab,pb-\omega
baa,pc,pd,\,\text{class }3\rangle  \tag{7.3511}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega bab,pb-xbaa-bab,pc,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.3512}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.3513}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-baa,pc-baa,pd,\,\text{class 
}3\rangle  \tag{7.3514}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-\omega baa,pc-baa,pd,\,%
\text{class }3\rangle  \tag{7.3515}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-bab,pc-baa,pd,\,\text{class 
}3\rangle  \tag{7.3516}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-\omega bab,pc-baa,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3517}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-\omega ^{2}bab,pc-baa,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3518}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-baa,pb-xbab,pc-baa,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3519}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega baa,pb-xbab,pc-baa,pd,\,%
\text{class }3\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.3520}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega
^{2}baa,pb-xbab,pc-baa,pd,\,\text{class }3\rangle \;(\text{all }x,\;p=1\func{%
mod}3)  \tag{7.3521}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-baa,pb-xbaa-bab,pc-baa,pd,\,%
\text{class }3\rangle \;(x\neq 0,\,x\sim x^{\prime }\text{ if }%
x^{3}=x^{\prime 3}\func{mod}p)  \tag{7.3522}
\end{equation}

In the next two presentations we have $\;x\neq 0,\,x\sim x^{\prime }$ if $%
x^{3}=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3$:

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega baa,pb-xbaa-\omega
bab,pc-baa,pd,\,\text{class }3\rangle  \tag{7.3523}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega ^{2}baa,pb-xbaa-\omega
^{2}bab,pc-baa,pd,\,\text{class }3\rangle  \tag{7.3524}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-bab,pb-xbaa-ybab,pc-baa,pd,\,%
\text{class }3\rangle \;(\text{all }x,\,y,\,y\sim -y,\;p=3\func{mod}4) 
\tag{7.3525}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega
bab,pb-xbaa-ybab,pc-baa,pd,\,\text{class }3\rangle \;(\text{all }%
x,\,y,\,y\sim -y,\;p=3\func{mod}4)  \tag{7.3526}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-bab,pb-xbaa,pc-baa,pd,\,\text{%
class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.3527}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega bab,pb-xbaa,pc-baa,pd,\,%
\text{class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4) 
\tag{7.3528}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega
^{2}bab,pb-xbaa,pc-baa,pd,\,\text{class }3\rangle \;(\text{all }x,\,x\sim
-x,\;p=1\func{mod}4)  \tag{7.3529}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega
^{3}bab,pb-xbaa,pc-baa,pd,\,\text{class }3\rangle \;(\text{all }x,\,x\sim
-x,\;p=1\func{mod}4)  \tag{7.3530}
\end{equation}

In the next four presentations we have $\;$all $x,\,y\neq 0,\,y\sim
y^{\prime }$ if $y^{4}=y^{\prime 4}\func{mod}p,\;p=1\func{mod}4$:

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-bab,pb-xbaa-ybab,pc-baa,pd,\,%
\text{class }3\rangle  \tag{7.3531}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega
bab,pb-xbaa-ybab,pc-baa,pd,\,\text{class }3\rangle  \tag{7.3532}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega
^{2}bab,pb-xbaa-ybab,pc-baa,pd,\,\text{class }3\rangle  \tag{7.3533}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega
^{3}bab,pb-xbaa-ybab,pc-baa,pd,\,\text{class }3\rangle  \tag{7.3534}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-baa-xbab,pb-ybaa-zbab,pc-bab,pd,\,%
\text{class }3\rangle \;(\text{all }x,\,y,z)  \tag{7.3535}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbab,pb-ybaa-bab,pc-bab,pd,\,%
\text{class }3\rangle \;(\text{all }x,\,y)  \tag{7.3536}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-xbaa,pc-bab,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3537}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-bab,pb-xbaa,pc-bab,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3538}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega bab,pb-xbaa,pc-bab,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.3539}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbab,pb,pc,pd-baa,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3540}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbab,pb-baa,pc,pd-baa,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3541}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbab,pb-\omega baa,pc,pd-baa,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.3542}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbab,pb-ybaa-bab,pc,pd-baa,\,%
\text{class }3\rangle \;(x\neq -2,\text{ all }y)  \tag{7.3543}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbab,pb,pc,pd-\omega baa,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.3544}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbab,pb-baa,pc,pd-\omega baa,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.3545}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbab,pb-\omega
baa,pc,pd-\omega baa,\,\text{class }3\rangle \;(\text{all }x)  \tag{7.3546}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbab,pb-ybaa-bab,pc,pd-\omega
baa,\,\text{class }3\rangle \;(x\neq -2\omega ,\text{ all }y)  \tag{7.3547}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-baa+2bab,pb-xbaa,pc,pd-baa,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.3548}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-baa+2\omega
bab,pb-xbaa,pc,pd-\omega baa,\,\text{class }3\rangle \;(\text{all }x) 
\tag{7.3549}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbab,pb-ybaa-zbab,pc-bab,pd-baa,\,%
\text{class }3\rangle \;(x\neq -2,\text{ all }y,z,z\sim -z)  \tag{7.3550}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbab,pb-ybaa-zbab,pc-bab,pd-\omega
baa,\,\text{class }3\rangle \;(x\neq -2\omega ,\text{ all }y,z,\,z\sim -z) 
\tag{7.3551}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbaa+2bab,pb-ybaa,pc-bab,pd-baa,\,%
\text{class }3\rangle \;(\text{all }x,\,y,\,x\sim -x)  \tag{7.3552}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbaa+2\omega
bab,pb-ybaa,pc-bab,pd-\omega baa,\,\text{class }3\rangle \;(\text{all }%
x,\,y,\,x\sim -x)  \tag{7.3553}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbaa-bab,pb-ybaa-zbab,pc,pd-bab,\,%
\text{class }3\rangle \;(\text{all }x,\,y,z)  \tag{7.3554}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbaa-\omega
bab,pb-ybaa-zbab,pc,pd-bab,\,\text{class }3\rangle \;(\text{all }x,\,y,z) 
\tag{7.3555}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-baa,pb-xbaa-ybab,pc,pd-bab,\,%
\text{class }3\rangle \;(\text{all }x,\,y,\,x\sim -x,\;p=1\func{mod}4) 
\tag{7.3556}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega
baa,pb-xbaa-ybab,pc,pd-bab,\,\text{class }3\rangle \;(\text{all }%
x,\,y,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.3557}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega
^{2}baa,pb-xbaa-ybab,pc,pd-bab,\,\text{class }3\rangle \;(\text{all }%
x,\,y,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.3558}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega
^{3}baa,pb-xbaa-ybab,pc,pd-bab,\,\text{class }3\rangle \;(\text{all }%
x,\,y,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.3559}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-baa,pb-xbaa-ybab,pc,pd-bab,\,%
\text{class }3\rangle \;(\text{all }x,\,y,\;p=3\func{mod}4)  \tag{7.3560}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-\omega
baa,pb-xbaa-ybab,pc,pd-bab,\,\text{class }3\rangle \;(\text{all }x,\,y,\;p=3%
\func{mod}4)  \tag{7.3561}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-xbaa-bab,pc,pd-bab,\,\text{%
class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.3562}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-xbaa-\omega bab,pc,pd-bab,\,%
\text{class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4) 
\tag{7.3563}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-xbaa-\omega
^{2}bab,pc,pd-bab,\,\text{class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1%
\func{mod}4)  \tag{7.3564}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-xbaa-\omega
^{3}bab,pc,pd-bab,\,\text{class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1%
\func{mod}4)  \tag{7.3565}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-xbaa-bab,pc,pd-bab,\,\text{%
class }3\rangle \;(\text{all }x,\;p=3\func{mod}4)  \tag{7.3566}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-xbaa-\omega bab,pc,pd-bab,\,%
\text{class }3\rangle \;(\text{all }x,\;p=3\func{mod}4)  \tag{7.3567}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb,pc,pd-bab,\,\text{class }%
3\rangle  \tag{7.3568}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-baa,pc,pd-bab,\,\text{class 
}3\rangle  \tag{7.3569}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-\omega baa,pc,pd-bab,\,%
\text{class }3\rangle  \tag{7.3570}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-\omega ^{2}baa,pc,pd-bab,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3571}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-\omega ^{3}baa,pc,pd-bab,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3572}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-\omega ^{4}baa,pc,pd-bab,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3573}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-\omega ^{5}baa,pc,pd-bab,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3574}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbaa-ybab,pb-zbaa-tbab,pc-baa,pd-bab,\,%
\text{class }3\rangle \;(\text{all }x,\,y,z,t,\;p\neq 1\func{mod}5) 
\tag{7.3575}
\end{equation}

In the next 5 presentations we have $\;x\neq 0,\,x\sim x^{\prime }$ if $%
x^{5}=x^{\prime 5}\func{mod}p,$ all $y,z,t,\;p=1\func{mod}5$:

\begin{equation}
\langle
a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbaa-ybab,pb-zbaa-tbab,pc-baa,pd-bab,\,%
\text{class }3\rangle  \tag{7.3576}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbaa-ybab,pb-zbaa-tbab,pc-\omega
baa,pd-bab,\,\text{class }3\rangle  \tag{7.3577}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbaa-ybab,pb-zbaa-tbab,pc-\omega
^{2}baa,pd-bab,\,\text{class }3\rangle  \tag{7.3578}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbaa-ybab,pb-zbaa-tbab,pc-\omega
^{3}baa,pd-bab,\,\text{class }3\rangle  \tag{7.3579}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbaa-ybab,pb-zbaa-tbab,pc-\omega
^{4}baa,pd-bab,\,\text{class }3\rangle  \tag{7.3580}
\end{equation}

In the next 5 presentations we have $\;x\neq 0,\,x\sim x^{\prime }$ if $%
x^{5}=x^{\prime 5}\func{mod}p,$ all $y,z,\;p=1\func{mod}5$:

\begin{equation}
\langle
a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbab,pb-ybaa-zbab,pc-baa,pd-bab,\,%
\text{class }3\rangle  \tag{7.3581}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbab,pb-ybaa-zbab,pc-\omega
baa,pd-bab,\,\text{class }3\rangle  \tag{7.3582}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbab,pb-ybaa-zbab,pc-\omega
^{2}baa,pd-bab,\,\text{class }3\rangle  \tag{7.3583}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbab,pb-ybaa-zbab,pc-\omega
^{3}baa,pd-bab,\,\text{class }3\rangle  \tag{7.3584}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa-xbab,pb-ybaa-zbab,pc-\omega
^{4}baa,pd-bab,\,\text{class }3\rangle  \tag{7.3585}
\end{equation}

In the next 5 presentations we have $\;x\neq 0,\,x\sim x^{\prime }$ if $%
x^{5}=x^{\prime 5}\func{mod}p,$ all $y,\;p=1\func{mod}5$:

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-xbaa-ybab,pc-baa,pd-bab,\,%
\text{class }3\rangle  \tag{7.3586}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-xbaa-ybab,pc-\omega
baa,pd-bab,\,\text{class }3\rangle  \tag{7.3587}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-xbaa-ybab,pc-\omega
^{2}baa,pd-bab,\,\text{class }3\rangle  \tag{7.3588}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-xbaa-ybab,pc-\omega
^{3}baa,pd-bab,\,\text{class }3\rangle  \tag{7.3589}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-xbaa-ybab,pc-\omega
^{4}baa,pd-bab,\,\text{class }3\rangle  \tag{7.3590}
\end{equation}

And in the next 5 presentations we have $\;x\neq 0,\,x\sim x^{\prime }$ if $%
x^{5}=x^{\prime 5}\func{mod}p,\;p=1\func{mod}5$:

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-xbab,pc-baa,pd-bab,\,\text{%
class }3\rangle  \tag{7.3591}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-xbab,pc-\omega baa,pd-bab,\,%
\text{class }3\rangle  \tag{7.3592}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-xbab,pc-\omega
^{2}baa,pd-bab,\,\text{class }3\rangle  \tag{7.3593}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-xbab,pc-\omega
^{3}baa,pd-bab,\,\text{class }3\rangle  \tag{7.3594}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb-xbab,pc-\omega
^{4}baa,pd-bab,\,\text{class }3\rangle  \tag{7.3595}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb,pc-baa,pd-bab,\,\text{class 
}3\rangle \;(p=1 \func{mod} 5)  \tag{7.3596}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb,pc-\omega baa,pd-bab,\,%
\text{class }3\rangle \;(p=1\func{mod}5)  \tag{7.3597}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb,pc-\omega ^{2}baa,pd-bab,\,%
\text{class }3\rangle \;(p=1\func{mod}5)  \tag{7.3598}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb,pc-\omega ^{3}baa,pd-bab,\,%
\text{class }3\rangle \;(p=1\func{mod}5)  \tag{7.3599}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-bab,cb,da,db,dc-baa,pa,pb,pc-\omega ^{4}baa,pd-bab,\,%
\text{class }3\rangle \;(p=1\func{mod}5)  \tag{7.3600}
\end{equation}

\subsection{Descendants of 6.10}

8 algebras

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc,pa,pb,pc-ba,\,\text{class }3\rangle 
\tag{7.3601}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db-p^2d,dc,pa,pb,pc-ba,\,\text{class }3\rangle 
\tag{7.3602}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-p^2d,pa,pb,pc-ba,\,\text{class }3\rangle 
\tag{7.3603}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db-p^2d,dc-p^2d,pa,pb,pc-ba,\,\text{class }%
3\rangle  \tag{7.3604}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-p^2d,cb,da,db,dc,pa,pb,pc-ba,\,\text{class }3\rangle 
\tag{7.3605}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-p^2d,cb,da,db-p^2d,dc,pa,pb,pc-ba,\,\text{class }%
3\rangle  \tag{7.3606}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-p^2d,cb,da,db,dc-p^2d,pa,pb,pc-ba,\,\text{class }%
3\rangle  \tag{7.3607}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-p^2d,cb,da,db-p^2d,dc-p^2d,pa,pb,pc-ba,\,\text{class }%
3\rangle  \tag{7.3608}
\end{equation}

\subsection{Descendants of 6.11}

39 algebras

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,ca,cb,da,db,dc,pb,pc,pd,\,\text{class }3\rangle 
\tag{7.3609}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,ca-p^2a,cb,da,db,dc,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3610}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,ca,cb-p^2a,da,db,dc,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3611}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,ca,cb-p^2a,da-p^2a,db,dc,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3612}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,ca,cb,da,db,dc-p^2a,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3613}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dc-bab,p^2a-bab,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3614}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dc,p^2a-bab,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3615}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dc,p^2a-bab,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3616}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dc-bab,p^{2}a-\omega bab,pb,pc,pd,\,%
\text{class }3\rangle  \tag{7.3617}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dc,p^{2}a-\omega bab,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3618}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dc,p^{2}a-\omega bab,pb,pc,pd,\,%
\text{class }3\rangle  \tag{7.3619}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dc-bab,p^2a,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3620}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dc-bab,p^2a,pb,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.3621}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dc-bab,p^2a,pb-bab,pc,pd,\,\text{class }%
3\rangle  \tag{7.3622}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dc,p^2a,pb,pc,pd,\,\text{class }3\rangle
\tag{7.3623}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dc,p^2a,pb-bab,pc,pd,\,\text{class }%
3\rangle  \tag{7.3624}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dc,p^2a,pb,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.3625}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dc,p^2a,pb,pc,pd-bab,\,\text{class }%
3\rangle  \tag{7.3626}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dc,p^2a,pb,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.3627}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dc,p^2a,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3628}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dc,p^2a,pb-bab,pc,pd,\,\text{class }%
3\rangle  \tag{7.3629}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc,p^2a-baa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3630}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baa,da,db,dc,p^2a-baa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3631}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baa,p^2a-baa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3632}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baa,p^2a,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3633}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baa,p^2a,pb-baa,pc,pd,\,\text{class }%
3\rangle  \tag{7.3634}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baa,p^{2}a,pb-\omega baa,pc,pd,\,%
\text{class }3\rangle  \tag{7.3635}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baa,p^2a,pb,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.3636}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baa,p^2a,pb-baa,pc-baa,pd,\,\text{%
class }3\rangle  \tag{7.3637}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baa,p^{2}a,pb-\omega baa,pc-baa,pd,\,%
\text{class }3\rangle  \tag{7.3638}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc,p^2a,pb,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.3639}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc,p^2a,pb,pc,pd,\,\text{class }3\rangle
\tag{7.3640}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc,p^2a,pb-baa,pc,pd,\,\text{class }%
3\rangle  \tag{7.3641}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc,p^{2}a,pb-\omega baa,pc,pd,\,\text{%
class }3\rangle  \tag{7.3642}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baa,da,db,dc,p^2a,pb,pc,pd-baa,\,\text{class }%
3\rangle  \tag{7.3643}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baa,da,db,dc,p^2a,pb,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.3644}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baa,da,db,dc,p^2a,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3645}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baa,da,db,dc,p^2a,pb-baa,pc,pd,\,\text{class }%
3\rangle  \tag{7.3646}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baa,da,db,dc,p^{2}a,pb-\omega baa,pc,pd,\,%
\text{class }3\rangle  \tag{7.3647}
\end{equation}

\subsection{Descendants of 6.12}

8 algebras

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dc,pb-ba,pc,pd,\,\text{class }3\rangle 
\tag{7.3648}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-p^2a,cb,da,db,dc,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3649}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb-p^2a,da-p^2a,db,dc,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3650}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb-p^2a,da,db,dc,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3651}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dc-p^2a,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3652}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc,p^2a,pb-ba,pc,pd,\,\text{class }3\rangle 
\tag{7.3653}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-baa,cb,da,db,dc,p^2a,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3654}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-baa,p^2a,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3655}
\end{equation}

\subsection{Descendants of 6.13}

11 algebras

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc,pb,pc-ba,pd,\,\text{class }3\rangle 
\tag{7.3656}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-p^2a,cb,da,db,dc,pb,pc-ba,pd,\,\text{class }3\rangle 
\tag{7.3657}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-p^2a,da,db,dc,pb,pc-ba,pd,\,\text{class }3\rangle 
\tag{7.3658}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-\omega p^{2}a,da,db,dc,pb,pc-ba,pd,\,\text{class }%
3\rangle  \tag{7.3659}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da-p^2a,db,dc,pb,pc-ba,pd,\,\text{class }3\rangle 
\tag{7.3660}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-p^2a,da-p^2a,db,dc,pb,pc-ba,pd,\,\text{class }%
3\rangle  \tag{7.3661}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-\omega p^{2}a,da-p^{2}a,db,dc,pb,pc-ba,pd,\,\text{%
class }3\rangle  \tag{7.3662}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db-p^2a,dc,pb,pc-ba,pd,\,\text{class }3\rangle 
\tag{7.3663}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-p^2a,cb,da,db-p^2a,dc,pb,pc-ba,pd,\,\text{class }%
3\rangle  \tag{7.3664}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-p^2a,pb,pc-ba,pd,\,\text{class }3\rangle 
\tag{7.3665}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db-p^2a,dc-p^2a,pb,pc-ba,pd,\,\text{class }%
3\rangle  \tag{7.3666}
\end{equation}

\subsection{Descendants of 6.14}

39 algebras

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,ca,cb,da,db,dc,pa,pb,pd,\,\text{class }3\rangle 
\tag{7.3667}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,ca-p^2c,cb,da,db,dc,pa,pb,pd,\,\text{class }%
3\rangle  \tag{7.3668}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,ca,cb,da-p^2c,db,dc,pa,pb,pd,\,\text{class }%
3\rangle  \tag{7.3669}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,ca,cb-p^2c,da-p^2c,db,dc,pa,pb,pd,\,\text{class }%
3\rangle  \tag{7.3670}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,ca,cb,da,db,dc-p^2c,pa,pb,pd,\,\text{class }%
3\rangle  \tag{7.3671}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,ca,cb,da-p^2c,db,dc-p^2c,pa,pb,pd,\,\text{class }%
3\rangle  \tag{7.3672}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc,pa,pb,p^2c,pd-baa,\,\text{class }%
3\rangle  \tag{7.3673}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc,pa,pb-baa,p^2c,pd,\,\text{class }%
3\rangle  \tag{7.3674}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc,pa,pb-\omega baa,p^{2}c,pd,\,\text{%
class }3\rangle  \tag{7.3675}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc,pa,pb,p^2c,pd,\,\text{class }3\rangle
\tag{7.3676}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc,pa-baa,pb,p^2c,pd,\,\text{class }%
3\rangle  \tag{7.3677}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baa,da,db,dc,pa,pb,p^2c,pd-baa,\,\text{class }%
3\rangle  \tag{7.3678}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baa,da,db,dc,pa,pb-baa,p^2c,pd,\,\text{class }%
3\rangle  \tag{7.3679}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baa,da,db,dc,pa,pb-\omega baa,p^{2}c,pd,\,%
\text{class }3\rangle  \tag{7.3680}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baa,da,db,dc,pa,pb,p^2c,pd,\,\text{class }%
3\rangle  \tag{7.3681}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baa,da,db,dc,pa-baa,pb,p^2c,pd,\,\text{class }%
3\rangle  \tag{7.3682}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db-baa,dc,pa,pb,p^2c,pd-baa,\,\text{class }%
3\rangle  \tag{7.3683}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db-baa,dc,pa,pb-baa,p^2c,pd,\,\text{class }%
3\rangle  \tag{7.3684}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db-baa,dc,pa,pb-\omega baa,p^{2}c,pd,\,%
\text{class }3\rangle  \tag{7.3685}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db-baa,dc,pa,pb,p^2c,pd,\,\text{class }%
3\rangle  \tag{7.3686}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db-baa,dc,pa-baa,pb,p^2c,pd,\,\text{class }%
3\rangle  \tag{7.3687}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baa,pa,pb,p^2c,pd-baa,\,\text{class }%
3\rangle  \tag{7.3688}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baa,pa,pb-baa,p^2c,pd-baa,\,\text{%
class }3\rangle  \tag{7.3689}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baa,pa,pb-\omega baa,p^{2}c,pd-baa,\,%
\text{class }3\rangle  \tag{7.3690}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baa,pa,pb-baa,p^2c,pd,\,\text{class }%
3\rangle  \tag{7.3691}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baa,pa,pb-\omega baa,p^{2}c,pd,\,%
\text{class }3\rangle  \tag{7.3692}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baa,pa,pb,p^2c,pd,\,\text{class }%
3\rangle  \tag{7.3693}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baa,pa-baa,pb,p^2c,pd,\,\text{class }%
3\rangle  \tag{7.3694}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db-baa,dc-baa,pa,pb,p^2c,pd-baa,\,\text{%
class }3\rangle  \tag{7.3695}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db-baa,dc-baa,pa,pb-baa,p^2c,pd,\,\text{%
class }3\rangle  \tag{7.3696}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db-baa,dc-baa,pa,pb-\omega baa,p^{2}c,pd,\,%
\text{class }3\rangle  \tag{7.3697}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db-baa,dc-baa,pa,pb,p^2c,pd,\,\text{class }%
3\rangle  \tag{7.3698}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db-baa,dc-baa,pa-baa,pb,p^2c,pd,\,\text{%
class }3\rangle  \tag{7.3699}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc,pa,pb,p^2c-baa,pd,\,\text{class }%
3\rangle  \tag{7.3700}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baa,da,db,dc,pa,pb,p^2c-baa,pd,\,\text{class }%
3\rangle  \tag{7.3701}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db-baa,dc,pa,pb,p^2c-baa,pd,\,\text{class }%
3\rangle  \tag{7.3702}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baa,pa,pb,p^2c-baa,pd,\,\text{class }%
3\rangle  \tag{7.3703}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db-baa,dc-baa,pa,pb,p^2c-baa,pd,\,\text{%
class }3\rangle  \tag{7.3704}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db-\omega baa,dc-baa,pa,pb,p^{2}c-baa,pd,\,%
\text{class }3\rangle  \tag{7.3705}
\end{equation}

\subsection{Descendants of 6.15}

$2p+6$ algebras

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-xp^2c,da,db,dc,pa-ba,pb,pd,\,\text{class }3\rangle
\;(\text{all }x)  \tag{7.3706}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-p^2c,cb,da,db,dc,pa-ba,pb,pd,\,\text{class }3\rangle 
\tag{7.3707}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-xp^2c,da-p^2c,db,dc,pa-ba,pb,pd,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.3708}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db-p^2c,dc,pa-ba,pb,pd,\,\text{class }3\rangle 
\tag{7.3709}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-p^2c,cb,da,db-p^2c,dc,pa-ba,pb,pd,\,\text{class }%
3\rangle  \tag{7.3710}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-p^2c,pa-ba,pb,pd,\,\text{class }3\rangle 
\tag{7.3711}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da-p^2c,db,dc-p^2c,pa-ba,pb,pd,\,\text{class }%
3\rangle  \tag{7.3712}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db-p^2c,dc-p^2c,pa-ba,pb,pd,\,\text{class }%
3\rangle  \tag{7.3713}
\end{equation}

\subsection{Descendants of 6.16}

6 algebras

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-ba,pb,pc,pd,\,\text{class }3\rangle 
\tag{7.3714}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-p^2a,cb,da,db,dc-ba,pb,pc,pd,\,\text{class }3\rangle 
\tag{7.3715}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-ba-p^2a,pb,pc,pd,\,\text{class }3\rangle 
\tag{7.3716}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-p^2a,da,db,dc-ba,pb,pc,pd,\,\text{class }3\rangle 
\tag{7.3717}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-p^2a,da-p^2a,db,dc-ba,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3718}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-p^{2}a,da-\omega p^{2}a,db,dc-ba,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3719}
\end{equation}

\subsection{Descendants of 6.17}

$p+7$ algebras

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-ba,pb-ba,pc,pd,\,\text{class }3\rangle 
\tag{7.3720}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-p^2a,cb,da,db,dc-ba,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3721}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-ba-p^2a,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3722}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-p^2a,cb,da,db,dc-ba-p^2a,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3723}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-p^2a,da,db,dc-ba,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3724}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-p^2a,da-p^2a,db,dc-ba,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3725}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-p^{2}a,da-\omega p^{2}a,db,dc-ba,pb-ba,pc,pd,\,%
\text{class }3\rangle  \tag{7.3726}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-p^2a,da-xp^2a,db,dc-ba-p^2a,pb-ba,pc,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3727}
\end{equation}

\subsection{Descendants of 6.18}

$3p+5$ algebras

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da-xp^2a,db,dc-ba,pb,pc-ba,pd,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.3728}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-p^2a,cb,da,db,dc-ba,pb,pc-ba,pd,\,\text{class }%
3\rangle  \tag{7.3729}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db,dc-ba-p^2a,pb,pc-ba,pd,\,\text{class }%
3\rangle  \tag{7.3730}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb,da,db-p^2a,dc-ba,pb,pc-ba,pd,\,\text{class }%
3\rangle  \tag{7.3731}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-p^2a,cb,da,db-p^2a,dc-ba,pb,pc-ba,pd,\,\text{class }%
3\rangle  \tag{7.3732}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca-\omega p^{2}a,cb,da,db-p^{2}a,dc-ba,pb,pc-ba,pd,\,%
\text{class }3\rangle  \tag{7.3733}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-p^2a,da-xp^2a,db,dc-ba,pb,pc-ba,pd,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.3734}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-\omega p^{2}a,da-xp^{2}a,db,dc-ba,pb,pc-ba,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.3735}
\end{equation}

\subsection{Descendants of 6.19}

97 algebras

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc,pa,pb,pc,pd-bab,\,\text{%
class }3\rangle  \tag{7.3736}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc,pa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3737}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc,pa,pb-bab,pc,pd,\,\text{%
class }3\rangle  \tag{7.3738}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc,pa-bab,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3739}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc,pa-bab,pb-bab,pc,pd,\,\text{%
class }3\rangle  \tag{7.3740}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc,pa-\omega bab,pb,pc,pd,\,%
\text{class }3\rangle  \tag{7.3741}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc,pa-\omega bab,pb-bab,pc,pd,\,%
\text{class }3\rangle  \tag{7.3742}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc,pa,pb,pc-bab,pd,\,\text{%
class }3\rangle  \tag{7.3743}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc-bab,pa,pb,pc,pd-bab,\,\text{%
class }3\rangle  \tag{7.3744}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc-bab,pa,pb,pc-bab,pd,\,\text{%
class }3\rangle  \tag{7.3745}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc-bab,pa-bab,pb,pc-bab,pd,\,%
\text{class }3\rangle  \tag{7.3746}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc-bab,pa-\omega
bab,pb,pc-bab,pd,\,\text{class }3\rangle  \tag{7.3747}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc-bab,pa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3748}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc-bab,pa,pb-bab,pc,pd,\,\text{%
class }3\rangle  \tag{7.3749}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc-bab,pa-bab,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3750}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc-bab,pa-bab,pb-bab,pc,pd,\,%
\text{class }3\rangle  \tag{7.3751}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc-bab,pa-\omega bab,pb,pc,pd,\,%
\text{class }3\rangle  \tag{7.3752}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da,db,dc-bab,pa-\omega
bab,pb-bab,pc,pd,\,\text{class }3\rangle  \tag{7.3753}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da-bab,db,dc,pa,pb,pc,pd-bab,\,\text{%
class }3\rangle  \tag{7.3754}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da-bab,db,dc,pa,pb,pc-bab,pd,\,\text{%
class }3\rangle  \tag{7.3755}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da-bab,db,dc,pa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3756}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da-bab,db,dc,pa,pb-bab,pc,pd,\,\text{%
class }3\rangle  \tag{7.3757}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da-bab,db,dc,pa-bab,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3758}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da-bab,db,dc,pa-bab,pb-bab,pc,pd,\,%
\text{class }3\rangle  \tag{7.3759}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da-bab,db,dc,pa-\omega bab,pb,pc,pd,\,%
\text{class }3\rangle  \tag{7.3760}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac,da-bab,db,dc,pa-\omega
bab,pb-bab,pc,pd,\,\text{class }3\rangle  \tag{7.3761}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa-bab,cac,da,db,dc,pa,pb,pc,pd-bab,\,\text{%
class }3\rangle  \tag{7.3762}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa-bab,cac,da,db,dc,pa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3763}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa-bab,cac,da,db,dc,pa-bab,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3764}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa-bab,cac,da,db,dc,pa-\omega bab,pb,pc,pd,\,%
\text{class }3\rangle  \tag{7.3765}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa-bab,cac,da,db,dc,pa,pb-bab,pc,pd,\,\text{%
class }3\rangle  \tag{7.3766}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa-bab,cac,da,db,dc,pa,pb,pc-bab,pd,\,\text{%
class }3\rangle  \tag{7.3767}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa-bab,cac,da,db,dc,pa,pb,pc-\omega bab,pd,\,%
\text{class }3\rangle  \tag{7.3768}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa-bab,cac,da,db,dc-bab,pa,pb,pc,pd-bab,\,%
\text{class }3\rangle  \tag{7.3769}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa-bab,cac,da,db,dc-bab,pa,pb,pc-bab,pd,\,%
\text{class }3\rangle  \tag{7.3770}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa-bab,cac,da,db,dc-bab,pa,pb,pc-\omega
bab,pd,\,\text{class }3\rangle  \tag{7.3771}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa-bab,cac,da,db,dc-bab,pa,pb-bab,pc,pd,\,%
\text{class }3\rangle  \tag{7.3772}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa-bab,cac,da,db,dc-bab,pa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3773}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa-bab,cac,da,db,dc-bab,pa-bab,pb,pc,pd,\,%
\text{class }3\rangle  \tag{7.3774}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa-bab,cac,da,db,dc-bab,pa-\omega
bab,pb,pc,pd,\,\text{class }3\rangle  \tag{7.3775}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+bab,da,db,dc,pa,pb,pc,pd-bab,\,\text{%
class }3\rangle  \tag{7.3776}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+bab,da,db,dc,pa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3777}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+bab,da,db,dc,pa-bab,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3778}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+bab,da,db,dc,pa,pb-bab,pc,pd,\,\text{%
class }3\rangle  \tag{7.3779}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+bab,da,db,dc,pa-bab,pb-bab,pc,pd,\,%
\text{class }3\rangle  \tag{7.3780}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+bab,da,db,dc,pa-\omega
bab,pb-bab,pc,pd,\,\text{class }3\rangle  \tag{7.3781}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+bab,da,db,dc,pa,pb-bab,pc-bab,pd,\,%
\text{class }3\rangle  \tag{7.3782}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,cb,baa,bac,caa,cac+bab,da,db,dc,pa-bab,pb-bab,pc-bab,pd,\,\text{%
class }3\rangle  \tag{7.3783}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+\omega bab,da,db,dc,pa,pb,pc,pd-bab,\,%
\text{class }3\rangle  \tag{7.3784}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+\omega bab,da,db,dc,pa,pb,pc,pd,\,%
\text{class }3\rangle  \tag{7.3785}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+\omega bab,da,db,dc,pa-bab,pb,pc,pd,\,%
\text{class }3\rangle  \tag{7.3786}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+\omega bab,da,db,dc,pa,pb-bab,pc,pd,\,%
\text{class }3\rangle  \tag{7.3787}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+\omega
bab,da,db,dc,pa-bab,pb-bab,pc,pd,\,\text{class }3\rangle  \tag{7.3788}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+\omega bab,da,db,dc,pa-\omega
bab,pb-bab,pc,pd,\,\text{class }3\rangle  \tag{7.3789}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+bab,da-bab,db,dc,pa,pb,pc,pd-bab,\,%
\text{class }3\rangle  \tag{7.3790}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+bab,da-bab,db,dc,pa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3791}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+bab,da-bab,db,dc,pa-bab,pb,pc,pd,\,%
\text{class }3\rangle  \tag{7.3792}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+bab,da-bab,db,dc,pa,pb-bab,pc,pd,\,%
\text{class }3\rangle  \tag{7.3793}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,cb,baa,bac,caa,cac+bab,da-bab,db,dc,pa-bab,pb-bab,pc,pd,\,\text{%
class }3\rangle  \tag{7.3794}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+bab,da-bab,db,dc,pa-\omega
bab,pb-bab,pc,pd,\,\text{class }3\rangle  \tag{7.3795}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,cb,baa,bac,caa,cac+bab,da-bab,db,dc,pa,pb-bab,pc-bab,pd,\,\text{%
class }3\rangle  \tag{7.3796}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,cb,baa,bac,caa,cac+bab,da-bab,db,dc,pa-bab,pb-bab,pc-bab,pd,\,%
\text{class }3\rangle  \tag{7.3797}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+\omega
bab,da-bab,db,dc,pa,pb,pc,pd-bab,\,\text{class }3\rangle  \tag{7.3798}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+\omega bab,da-bab,db,dc,pa,pb,pc,pd,\,%
\text{class }3\rangle  \tag{7.3799}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+\omega
bab,da-bab,db,dc,pa-bab,pb,pc,pd,\,\text{class }3\rangle  \tag{7.3800}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+\omega
bab,da-bab,db,dc,pa,pb-bab,pc,pd,\,\text{class }3\rangle  \tag{7.3801}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+\omega
bab,da-bab,db,dc,pa-bab,pb-bab,pc,pd,\,\text{class }3\rangle  \tag{7.3802}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,baa,bac,caa,cac+\omega bab,da-bab,db,dc,pa-\omega
bab,pb-bab,pc,pd,\,\text{class }3\rangle  \tag{7.3803}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db,dc,pa,pb,pc,pd-baa,\,\text{%
class }3\rangle  \tag{7.3804}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db,dc,pa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3805}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db,dc,pa-baa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3806}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db,dc,pa,pb-baa,pc,pd,\,\text{%
class }3\rangle  \tag{7.3807}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db,dc,pa,pb-\omega baa,pc,pd,\,%
\text{class }3\rangle  \tag{7.3808}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db,dc,pa,pb,pc-baa,pd,\,\text{%
class }3\rangle  \tag{7.3809}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db-baa,dc,pa,pb,pc,pd-baa,\,\text{%
class }3\rangle  \tag{7.3810}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db-baa,dc,pa,pb,pc-baa,pd-baa,\,%
\text{class }3\rangle  \tag{7.3811}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db-baa,dc,pa,pb,pc-baa,pd,\,\text{%
class }3\rangle  \tag{7.3812}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db-baa,dc,pa,pb-baa,pc,pd,\,\text{%
class }3\rangle  \tag{7.3813}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db-baa,dc,pa,pb-\omega baa,pc,pd,\,%
\text{class }3\rangle  \tag{7.3814}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db-baa,dc,pa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3815}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db-baa,dc,pa-baa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3816}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db,dc-baa,pa,pb,pc,pd-baa,\,\text{%
class }3\rangle  \tag{7.3817}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db,dc-baa,pa,pb-baa,pc,pd-baa,\,%
\text{class }3\rangle  \tag{7.3818}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db,dc-baa,pa,pb-\omega
baa,pc,pd-baa,\,\text{class }3\rangle  \tag{7.3819}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db,dc-baa,pa,pb,pc-baa,pd,\,\text{%
class }3\rangle  \tag{7.3820}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db,dc-baa,pa,pb-baa,pc-baa,pd,\,%
\text{class }3\rangle  \tag{7.3821}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db,dc-baa,pa,pb-\omega
baa,pc-baa,pd,\,\text{class }3\rangle  \tag{7.3822}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db,dc-baa,pa,pb-baa,pc,pd,\,\text{%
class }3\rangle  \tag{7.3823}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db,dc-baa,pa,pb-\omega baa,pc,pd,\,%
\text{class }3\rangle  \tag{7.3824}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db,dc-baa,pa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3825}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,bab,bac,caa,cac,da,db,dc-baa,pa-baa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3826}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb-baa,bab,bac,caa,cac,da,db,dc,pa,pb,pc,pd-baa,\,\text{%
class }3\rangle  \tag{7.3827}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb-baa,bab,bac,caa,cac,da,db,dc,pa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3828}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb-baa,bab,bac,caa,cac,da,db,dc,pa-baa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.3829}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb-baa,bab,bac,caa,cac,da,db,dc,pa,pb-baa,pc,pd,\,\text{%
class }3\rangle  \tag{7.3830}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb-baa,bab,bac,caa,cac,da,db,dc,pa,pb-\omega baa,pc,pd,\,%
\text{class }3\rangle  \tag{7.3831}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb-baa,bab,bac,caa,cac,da,db,dc,pa,pb,pc-baa,pd,\,\text{%
class }3\rangle  \tag{7.3832}
\end{equation}

\subsection{Descendants of 6.20}

$6p+35$ algebras

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,caa,pa-ba,pb,pc,pd-cac,\,\text{class }%
3\rangle  \tag{7.3833}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,caa,pa-ba,pb-cac,pc,pd,\,\text{class }%
3\rangle  \tag{7.3834}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,caa,pa-ba,pb,pc,pd,\,\text{class }3\rangle 
\tag{7.3835}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,caa,pa-ba,pb,pc-cac,pd,\,\text{class }%
3\rangle  \tag{7.3836}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,caa,pa-ba-cac,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3837}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,caa,pa-ba-cac,pb,pc-cac,pd,\,\text{class }%
3\rangle  \tag{7.3838}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,caa,pa-ba-\omega cac,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3839}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,caa,pa-ba-\omega cac,pb,pc-cac,pd,\,\text{%
class }3\rangle  \tag{7.3840}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da-cac,db,dc,caa,pa-ba,pb,pc,pd-cac,\,\text{class }%
3\rangle  \tag{7.3841}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da-cac,db,dc,caa,pa-ba,pb-cac,pc,pd,\,\text{class }%
3\rangle  \tag{7.3842}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da-cac,db,dc,caa,pa-ba,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3843}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da-cac,db,dc,caa,pa-ba,pb,pc-cac,pd,\,\text{class }%
3\rangle  \tag{7.3844}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db-cac,dc,caa,pa-ba,pb,pc,pd-xcac,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.3845}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db-cac,dc,caa,pa-ba,pb-cac,pc,pd,\,\text{class }%
3\rangle  \tag{7.3846}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db-cac,dc,caa,pa-ba,pb,pc-cac,pd,\,\text{class }%
3\rangle  \tag{7.3847}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db-cac,dc,caa,pa-ba-cac,pb,pc,pd+cac,\,\text{class 
}3\rangle  \tag{7.3848}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db-cac,dc,caa,pa-ba-\omega cac,pb,pc,pd+cac,\,%
\text{class }3\rangle  \tag{7.3849}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db-caa,dc,cac,pa-ba,pb,pc,pd-xcaa,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.3850}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db-caa,dc,cac,pa-ba,pb,pc-caa,pd-xcaa,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3851}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db-caa,dc,cac,pa-ba,pb,pc-\omega caa,pd-xcaa,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.3852}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db-caa,dc,cac,pa-ba,pb-caa,pc,pd,\,\text{class }%
3\rangle  \tag{7.3853}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db-caa,dc,cac,pa-ba,pb-caa,pc-caa,pd,\,\text{class 
}3\rangle  \tag{7.3854}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db-caa,dc,cac,pa-ba,pb-caa,pc-\omega caa,pd,\,%
\text{class }3\rangle  \tag{7.3855}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc-caa,cac,pa-ba,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.3856}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc-caa,cac,pa-ba,pb-caa,pc,pd,\,\text{class }%
3\rangle  \tag{7.3857}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc-caa,cac,pa-ba,pb,pc,pd-caa,\,\text{class }%
3\rangle  \tag{7.3858}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc-caa,cac,pa-ba,pb-caa,pc,pd-caa,\,\text{class 
}3\rangle  \tag{7.3859}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc-caa,cac,pa-ba,pb-\omega caa,pc,pd-caa,\,%
\text{class }3\rangle  \tag{7.3860}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc-caa,cac,pa-ba,pb,pc-caa,pd,\,\text{class }%
3\rangle  \tag{7.3861}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc-caa,cac,pa-ba,pb,pc-\omega caa,pd,\,\text{%
class }3\rangle  \tag{7.3862}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb-caa,da,db,dc,cac,pa-ba,pb,pc,pd-caa,\,\text{class }%
3\rangle  \tag{7.3863}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb-caa,da,db,dc,cac,pa-ba,pb,pc-xcaa,pd,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.3864}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb-caa,da,db,dc,cac,pa-ba,pb-caa,pc,pd,\,\text{class }%
3\rangle  \tag{7.3865}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb-\omega caa,da,db,dc,cac,pa-ba,pb,pc,pd-caa,\,\text{%
class }3\rangle  \tag{7.3866}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb-\omega caa,da,db,dc,cac,pa-ba,pb,pc-xcaa,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3867}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb-\omega caa,da,db,dc,cac,pa-ba,pb-caa,pc,pd,\,\text{%
class }3\rangle  \tag{7.3868}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,cac,pa-ba,pb,pc,pd-caa,\,\text{class }%
3\rangle  \tag{7.3869}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,cac,pa-ba,pb,pc,pd,\,\text{class }3\rangle 
\tag{7.3870}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,cac,pa-ba,pb-caa,pc,pd,\,\text{class }%
3\rangle  \tag{7.3871}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,cac,pa-ba,pb,pc-caa,pd,\,\text{class }%
3\rangle  \tag{7.3872}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,cac,pa-ba,pb,pc-\omega caa,pd,\,\text{class }%
3\rangle  \tag{7.3873}
\end{equation}

\subsection{Descendants of 6.21}

$13p+27$ algebras

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db,dc,pa,pb-ba,pc,pd-cac,\,\text{class }%
3\rangle  \tag{7.3874}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db,dc,pa,pb-ba,pc,pd,\,\text{class }3\rangle 
\tag{7.3875}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db,dc,pa-cac,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3876}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db,dc,pa-\omega cac,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3877}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db,dc,pa-xcac,pb-ba,pc-cac,pd,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.3878}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db,dc,pa,pb-ba-cac,pc,pd,\,\text{class }%
3\rangle  \tag{7.3879}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db,dc,pa-cac,pb-ba-cac,pc,pd,\,\text{class }%
3\rangle  \tag{7.3880}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db,dc,pa-\omega cac,pb-ba-cac,pc,pd,\,\text{%
class }3\rangle  \tag{7.3881}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db,dc,pa-xcac,pb-ba-cac,pc-cac,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3882}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db,dc,pa,pb-ba,pc,pd-xcac,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.3883}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db,dc,pa,pb-ba-cac,pc,pd-cac,\,\text{class 
}3\rangle  \tag{7.3884}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db,dc,pa-cac,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3885}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db,dc,pa-\omega cac,pb-ba,pc,pd,\,\text{%
class }3\rangle  \tag{7.3886}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db,dc,pa-xcac,pb-ba,pc-cac,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3887}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-cac,dc,pa,pb-ba,pc,pd-cac,\,\text{class }%
3\rangle  \tag{7.3888}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-cac,dc,pa,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3889}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-cac,dc,pa-cac,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3890}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-cac,dc,pa-\omega cac,pb-ba,pc,pd,\,\text{%
class }3\rangle  \tag{7.3891}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-cac,dc,pa-xcac,pb-ba,pc-cac,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3892}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-cac,dc,pa,pb-ba,pc,pd-xcac,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3893}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-cac,dc,pa,pb-ba-cac,pc,pd-cac,\,\text{%
class }3\rangle  \tag{7.3894}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-cac,dc,pa,pb-ba-\omega cac,pc,pd-cac,\,%
\text{class }3\rangle  \tag{7.3895}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-cac,dc,pa-cac,pb-ba,pc,pd,\,\text{class 
}3\rangle  \tag{7.3896}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-cac,dc,pa-\omega cac,pb-ba,pc,pd,\,%
\text{class }3\rangle  \tag{7.3897}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-cac,dc,pa-xcac,pb-ba,pc-cac,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3898}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db,dc,pa,pb-ba,pc,pd-caa,\,\text{class }%
3\rangle  \tag{7.3899}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db,dc,pa-caa,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3900}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db,dc,pa,pb-ba,pc-xcaa,pd,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.3901}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db,dc-caa,pa,pb-ba,pc,pd-caa,\,\text{class }%
3\rangle  \tag{7.3902}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db,dc-caa,pa-caa,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3903}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db,dc-caa,pa,pb-ba,pc-xcaa,pd,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.3904}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-caa,dc,pa,pb-ba,pc-xcaa,pd-caa,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3905}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-caa,dc,pa-caa,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3906}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-caa,dc,pa,pb-ba,pc-xcaa,pd,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.3907}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-caa,dc-caa,pa,pb-ba,pc,pd-caa,\,\text{class 
}3\rangle  \tag{7.3908}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-caa,dc-caa,pa-caa,pb-ba,pc,pd,\,\text{class 
}3\rangle  \tag{7.3909}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-caa,dc-caa,pa,pb-ba,pc-xcaa,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3910}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db,dc,pa,pb-ba,pc,pd-caa,\,\text{class }%
3\rangle  \tag{7.3911}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db,dc,pa-caa,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.3912}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db,dc,pa,pb-ba,pc-xcaa,pd,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.3913}
\end{equation}

\subsection{Descendants of 6.23}

$3p+41+8\gcd (p-1,3)+2\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa,pb-ca,pc,pd-bab,\,\text{class }%
3\rangle  \tag{7.3914}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa,pb-ca,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.3915}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa,pb-ca,pc,pd,\,\text{class }3\rangle 
\tag{7.3916}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa-bab,pb-ca,pc,pd,\,\text{class }%
3\rangle  \tag{7.3917}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa-\omega bab,pb-ca,pc,pd,\,\text{class }%
3\rangle  \tag{7.3918}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa,pb-ca-bab,pc,pd,\,\text{class }%
3\rangle  \tag{7.3919}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa-bab,pb-ca-bab,pc,pd,\,\text{class }%
3\rangle  \tag{7.3920}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa-\omega bab,pb-ca-bab,pc,pd,\,\text{%
class }3\rangle  \tag{7.3921}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa,pb-ca,pc,pd-bab,\,\text{class }%
3\rangle  \tag{7.3922}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa,pb-ca,pc,pd,\,\text{class }%
3\rangle  \tag{7.3923}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa-bab,pb-ca,pc,pd,\,\text{class }%
3\rangle  \tag{7.3924}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa-\omega bab,pb-ca,pc,pd,\,\text{%
class }3\rangle  \tag{7.3925}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa,pb-ca,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.3926}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa-bab,pb-ca,pc-bab,pd,\,\text{class 
}3\rangle  \tag{7.3927}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa-\omega bab,pb-ca,pc-bab,pd,\,%
\text{class }3\rangle  \tag{7.3928}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa-\omega ^{2}bab,pb-ca,pc-bab,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.3929}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa-\omega ^{3}bab,pb-ca,pc-bab,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.3930}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa,pb-ca,pc,pd-bab,\,\text{class }%
3\rangle  \tag{7.3931}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa,pb-ca,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.3932}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa,pb-ca,pc,pd,\,\text{class }%
3\rangle  \tag{7.3933}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa-bab,pb-ca,pc,pd,\,\text{class }%
3\rangle  \tag{7.3934}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa-\omega bab,pb-ca,pc,pd,\,\text{%
class }3\rangle  \tag{7.3935}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc,pa,pb-ca,pc,pd-baa,\,\text{class }%
3\rangle  \tag{7.3936}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc,pa,pb-ca,pc,pd,\,\text{class }3\rangle 
\tag{7.3937}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc,pa-baa,pb-ca,pc,pd,\,\text{class }%
3\rangle  \tag{7.3938}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc,pa,pb-ca,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.3939}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc,pa,pb-ca,pc-\omega baa,pd,\,\text{class }%
3\rangle \;(p=1\func{mod}3)  \tag{7.3940}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc,pa,pb-ca,pc-\omega ^{2}baa,pd,\,\text{%
class }3\rangle \;(p=1\func{mod}3)  \tag{7.3941}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-baa,dc,pa,pb-ca,pc,pd-baa,\,\text{class }%
3\rangle  \tag{7.3942}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-baa,dc,pa,pb-ca,pc-baa,pd-baa,\,\text{class 
}3\rangle  \tag{7.3943}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-baa,dc,pa,pb-ca,pc-\omega baa,pd-baa,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3944}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-baa,dc,pa,pb-ca,pc-\omega ^{2}baa,pd-baa,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3945}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-baa,dc,pa,pb-ca,pc,pd,\,\text{class }%
3\rangle  \tag{7.3946}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-baa,dc,pa-baa,pb-ca,pc,pd,\,\text{class }%
3\rangle  \tag{7.3947}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-baa,dc,pa,pb-ca,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.3948}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-baa,dc,pa,pb-ca,pc-\omega baa,pd,\,\text{%
class }3\rangle \;(p=1\func{mod}3)  \tag{7.3949}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-baa,dc,pa,pb-ca,pc-\omega ^{2}baa,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3950}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc-baa,pa,pb-ca,pc,pd-baa,\,\text{class }%
3\rangle  \tag{7.3951}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc-baa,pa,pb-ca,pc,pd,\,\text{class }%
3\rangle  \tag{7.3952}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc-baa,pa-baa,pb-ca,pc,pd,\,\text{class }%
3\rangle  \tag{7.3953}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc-baa,pa,pb-ca,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.3954}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc-baa,pa,pb-ca,pc-\omega baa,pd,\,\text{%
class }3\rangle \;(p=1\func{mod}3)  \tag{7.3955}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc-baa,pa,pb-ca,pc-\omega ^{2}baa,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3956}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc,pa,pb-ca,pc,pd-baa,\,\text{class }%
3\rangle  \tag{7.3957}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc,pa,pb-ca,pc,pd,\,\text{class }%
3\rangle  \tag{7.3958}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc,pa-baa,pb-ca,pc,pd,\,\text{class }%
3\rangle  \tag{7.3959}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc,pa-\omega baa,pb-ca,pc,pd,\,\text{%
class }3\rangle  \tag{7.3960}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc,pa-\omega ^{2}baa,pb-ca,pc,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.3961}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc,pa-\omega ^{3}baa,pb-ca,pc,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.3962}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc,pa,pb-ca,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.3963}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc,pa,pb-ca,pc-\omega baa,pd,\,\text{%
class }3\rangle \;(p=1\func{mod}3)  \tag{7.3964}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc,pa,pb-ca,pc-\omega ^{2}baa,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3965}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da,db,dc,pa,pb-ca,pc,pd-baa,\,\text{class }%
3\rangle  \tag{7.3966}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da,db,dc,pa,pb-ca,pc,pd,\,\text{class }%
3\rangle  \tag{7.3967}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da,db,dc,pa-baa,pb-ca,pc,pd,\,\text{class }%
3\rangle  \tag{7.3968}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da,db,dc,pa-\omega baa,pb-ca,pc,pd,\,\text{%
class }3\rangle  \tag{7.3969}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da,db,dc,pa-xbaa,pb-ca-baa,pc,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3970}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da,db,dc,pa-xbaa,pb-ca-\omega baa,pc,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.3971}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da,db,dc,pa,pb-ca,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.3972}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da,db,dc,pa,pb-ca,pc-\omega baa,pd,\,\text{%
class }3\rangle \;(p=1\func{mod}3)  \tag{7.3973}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da,db,dc,pa,pb-ca,pc-\omega ^{2}baa,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3974}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da-baa,db,dc,pa,pb-ca,pc,pd-baa,\,\text{class 
}3\rangle  \tag{7.3975}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da-baa,db,dc,pa,pb-ca,pc,pd,\,\text{class }%
3\rangle  \tag{7.3976}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da-baa,db,dc,pa-baa,pb-ca,pc,pd,\,\text{class 
}3\rangle  \tag{7.3977}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da-baa,db,dc,pa-\omega baa,pb-ca,pc,pd,\,%
\text{class }3\rangle  \tag{7.3978}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da-baa,db,dc,pa,pb-ca,pc-baa,pd,\,\text{class 
}3\rangle  \tag{7.3979}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da-baa,db,dc,pa,pb-ca,pc-\omega baa,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3980}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da-baa,db,dc,pa,pb-ca,pc-\omega ^{2}baa,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3981}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da,db,dc-baa,pa,pb-ca,pc,pd-xbaa,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.3982}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da,db,dc-baa,pa-baa,pb-ca,pc,pd,\,\text{class 
}3\rangle  \tag{7.3983}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da,db,dc-baa,pa-\omega baa,pb-ca,pc,pd,\,%
\text{class }3\rangle  \tag{7.3984}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da,db,dc-baa,pa,pb-ca,pc-baa,pd,\,\text{class 
}3\rangle  \tag{7.3985}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da,db,dc-baa,pa,pb-ca,pc-\omega baa,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3986}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da,db,dc-baa,pa,pb-ca,pc-\omega ^{2}baa,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.3987}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da,db,dc-baa,pa,pb-ca-baa,pc,pd-baa,\,\text{%
class }3\rangle  \tag{7.3988}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baa,cb,da,db,dc-baa,pa,pb-ca-\omega baa,pc,pd-baa,\,%
\text{class }3\rangle  \tag{7.3989}
\end{equation}

\subsection{Descendants of 6.24}

5 algebras

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa-ba,pb-ca,pc,pd-bab,\,\text{class }3\rangle
\tag{7.3990}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa-ba,pb-ca,pc,pd,\,\text{class }3\rangle 
\tag{7.3991}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa-ba,pb-ca,pc-bab,pd,\,\text{class }3\rangle
\tag{7.3992}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc,pa-ba,pb-ca,pc-\omega bab,pd,\,\text{class }%
3\rangle  \tag{7.3993}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cb,da,db,dc-bab,pa-ba,pb-ca,pc,pd,\,\text{class }3\rangle
\tag{7.3994}
\end{equation}

\subsection{Descendants of 6.29}

$10p+69+\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa,pb,pc-bab,pd-ca,\,\text{class }%
3\rangle  \tag{7.3995}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa,pb,pc-bab,pd-ca-bab,\,\text{class }%
3\rangle  \tag{7.3996}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa,pb,pc,pd-ca,\,\text{class }3\rangle 
\tag{7.3997}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa-bab,pb,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.3998}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa-\omega bab,pb,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.3999}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa,pb-bab,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4000}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa-bab,pb-bab,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4001}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa-\omega bab,pb-bab,pc,pd-ca,\,\text{%
class }3\rangle  \tag{7.4002}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa,pb,pc,pd-ca-bab,\,\text{class }%
3\rangle  \tag{7.4003}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa-bab,pb,pc,pd-ca-bab,\,\text{class }%
3\rangle  \tag{7.4004}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa-\omega bab,pb,pc,pd-ca-bab,\,\text{%
class }3\rangle  \tag{7.4005}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa,pb-bab,pc,pd-ca-bab,\,\text{class }%
3\rangle  \tag{7.4006}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa-bab,pb-bab,pc,pd-ca-bab,\,\text{class 
}3\rangle  \tag{7.4007}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc,pa-\omega bab,pb-bab,pc,pd-ca-bab,\,%
\text{class }3\rangle  \tag{7.4008}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa,pb,pc-xbab,pd-ca-bab,\,\text{%
class }3\rangle \;(x \neq 0)  \tag{7.4009}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa,pb,pc-bab,pd-ca,\,\text{class }%
3\rangle  \tag{7.4010}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa,pb,pc-\omega bab,pd-ca,\,\text{%
class }3\rangle  \tag{7.4011}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa,pb,pc,pd-ca-bab,\,\text{class }%
3\rangle  \tag{7.4012}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa-bab,pb,pc,pd-ca-bab,\,\text{class 
}3\rangle  \tag{7.4013}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa-\omega bab,pb,pc,pd-ca-bab,\,%
\text{class }3\rangle  \tag{7.4014}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa-xbab,pb-bab,pc,pd-ca-bab,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.4015}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa,pb,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4016}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa-bab,pb,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4017}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa-\omega bab,pb,pc,pd-ca,\,\text{%
class }3\rangle  \tag{7.4018}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa,pb-bab,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4019}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa-bab,pb-bab,pc,pd-ca,\,\text{class 
}3\rangle  \tag{7.4020}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db,dc,pa-\omega bab,pb-bab,pc,pd-ca,\,%
\text{class }3\rangle  \tag{7.4021}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa-xbab,pb,pc-bab,pd-ca-bab,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.4022}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa,pb,pc-bab,pd-ca,\,\text{class }%
3\rangle  \tag{7.4023}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa-bab,pb,pc-bab,pd-ca,\,\text{class 
}3\rangle  \tag{7.4024}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa-\omega bab,pb,pc-bab,pd-ca,\,%
\text{class }3\rangle  \tag{7.4025}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa-xbab,pb,pc,pd-ca-bab,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.4026}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa-xbab,pb-bab,pc,pd-ca-bab,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.4027}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa,pb,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4028}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa-bab,pb,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4029}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa-\omega bab,pb,pc,pd-ca,\,\text{%
class }3\rangle  \tag{7.4030}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa,pb-bab,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4031}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa-bab,pb-bab,pc,pd-ca,\,\text{class 
}3\rangle  \tag{7.4032}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-bab,pa-\omega bab,pb-bab,pc,pd-ca,\,%
\text{class }3\rangle  \tag{7.4033}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-\omega bab,pa-xbab,pb,pc-bab,pd-ca-bab,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.4034}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-\omega bab,pa,pb,pc-bab,pd-ca,\,\text{%
class }3\rangle  \tag{7.4035}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-\omega bab,pa-bab,pb,pc-bab,pd-ca,\,%
\text{class }3\rangle  \tag{7.4036}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-\omega bab,pa-\omega
bab,pb,pc-bab,pd-ca,\,\text{class }3\rangle  \tag{7.4037}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-\omega bab,pa-xbab,pb,pc,pd-ca-bab,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.4038}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-\omega bab,pa-xbab,pb-bab,pc,pd-ca-bab,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.4039}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-\omega bab,pa,pb,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4040}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-\omega bab,pa-bab,pb,pc,pd-ca,\,\text{%
class }3\rangle  \tag{7.4041}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-\omega bab,pa-\omega bab,pb,pc,pd-ca,\,%
\text{class }3\rangle  \tag{7.4042}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-\omega bab,pa,pb-bab,pc,pd-ca,\,\text{%
class }3\rangle  \tag{7.4043}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-\omega bab,pa-bab,pb-bab,pc,pd-ca,\,%
\text{class }3\rangle  \tag{7.4044}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db,dc-\omega bab,pa-\omega
bab,pb-bab,pc,pd-ca,\,\text{class }3\rangle  \tag{7.4045}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc,pa,pb,pc-baa,pd-ca,\,\text{class }%
3\rangle  \tag{7.4046}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc,pa,pb-baa,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4047}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc,pa,pb-\omega baa,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4048}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc,pa,pb,pc,pd-ca,\,\text{class }3\rangle 
\tag{7.4049}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc,pa-baa,pb,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4050}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc-baa,pa,pb,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4051}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc-baa,pa-baa,pb,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4052}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc-baa,pa-\omega baa,pb,pc,pd-ca,\,\text{%
class }3\rangle  \tag{7.4053}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc-baa,pa,pb-baa,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4054}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc-baa,pa,pb-\omega baa,pc,pd-ca,\,\text{%
class }3\rangle  \tag{7.4055}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc-baa,pa,pb,pc-baa,pd-ca,\,\text{class }%
3\rangle  \tag{7.4056}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc-baa,pa,pb-baa,pc-baa,pd-ca,\,\text{class 
}3\rangle  \tag{7.4057}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db,dc-baa,pa,pb-\omega baa,pc-baa,pd-ca,\,%
\text{class }3\rangle  \tag{7.4058}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc-baa,pa,pb,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4059}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc-baa,pa-baa,pb,pc,pd-ca,\,\text{class 
}3\rangle  \tag{7.4060}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc-baa,pa-\omega baa,pb,pc,pd-ca,\,%
\text{class }3\rangle  \tag{7.4061}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc-baa,pa-\omega ^{2}baa,pb,pc,pd-ca,\,%
\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.4062}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc-baa,pa-\omega ^{3}baa,pb,pc,pd-ca,\,%
\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.4063}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc-baa,pa,pb,pc-baa,pd-ca,\,\text{class 
}3\rangle  \tag{7.4064}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc-baa,pa,pb,pc-\omega baa,pd-ca,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.4065}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc-baa,pa,pb,pc-\omega ^{2}baa,pd-ca,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.4066}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc-baa,pa,pb-baa,pc-xbaa,pd-ca,\,\text{%
class }3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.4067}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc-baa,pa,pb-\omega baa,pc-xbaa,pd-ca,\,%
\text{class }3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.4068}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc,pa,pb,pc-baa,pd-ca,\,\text{class }%
3\rangle  \tag{7.4069}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc,pa,pb-baa,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4070}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc,pa,pb-\omega baa,pc,pd-ca,\,\text{%
class }3\rangle  \tag{7.4071}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc,pa,pb,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4072}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db,dc,pa-baa,pb,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4073}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-baa,dc,pa,pb,pc-baa,pd-ca,\,\text{class }%
3\rangle  \tag{7.4074}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-baa,dc,pa,pb-baa,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4075}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-baa,dc,pa,pb-\omega baa,pc,pd-ca,\,\text{%
class }3\rangle  \tag{7.4076}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-baa,dc,pa,pb,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4077}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-baa,dc,pa-baa,pb,pc,pd-ca,\,\text{class }%
3\rangle  \tag{7.4078}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db-baa,dc,pa,pb,pc-baa,pd-ca,\,\text{class 
}3\rangle  \tag{7.4079}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db-baa,dc,pa,pb-xbaa,pc,pd-ca,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.4080}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db-baa,dc,pa-baa,pb,pc,pd-ca,\,\text{class 
}3\rangle  \tag{7.4081}
\end{equation}

\subsection{Descendants of 6.33}

$39+\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd,pa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.4082}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd,pa,pb-bab,pc,pd,\,\text{class }%
3\rangle  \tag{7.4083}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd,pa-bab,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.4084}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd,pa-\omega bab,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.4085}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd,pa,pb,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.4086}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd,pa,pb-bab,pc-bab,pd,\,\text{%
class }3\rangle  \tag{7.4087}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd,pa-bab,pb,pc-bab,pd,\,\text{%
class }3\rangle  \tag{7.4088}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd,pa-\omega bab,pb,pc-bab,pd,\,%
\text{class }3\rangle  \tag{7.4089}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da-bab,db,dcc,dcd,pa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.4090}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da-bab,db,dcc,dcd,pa,pb-bab,pc,pd,\,\text{%
class }3\rangle  \tag{7.4091}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da-bab,db,dcc,dcd,pa-bab,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.4092}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da-bab,db,dcc,dcd,pa-\omega bab,pb,pc,pd,\,%
\text{class }3\rangle  \tag{7.4093}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da-bab,db,dcc,dcd,pa,pb,pc-bab,pd,\,\text{%
class }3\rangle  \tag{7.4094}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da-bab,db,dcc,dcd,pa,pb-bab,pc-bab,pd,\,\text{%
class }3\rangle  \tag{7.4095}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da-bab,db,dcc,dcd,pa-bab,pb,pc-bab,pd,\,\text{%
class }3\rangle  \tag{7.4096}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da-bab,db,dcc,dcd,pa-\omega bab,pb,pc-bab,pd,\,%
\text{class }3\rangle  \tag{7.4097}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da-bab,db,dcc,dcd,pa,pb,pc,pd-bab,\,\text{%
class }3\rangle  \tag{7.4098}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da-bab,db,dcc,dcd,pa,pb-bab,pc,pd-bab,\,\text{%
class }3\rangle  \tag{7.4099}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da-bab,db,dcc,dcd,pa-bab,pb,pc,pd-bab,\,\text{%
class }3\rangle  \tag{7.4100}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da-bab,db,dcc,dcd,pa-\omega bab,pb,pc,pd-bab,\,%
\text{class }3\rangle  \tag{7.4101}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd-bab,pa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.4102}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd-bab,pa,pb,pc,pd-bab,\,\text{%
class }3\rangle  \tag{7.4103}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd-bab,pa,pb,pc-bab,pd,\,\text{%
class }3\rangle  \tag{7.4104}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd-bab,pa,pb,pc-\omega bab,pd,\,%
\text{class }3\rangle  \tag{7.4105}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd-bab,pa,pb-bab,pc,pd-bab,\,\text{%
class }3\rangle  \tag{7.4106}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd-bab,pa,pb-bab,pc-bab,pd,\,\text{%
class }3\rangle  \tag{7.4107}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd-bab,pa,pb-bab,pc-\omega bab,pd,\,%
\text{class }3\rangle  \tag{7.4108}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd-bab,pa-bab,pb,pc-bab,pd,\,\text{%
class }3\rangle  \tag{7.4109}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd-bab,pa-bab,pb,pc-\omega bab,pd,\,%
\text{class }3\rangle  \tag{7.4110}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca,cb,da,db,dcc,dcd-bab,pa-\omega bab,pb,pc-\omega
bab,pd,\,\text{class }3\rangle  \tag{7.4111}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dcc,dcd-bab,pa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.4112}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dcc,dcd-bab,pa,pb,pc,pd-bab,\,\text{%
class }3\rangle  \tag{7.4113}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dcc,dcd-bab,pa,pb,pc-bab,pd,\,\text{%
class }3\rangle  \tag{7.4114}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dcc,dcd-bab,pa,pb,pc-\omega bab,pd,\,%
\text{class }3\rangle  \tag{7.4115}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dcc,dcd-bab,pa,pb-bab,pc,pd-bab,\,%
\text{class }3\rangle  \tag{7.4116}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dcc,dcd-bab,pa,pb-bab,pc-bab,pd,\,%
\text{class }3\rangle  \tag{7.4117}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dcc,dcd-bab,pa,pb-bab,pc-\omega
bab,pd,\,\text{class }3\rangle  \tag{7.4118}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dcc,dcd-bab,pa-bab,pb,pc-bab,pd,\,%
\text{class }3\rangle  \tag{7.4119}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dcc,dcd-bab,pa-bab,pb,pc-\omega
bab,pd,\,\text{class }3\rangle  \tag{7.4120}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dcc,dcd-bab,pa-\omega
bab,pb,pc-\omega bab,pd,\,\text{class }3\rangle  \tag{7.4121}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dcc,dcd-bab,pa,pb-bab,pc,pd-\omega
bab,\,\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.4122}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,ca-bab,cb,da,db,dcc,dcd-bab,pa,pb-bab,pc,pd-\omega
^{2}bab,\,\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.4123}
\end{equation}

\subsection{Descendants of 6.34}

$5p+38+2\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba,pb,pc,pd,\,\text{class }3\rangle
\tag{7.4124}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba-cac,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.4125}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba-\omega cac,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.4126}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba,pb,pc-cac,pd,\,\text{class }%
3\rangle  \tag{7.4127}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba-cac,pb,pc-cac,pd,\,\text{class }%
3\rangle  \tag{7.4128}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba-\omega cac,pb,pc-cac,pd,\,\text{%
class }3\rangle  \tag{7.4129}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba,pb-cac,pc,pd,\,\text{class }%
3\rangle  \tag{7.4130}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba-cac,pb-cac,pc,pd,\,\text{class }%
3\rangle  \tag{7.4131}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba-\omega cac,pb-cac,pc,pd,\,\text{%
class }3\rangle  \tag{7.4132}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba,pb-cac,pc-cac,pd,\,\text{class }%
3\rangle  \tag{7.4133}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba-cac,pb-cac,pc-cac,pd,\,\text{%
class }3\rangle  \tag{7.4134}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba-\omega cac,pb-cac,pc-cac,pd,\,%
\text{class }3\rangle  \tag{7.4135}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.4136}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-cac,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.4137}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-\omega cac,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.4138}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba,pb,pc,pd-cac,\,\text{class }%
3\rangle  \tag{7.4139}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba+cac,pb,pc,pd-cac,\,\text{%
class }3\rangle  \tag{7.4140}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba+(1-\omega
)cac,pb,pc,pd-cac,\,\text{class }3\rangle  \tag{7.4141}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba,pb,pc-cac,pd,\,\text{class }%
3\rangle  \tag{7.4142}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-cac,pb,pc-cac,pd,\,\text{%
class }3\rangle  \tag{7.4143}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-\omega cac,pb,pc-cac,pd,\,%
\text{class }3\rangle  \tag{7.4144}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-xcac,pb,pc-cac,pd-cac,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.4145}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-xcac,pb-cac,pc,pd,\,\text{%
class }3\rangle \;(\text{all }x,\;p=3\func{mod}4)  \tag{7.4146}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-xcac,pb-cac,pc-cac,pd,\,%
\text{class }3\rangle \;(\text{all }x,\;p=3\func{mod}4)  \tag{7.4147}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-xcac,pb-\omega cac,pc,pd,\,%
\text{class }3\rangle \;(\text{all }x,\;p=3\func{mod}4)  \tag{7.4148}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-xcac,pb-\omega
cac,pc-cac,pd,\,\text{class }3\rangle \;(\text{all }x,\;p=3\func{mod}4) 
\tag{7.4149}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-xcac,pb-cac,pc,pd,\,\text{%
class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.4150}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-xcac,pb-cac,pc-cac,pd,\,%
\text{class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4) 
\tag{7.4151}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-xcac,pb-\omega cac,pc,pd,\,%
\text{class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4) 
\tag{7.4152}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-xcac,pb-\omega
cac,pc-cac,pd,\,\text{class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{%
mod}4)  \tag{7.4153}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-xcac,pb-\omega
^{2}cac,pc,pd,\,\text{class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{%
mod}4)  \tag{7.4154}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-xcac,pb-\omega
^{2}cac,pc-cac,pd,\,\text{class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1%
\func{mod}4)  \tag{7.4155}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-xcac,pb-\omega
^{3}cac,pc,pd,\,\text{class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{%
mod}4)  \tag{7.4156}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-xcac,pb-\omega
^{3}cac,pc-cac,pd,\,\text{class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1%
\func{mod}4)  \tag{7.4157}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba,pb,pc,pd,\,\text{class }3\rangle
\tag{7.4158}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba,pb,pc-caa,pd,\,\text{class }%
3\rangle  \tag{7.4159}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba,pb,pc-\omega caa,pd,\,\text{%
class }3\rangle  \tag{7.4160}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba,pb-caa,pc,pd,\,\text{class }%
3\rangle  \tag{7.4161}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba,pb-caa,pc-caa,pd,\,\text{class }%
3\rangle  \tag{7.4162}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba,pb-caa,pc-\omega caa,pd,\,\text{%
class }3\rangle  \tag{7.4163}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba-caa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.4164}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba-caa,pb-caa,pc,pd,\,\text{class }%
3\rangle  \tag{7.4165}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.4166}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba-caa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.4167}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba,pb-caa,pc,pd,\,\text{class }%
3\rangle  \tag{7.4168}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba-caa,pb-caa,pc,pd,\,\text{%
class }3\rangle  \tag{7.4169}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba,pb,pc-caa,pd,\,\text{class }%
3\rangle  \tag{7.4170}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba,pb,pc-\omega caa,pd,\,\text{%
class }3\rangle  \tag{7.4171}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba,pb-caa,pc-caa,pd,\,\text{%
class }3\rangle  \tag{7.4172}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba,pb-caa,pc-\omega caa,pd,\,%
\text{class }3\rangle  \tag{7.4173}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba,pb,pc,pd-caa,\,\text{class }%
3\rangle  \tag{7.4174}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba+caa,pb,pc,pd-caa,\,\text{%
class }3\rangle  \tag{7.4175}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba+(1-\omega
)caa,pb,pc,pd-caa,\,\text{class }3\rangle  \tag{7.4176}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba,pb,pc-caa,pd-caa,\,\text{%
class }3\rangle  \tag{7.4177}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba,pb,pc-\omega caa,pd-caa,\,%
\text{class }3\rangle  \tag{7.4178}
\end{equation}

\subsection{Descendants of 6.35}

$10p+26$ algebras

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa,pb-ba,pc,pd,\,\text{class }3\rangle
\tag{7.4179}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-caa,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.4180}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa,pb-ba-caa,pc,pd,\,\text{class }%
3\rangle  \tag{7.4181}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-caa,pb-ba-caa,pc,pd,\,\text{class }%
3\rangle  \tag{7.4182}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa,pb-ba,pc-caa,pd,\,\text{class }%
3\rangle  \tag{7.4183}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa,pb-ba,pc-\omega caa,pd,\,\text{%
class }3\rangle  \tag{7.4184}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa,pb-ba-caa,pc-caa,pd,\,\text{class }%
3\rangle  \tag{7.4185}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa,pb-ba-caa,pc-\omega caa,pd,\,\text{%
class }3\rangle  \tag{7.4186}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa,pb-ba,pc,pd-caa,\,\text{class }%
3\rangle  \tag{7.4187}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa+caa,pb-ba,pc,pd-caa,\,\text{class }%
3\rangle  \tag{7.4188}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa,pb-ba,pc-caa,pd-caa,\,\text{class }%
3\rangle  \tag{7.4189}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa,pb-ba,pc-\omega caa,pd-caa,\,\text{%
class }3\rangle  \tag{7.4190}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.4191}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-caa,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.4192}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa,pb-ba-caa,pc,pd,\,\text{class }%
3\rangle  \tag{7.4193}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-caa,pb-ba-caa,pc,pd,\,\text{%
class }3\rangle  \tag{7.4194}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa,pb-ba,pc-caa,pd,\,\text{class }%
3\rangle  \tag{7.4195}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa,pb-ba,pc-\omega caa,pd,\,\text{%
class }3\rangle  \tag{7.4196}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa,pb-ba-caa,pc-caa,pd,\,\text{%
class }3\rangle  \tag{7.4197}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa,pb-ba-caa,pc-\omega caa,pd,\,%
\text{class }3\rangle  \tag{7.4198}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa,pb-ba,pc,pd-caa,\,\text{class }%
3\rangle  \tag{7.4199}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa+caa,pb-ba,pc,pd-caa,\,\text{%
class }3\rangle  \tag{7.4200}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa,pb-ba,pc-caa,pd-caa,\,\text{%
class }3\rangle  \tag{7.4201}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa,pb-ba,pc-\omega caa,pd-caa,\,%
\text{class }3\rangle  \tag{7.4202}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-caa,pa,pb-ba,pc-xcaa,pd,\,\text{class 
}3\rangle \;(\text{all }x)  \tag{7.4203}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-caa,pa,pb-ba,pc-xcaa,pd-caa,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.4204}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-caa,pa,pb-ba-caa,pc-xcaa,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.4205}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-caa,pa,pb-ba-caa,pc-xcaa,pd-caa,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.4206}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-caa,pa-caa,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.4207}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-caa,pa-caa,pb-ba-caa,pc,pd,\,\text{%
class }3\rangle  \tag{7.4208}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-caa,pa-xcaa,pb-ba,pc,pd-caa,\,\text{%
class }3\rangle \;(x\neq 0,-2,\,x\sim -x-2)  \tag{7.4209}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-caa,pa-xcaa,pb-ba-caa,pc,pd-caa,\,%
\text{class }3\rangle \;(x \neq 0,-2, x \sim -x-2)  \tag{7.4210}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-\omega caa,pa,pb-ba,pc-xcaa,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.4211}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-\omega caa,pa,pb-ba,pc-xcaa,pd-caa,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.4212}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-\omega caa,pa,pb-ba-caa,pc-xcaa,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.4213}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-\omega
caa,pa,pb-ba-caa,pc-xcaa,pd-caa,\,\text{class }3\rangle \;(\text{all }x) 
\tag{7.4214}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-\omega caa,pa-caa,pb-ba,pc,pd,\,\text{%
class }3\rangle  \tag{7.4215}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-\omega caa,pa-caa,pb-ba-caa,pc,pd,\,%
\text{class }3\rangle  \tag{7.4216}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-\omega caa,pa-xcaa,pb-ba,pc,pd-caa,\,%
\text{class }3\rangle \;(x\neq 0,-2,\,x\sim -x-2)  \tag{7.4217}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-\omega
caa,pa-xcaa,pb-ba-caa,pc,pd-caa,\,\text{class }3\rangle \;(x\neq
0,-2,\,x\sim -x-2)  \tag{7.4218}
\end{equation}

\subsection{Descendants of 6.36}

$p^{3}+9p^{2}+20p+18+\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.4219}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba-cac,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.4220}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba-\omega cac,pb-ba,pc,pd,\,\text{%
class }3\rangle  \tag{7.4221}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba-xcac,pb-ba-cac,pc,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.4222}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba-xcac,pb-ba-\omega cac,pc,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.4223}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba,pb-ba,pc,pd-cac,\,\text{class }%
3\rangle  \tag{7.4224}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba+cac,pb-ba,pc,pd-cac,\,\text{%
class }3\rangle  \tag{7.4225}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba+(1-\omega
)cac,pb-ba,pc,pd-cac,\,\text{class }3\rangle  \tag{7.4226}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba,pb-ba,pc-cac,pd,\,\text{class }%
3\rangle  \tag{7.4227}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba-cac,pb-ba,pc-cac,pd,\,\text{%
class }3\rangle  \tag{7.4228}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba-\omega cac,pb-ba,pc-cac,pd,\,%
\text{class }3\rangle  \tag{7.4229}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba-xcac,pb-ba-cac,pc-cac,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.4230}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba-xcac,pb-ba-\omega
cac,pc-cac,pd,\,\text{class }3\rangle \;(\text{all }x)  \tag{7.4231}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba,dc,pa-ba-xcac,pb-ba,pc-cac,pd-cac,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.4232}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba-xcac,pb-ba,pc-ycac,pd-cac,\,%
\text{class }3\rangle \;(\text{all }x,\,y,\,y\sim -y,\;p=3\func{mod}4) 
\tag{7.4233}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba-xcac,pb-ba,pc-ycac,pd-\omega cac,\,%
\text{class }3\rangle \;(\text{all }x,\,y,\,y\sim -y,\;p=3\func{mod}4) 
\tag{7.4234}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba-xcac,pb-ba,pc-ycac,pd-cac,\,%
\text{class }3\rangle \;(x\neq 0,\,x\sim -x,\text{ all }y,\,y\sim -y,\;p=1%
\func{mod}4)  \tag{7.4235}
\end{equation}

In the next presentation we have $\;x\neq 1-\omega ,\,x\sim -x+2(1-\omega ),$
all $y,\,y\sim -y,\;p=1\func{mod}4$:

\begin{equation}
\langle
a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba-xcac,pb-ba,pc-ycac,pd-\omega cac,\,%
\text{class }3\rangle  \tag{7.4236}
\end{equation}

In the next presentation we have $\;x\neq 1-\omega ^{2},\,x\sim
-x+2(1-\omega ^{2}),$ all $y,\,y\sim -y,\;p=1\func{mod}4$:

\begin{equation}
\langle
a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba-xcac,pb-ba,pc-ycac,pd-\omega
^{2}cac,\,\text{class }3\rangle  \tag{7.4237}
\end{equation}

In the next presentation we have $\;x\neq 1-\omega ^{3},\,x\sim
-x+2(1-\omega ^{3}),$ all $y,\,y\sim -y,\;p=1\func{mod}4$:

\begin{equation}
\langle
a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba-xcac,pb-ba,pc-ycac,pd-\omega
^{3}cac,\,\text{class }3\rangle  \tag{7.4238}
\end{equation}

In the next four presentations we have $\;$all $y,\,y\sim y^{\prime }$ if $%
y^{4}=y^{\prime 4}\func{mod}p,\;p=1\func{mod}4$:

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba,pb-ba,pc-ycac,pd-cac,\,%
\text{class }3\rangle  \tag{7.4239}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba-(1-\omega
)cac,pb-ba,pc-ycac,pd-\omega cac,\,\text{class }3\rangle  \tag{7.4240}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba-(1-\omega
^{2})cac,pb-ba,pc-ycac,pd-\omega ^{2}cac,\,\text{class }3\rangle 
\tag{7.4241}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba-(1-\omega
^{3})cac,pb-ba,pc-ycac,pd-\omega ^{3}cac,\,\text{class }3\rangle 
\tag{7.4242}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba-cac,pb-ba-cac,pc,pd,\,\text{%
class }3\rangle  \tag{7.4243}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba-cac,pb-ba-cac,pc-cac,pd,\,%
\text{class }3\rangle  \tag{7.4244}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba-cac,pb-ba-cac,pc-\omega
cac,pd,\,\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.4245}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba-cac,pb-ba-cac,pc-\omega
^{2}cac,pd,\,\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.4246}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba-cac,pb-ba-2cac,pc-xcac,pd,\,%
\text{class }3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.4247}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba-cac,pb-ba-(1+\omega
)cac,pc-xcac,pd,\,\text{class }3\rangle \;(\text{all }x,\,x\sim -x) 
\tag{7.4248}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba-2cac,pb-ba-xcac,pc-ycac,pd,\,\text{%
class }3\rangle \;(\text{all }x,\text{ all }y,\,y\sim -y)  \tag{7.4249}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da,db-ba-cac,dc,pa-ba-(1+\omega
)cac,pb-ba-xcac,pc-ycac,pd,\,\text{class }3\rangle \;(\text{all }x,\text{
all }y,\,y\sim -y)  \tag{7.4250}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-xcac,pb-ba-ycac,pc-zcac,pd-cac,\,%
\text{class }3\rangle \;(\text{all }x,\,y,z,\,z\sim -z)  \tag{7.4251}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-xcac,pb-ba-ycac,pc,pd,\,%
\text{class }3\rangle \;(\text{all }x,\,y)  \tag{7.4252}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-cac,db-ba,dc,pa-ba-xcac,pb-ba-ycac,pc-cac,pd,\,%
\text{class }3\rangle \;(\text{all }x,\,y)  \tag{7.4253}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-\omega
cac,db-ba,dc,pa-ba-xcac,pb-ba-ycac,pc-zcac,pd-cac,\,\text{class }3\rangle \;(%
\text{all }x,\,y,z,\,z\sim -z)  \tag{7.4254}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-\omega
cac,db-ba,dc,pa-ba-xcac,pb-ba-ycac,pc,pd,\,\text{class }3\rangle \;(\text{%
all }x,\,y)  \tag{7.4255}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,caa,cb,da-\omega
cac,db-ba,dc,pa-ba-xcac,pb-ba-ycac,pc-cac,pd,\,\text{class }3\rangle \;(%
\text{all }x,\,y)  \tag{7.4256}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.4257}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba,pb-ba-caa,pc,pd,\,\text{class }%
3\rangle  \tag{7.4258}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba-caa,pb-ba-xcaa,pc,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.4259}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba,pb-ba,pc,pd-caa,\,\text{class }%
3\rangle  \tag{7.4260}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba+caa,pb-ba,pc,pd-caa,\,\text{%
class }3\rangle  \tag{7.4261}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba,pb-ba,pc-caa,pd,\,\text{class }%
3\rangle  \tag{7.4262}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba,pb-ba-caa,pc-caa,pd,\,\text{%
class }3\rangle  \tag{7.4263}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba,pb-ba,pc-\omega caa,pd,\,\text{%
class }3\rangle  \tag{7.4264}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba,pb-ba-caa,pc-\omega caa,pd,\,%
\text{class }3\rangle  \tag{7.4265}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba,pb-ba,pc-caa,pd-caa,\,\text{%
class }3\rangle  \tag{7.4266}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc,pa-ba,pb-ba,pc-\omega caa,pd-caa,\,%
\text{class }3\rangle  \tag{7.4267}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.4268}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba,pb-ba-caa,pc,pd,\,\text{%
class }3\rangle  \tag{7.4269}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba-caa,pb-ba-xcaa,pc,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.4270}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba,pb-ba,pc-xcaa,pd,\,\text{%
class }3\rangle \;(x \neq 0)  \tag{7.4271}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba,pb-ba-caa,pc-xcaa,pd,\,%
\text{class }3\rangle \;(x \neq 0)  \tag{7.4272}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba-xcaa,pb-ba,pc,pd-caa,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.4273}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb-caa,da,db-ba,dc,pa-ba,pb-ba,pc-xcaa,pd-caa,\,%
\text{class }3\rangle \;(x \neq 0)  \tag{7.4274}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-caa,pa-ba,pb-ba,pc,pd,\,\text{class }%
3\rangle  \tag{7.4275}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-caa,pa-ba,pb-ba-caa,pc,pd,\,\text{%
class }3\rangle  \tag{7.4276}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-caa,pa-ba-caa,pb-ba-xcaa,pc,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.4277}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-caa,pa-ba,pb-ba,pc-xcaa,pd,\,\text{%
class }3\rangle \;(x \neq 0)  \tag{7.4278}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-caa,pa-ba,pb-ba-caa,pc-xcaa,pd,\,%
\text{class }3\rangle \;(x \neq 0)  \tag{7.4279}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-caa,pa-ba-ycaa,pb-ba-xcaa,pc,pd-caa,\,%
\text{class }3\rangle \;(x\neq 0,\,x\sim -x,\text{ all }y)  \tag{7.4280}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-caa,pa-ba-ycaa,pb-ba,pc,pd-caa,\,%
\text{class }3\rangle \;(\text{all }y,\,y\sim -y-2)  \tag{7.4281}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-caa,pa-ba,pb-ba-ycaa,pc-xcaa,pd-caa,\,%
\text{class }3\rangle \;(x\neq 0,\text{ all }y,\,y\sim -y)  \tag{7.4282}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-\omega caa,pa-ba,pb-ba,pc,pd,\,\text{%
class }3\rangle  \tag{7.4283}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-\omega caa,pa-ba,pb-ba-caa,pc,pd,\,%
\text{class }3\rangle  \tag{7.4284}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-\omega
caa,pa-ba-caa,pb-ba-xcaa,pc,pd,\,\text{class }3\rangle \;(\text{all }x) 
\tag{7.4285}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-\omega caa,pa-ba,pb-ba,pc-xcaa,pd,\,%
\text{class }3\rangle \;(x\neq 0)  \tag{7.4286}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-\omega
caa,pa-ba,pb-ba-caa,pc-xcaa,pd,\,\text{class }3\rangle \;(x\neq 0) 
\tag{7.4287}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-\omega
caa,pa-ba-ycaa,pb-ba-xcaa,pc,pd-caa,\,\text{class }3\rangle \;(x\neq
0,\,x\sim -x,\text{ all }y)  \tag{7.4288}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-\omega
caa,pa-ba-ycaa,pb-ba,pc,pd-caa,\,\text{class }3\rangle \;(\text{all }%
y,\,y\sim -y-2)  \tag{7.4289}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,cac,cb,da,db-ba,dc-\omega
caa,pa-ba,pb-ba-ycaa,pc-xcaa,pd-caa,\,\text{class }3\rangle \;(x\neq 0,\text{
all }y,\,y\sim -y)  \tag{7.4290}
\end{equation}

\subsection{Descendants of 6.48}

$27+3\gcd (p-1,3)+2\gcd (p-1,4)$ algebras%
\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc,pa,pb,pc-baa,pd,\,\text{class 
}3\rangle  \tag{7.4291}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc,pa,pb,pc,pd-baa,\,\text{class 
}3\rangle  \tag{7.4292}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc,pa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.4293}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc,pa-baa,pb,pc,pd,\,\text{class 
}3\rangle  \tag{7.4294}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc,pa,pb-baa,pc,pd,\,\text{class 
}3\rangle  \tag{7.4295}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc,pa,pb-\omega baa,pc,pd,\,%
\text{class }3\rangle  \tag{7.4296}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb-baa,da,db-ca,dc,pa,pb,pc-baa,pd,\,\text{%
class }3\rangle  \tag{7.4297}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb-baa,da,db-ca,dc,pa,pb,pc,pd-baa,\,\text{%
class }3\rangle  \tag{7.4298}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb-baa,da,db-ca,dc,pa,pb,pc,pd,\,\text{class 
}3\rangle  \tag{7.4299}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb-baa,da,db-ca,dc,pa-baa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.4300}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb-baa,da,db-ca,dc,pa,pb-baa,pc,pd,\,\text{%
class }3\rangle  \tag{7.4301}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb-baa,da,db-ca,dc,pa,pb-\omega baa,pc,pd,\,%
\text{class }3\rangle  \tag{7.4302}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-baa,pa,pb,pc-baa,pd,\,\text{%
class }3\rangle  \tag{7.4303}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-baa,pa,pb,pc-\omega baa,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.4304}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-baa,pa,pb,pc-\omega
^{2}baa,pd,\,\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.4305}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-baa,pa-baa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.4306}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-baa,pa,pb,pc,pd,\,\text{class 
}3\rangle  \tag{7.4307}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-baa,pa,pb,pc,pd-baa,\,\text{%
class }3\rangle  \tag{7.4308}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-baa,pa,pb-baa,pc,pd,\,\text{%
class }3\rangle  \tag{7.4309}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-baa,pa,pb-baa,pc,pd-baa,\,%
\text{class }3\rangle  \tag{7.4310}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-baa,pa,pb-\omega baa,pc,pd,\,%
\text{class }3\rangle  \tag{7.4311}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-baa,pa,pb-\omega
baa,pc,pd-baa,\,\text{class }3\rangle  \tag{7.4312}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-baa,pa,pb-\omega
^{2}baa,pc,pd,\,\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.4313}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-baa,pa,pb-\omega
^{2}baa,pc,pd-baa,\,\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.4314}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-baa,pa,pb-\omega
^{3}baa,pc,pd,\,\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.4315}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-baa,pa,pb-\omega
^{3}baa,pc,pd-baa,\,\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.4316}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb,da,db-ca,dc,pa,pb,pc-bad,pd,\,\text{class 
}3\rangle  \tag{7.4317}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb,da,db-ca,dc,pa,pb,pc-\omega bad,pd,\,%
\text{class }3\rangle  \tag{7.4318}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb,da,db-ca,dc,pa-bad,pb,pc,pd,\,\text{class 
}3\rangle  \tag{7.4319}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb,da,db-ca,dc,pa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.4320}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb,da,db-ca,dc,pa,pb,pc,pd-bad,\,\text{class 
}3\rangle  \tag{7.4321}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb,da,db-ca,dc,pa,pb-bad,pc,pd,\,\text{class 
}3\rangle  \tag{7.4322}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb,da,db-ca,dc,pa,pb-bad,pc,pd-bad,\,\text{%
class }3\rangle  \tag{7.4323}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb-bad,da,db-ca,dc,pa,pb,pc-bad,pd,\,\text{%
class }3\rangle  \tag{7.4324}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb-bad,da,db-ca,dc,pa,pb,pc-\omega bad,pd,\,%
\text{class }3\rangle  \tag{7.4325}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb-bad,da,db-ca,dc,pa-bad,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.4326}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb-bad,da,db-ca,dc,pa,pb,pc,pd,\,\text{class 
}3\rangle  \tag{7.4327}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb-bad,da,db-ca,dc,pa,pb-bad,pc,pd,\,\text{%
class }3\rangle  \tag{7.4328}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb-bad,da,db-ca,dc,pa,pb,pc,pd-bad,\,\text{%
class }3\rangle  \tag{7.4329}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb-bad,da,db-ca,dc,pa,pb-bad,pc,pd-bad,\,%
\text{class }3\rangle  \tag{7.4330}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb-bad,da,db-ca,dc,pa,pb,pc,pd-\omega bad,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.4331}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb-bad,da,db-ca,dc,pa,pb-bad,pc,pd-\omega
bad,\,\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.4332}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb-bad,da,db-ca,dc,pa,pb,pc,pd-\omega
^{2}bad,\,\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.4333}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,bab,bac,cb-bad,da,db-ca,dc,pa,pb-bad,pc,pd-\omega
^{2}bad,\,\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.4334}
\end{equation}

\subsection{Descendants of 6.51}

$8p+15+(2p+5)\gcd (p-1,3)+(p+2)\gcd (p-1,4)+\gcd (p-1,5)$ algebras

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa-ca,pb,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.4335}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa-ca,pb,pc,pd-bab,\,\text{class }%
3\rangle  \tag{7.4336}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa-ca,pb,pc-bab,pd-bab,\,\text{class }%
3\rangle  \tag{7.4337}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa-ca,pb,pc,pd-\omega bab,\,\text{%
class }3\rangle \;(p=1\func{mod}3)  \tag{7.4338}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa-ca,pb,pc-bab,pd-\omega bab,\,\text{%
class }3\rangle \;(p=1\func{mod}3)  \tag{7.4339}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa-ca,pb,pc,pd-\omega ^{2}bab,\,\text{%
class }3\rangle \;(p=1\func{mod}3)  \tag{7.4340}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa-ca,pb,pc-bab,pd-\omega ^{2}bab,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.4341}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa-ca,pb,pc,pd,\,\text{class }3\rangle
\tag{7.4342}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa-ca-bab,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.4343}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa-ca-\omega bab,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.4344}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa-ca,pb-bab,pc,pd,\,\text{class }%
3\rangle  \tag{7.4345}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa-ca-bab,pb-bab,pc,pd,\,\text{class }%
3\rangle  \tag{7.4346}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa-ca-\omega bab,pb-bab,pc,pd,\,\text{%
class }3\rangle  \tag{7.4347}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-ca,pb,pc-xbab,pd-bab,\,\text{%
class }3\rangle \;(\text{all }x,\,x \sim x^{\prime }\text{ if }x^3=x^{\prime
3}\func{mod}p)  \tag{7.4348}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-ca,pb,pc-xbab,pd-\omega bab,\,%
\text{class }3\rangle \;(\text{all }x,\,x \sim x^{\prime }\text{ if }%
x^3=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.4349}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-ca,pb,pc-xbab,pd-\omega
^{2}bab,\,\text{class }3\rangle \;(\text{all }x,\,x\sim x^{\prime}\text{ if }%
x^3 =x^{\prime 3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.4350}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-ca,pb,pc-bab,pd,\,\text{class }%
3\rangle  \tag{7.4351}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-ca,pb,pc-\omega bab,pd,\,\text{%
class }3\rangle \;(p=1\func{mod}5)  \tag{7.4352}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-ca,pb,pc-\omega ^{2}bab,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}5)  \tag{7.4353}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-ca,pb,pc-\omega ^{3}bab,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}5)  \tag{7.4354}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-ca,pb,pc-\omega ^{4}bab,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}5)  \tag{7.4355}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-ca-xbab,pb-bab,pc,pd,\,\text{%
class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4)  \tag{7.4356}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-ca-xbab,pb-\omega bab,pc,pd,\,%
\text{class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{mod}4) 
\tag{7.4357}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-ca-xbab,pb-\omega
^{2}bab,pc,pd,\,\text{class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{%
mod}4)  \tag{7.4358}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-ca-xbab,pb-\omega
^{3}bab,pc,pd,\,\text{class }3\rangle \;(\text{all }x,\,x\sim -x,\;p=1\func{%
mod}4)  \tag{7.4359}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-ca-xbab,pb-bab,pc,pd,\,\text{%
class }3\rangle \;(\text{all }x,\;p=3\func{mod}4)  \tag{7.4360}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-ca-xbab,pb-\omega bab,pc,pd,\,%
\text{class }3\rangle \;(\text{all }x,\;p=3\func{mod}4)  \tag{7.4361}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-ca,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.4362}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-ca-bab,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.4363}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-ca-\omega bab,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.4364}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-bab,pa-ca,pb,pc,pd-xbab,\,\text{class 
}3\rangle \;(\text{all }x)  \tag{7.4365}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-bab,pa-ca,pb,pc-bab,pd-xbab,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.4366}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-bab,pa-ca,pb-bab,pc,pd,\,\text{class }%
3\rangle  \tag{7.4367}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-bab,pa-ca-ybab,pb,pc,pd+bab,\,\text{%
class }3\rangle \;(y\neq 0,\,y\sim y^{\prime }\text{ if }y^{3}=y^{\prime 3}%
\func{mod}p)  \tag{7.4368}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-bab,pa-ca-ybab,pb,pc-bab,pd+bab,\,%
\text{class }3\rangle \;(y\neq 0,\,y\sim y^{\prime }\text{ if }%
y^{3}=y^{\prime 3}\func{mod}p)  \tag{7.4369}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-\omega bab,pa-ca,pb,pc,pd-xbab,\,%
\text{class }3\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.4370}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-\omega bab,pa-ca,pb,pc-bab,pd-xbab,\,%
\text{class }3\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.4371}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-\omega bab,pa-ca,pb-bab,pc,pd,\,\text{%
class }3\rangle \;(p=1\func{mod}3)  \tag{7.4372}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-\omega bab,pa-ca-ybab,pb,pc,pd+\omega
bab,\,\text{class }3\rangle \;(y\neq 0,\,y\sim y^{\prime }\text{ if }%
y^{3}=y^{\prime 3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.4373}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-\omega
bab,pa-ca-ybab,pb,pc-bab,pd+\omega bab,\,\text{class }3\rangle \;(y\neq
0,\,y\sim y^{\prime }\text{ if }y^{3}=y^{\prime 3}\func{mod}p,\;p=1\func{mod}%
3)  \tag{7.4374}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-\omega ^{2}bab,pa-ca,pb,pc,pd-xbab,\,%
\text{class }3\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.4375}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-\omega
^{2}bab,pa-ca,pb,pc-bab,pd-xbab,\,\text{class }3\rangle \;(\text{all }x,\;p=1%
\func{mod}3)  \tag{7.4376}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-\omega ^{2}bab,pa-ca,pb-bab,pc,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.4377}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-\omega
^{2}bab,pa-ca-ybab,pb,pc,pd+\omega ^{2}bab,\,\text{class }3\rangle \;(y\neq
0,\,y\sim y^{\prime }\text{ if }y^{3}=y^{\prime 3}\func{mod}p,\;p=1\func{mod}%
3)  \tag{7.4378}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-\omega
^{2}bab,pa-ca-ybab,pb,pc-bab,pd+\omega ^{2}bab,\,\text{class }3\rangle
\;(y\neq 0,\,y\sim y^{\prime }\text{ if }y^{3}=y^{\prime 3}\func{mod}p,\;p=1%
\func{mod}3)  \tag{7.4379}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc,pa-ca,pb,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.4380}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc,pa-ca,pb,pc,pd-baa,\,\text{class }%
3\rangle  \tag{7.4381}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc,pa-ca,pb,pc,pd,\,\text{class }3\rangle
\tag{7.4382}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc,pa-ca-baa,pb,pc,pd,\,\text{class }%
3\rangle  \tag{7.4383}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc,pa-ca,pb-baa,pc,pd,\,\text{class }%
3\rangle  \tag{7.4384}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc,pa-ca,pb-\omega baa,pc,pd,\,\text{%
class }3\rangle  \tag{7.4385}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db-ca,dc,pa-ca,pb,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.4386}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db-ca,dc,pa-ca,pb,pc,pd-baa,\,\text{class }%
3\rangle  \tag{7.4387}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db-ca,dc,pa-ca,pb,pc,pd-\omega baa,\,\text{%
class }3\rangle \;(p=1\func{mod}4)  \tag{7.4388}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db-ca,dc,pa-ca,pb-xbaa,pc,pd,\,\text{class 
}3\rangle \;(\text{all }x)  \tag{7.4389}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db-ca,dc,pa-ca-baa,pb+baa,pc,pd,\,\text{%
class }3\rangle  \tag{7.4390}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-\omega baa,da,db-ca,dc,pa-ca,pb,pc-baa,pd,\,\text{%
class }3\rangle  \tag{7.4391}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-\omega baa,da,db-ca,dc,pa-ca,pb,pc,pd-baa,\,\text{%
class }3\rangle  \tag{7.4392}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-\omega baa,da,db-ca,dc,pa-ca,pb,pc,pd-\omega
baa,\,\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.4393}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-\omega baa,da,db-ca,dc,pa-ca,pb-xbaa,pc,pd,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.4394}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-\omega baa,da,db-ca,dc,pa-ca-baa,pb+\omega
baa,pc,pd,\,\text{class }3\rangle  \tag{7.4395}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc-baa,pa-ca,pb,pc-baa,pd,\,\text{class }%
3\rangle  \tag{7.4396}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc-baa,pa-ca,pb,pc-\omega baa,pd,\,\text{%
class }3\rangle \;(p=1\func{mod}3)  \tag{7.4397}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc-baa,pa-ca,pb,pc-\omega ^{2}baa,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.4398}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc-baa,pa-ca,pb,pc,pd-xbaa,\,\text{class 
}3\rangle \;(\text{all }x)  \tag{7.4399}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc-baa,pa-ca,pb-baa,pc,pd-xbaa,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.4400}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc-baa,pa-ca,pb-\omega baa,pc,pd-xbaa,\,%
\text{class }3\rangle \;(\text{all }x)  \tag{7.4401}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc-baa,pa-ca,pb-\omega
^{2}baa,pc,pd-xbaa,\,\text{class }3\rangle \;(\text{all }x,\;p=1\func{mod}4)
\tag{7.4402}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc-baa,pa-ca,pb-\omega
^{3}baa,pc,pd-xbaa,\,\text{class }3\rangle \;(\text{all }x,\;p=1\func{mod}4)
\tag{7.4403}
\end{equation}

\subsection{Descendants of 6.52}

$4p^{2}+15p+15+(p+1)\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa,pb,pc-ca,pd-bab,\,\text{class }%
3\rangle  \tag{7.4404}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa,pb-bab,pc-ca,pd,\,\text{class }%
3\rangle  \tag{7.4405}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa,pb-bab,pc-ca-bab,pd,\,\text{class }%
3\rangle  \tag{7.4406}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa,pb,pc-ca,pd,\,\text{class }3\rangle
\tag{7.4407}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa-bab,pb,pc-ca,pd,\,\text{class }%
3\rangle  \tag{7.4408}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa-\omega bab,pb,pc-ca,pd,\,\text{%
class }3\rangle  \tag{7.4409}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa,pb,pc-ca-bab,pd,\,\text{class }%
3\rangle  \tag{7.4410}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa-bab,pb,pc-ca-bab,pd,\,\text{class }%
3\rangle  \tag{7.4411}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc,pa-\omega bab,pb,pc-ca-bab,pd,\,\text{%
class }3\rangle  \tag{7.4412}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-bab,pa,pb,pc-ca,pd-bab,\,\text{class }%
3\rangle  \tag{7.4413}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-bab,pa,pb,pc-ca,pd-\omega bab,\,\text{%
class }3\rangle \;(p=1\func{mod}3)  \tag{7.4414}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-bab,pa,pb,pc-ca,pd-\omega ^{2}bab,\,%
\text{class }3\rangle \;(p=1\func{mod}3)  \tag{7.4415}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-bab,pa-bab,pb,pc-ca,pd,\,\text{class }%
3\rangle  \tag{7.4416}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-bab,pa-\omega bab,pb,pc-ca,pd,\,\text{%
class }3\rangle  \tag{7.4417}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-bab,pa-\omega ^{2}bab,pb,pc-ca,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.4418}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-bab,pa-\omega ^{3}bab,pb,pc-ca,pd,\,%
\text{class }3\rangle \;(p=1\func{mod}4)  \tag{7.4419}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-bab,pa,pb,pc-ca,pd,\,\text{class }%
3\rangle  \tag{7.4420}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-bab,pa,pb-bab,pc-ca,pd,\,\text{class }%
3\rangle  \tag{7.4421}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da,db-ca,dc-bab,pa,pb-\omega bab,pc-ca,pd,\,\text{%
class }3\rangle  \tag{7.4422}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa,pb,pc-ca,pd-xbab,\,\text{class 
}3\rangle \;(\text{all }x)  \tag{7.4423}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa,pb,pc-ca-bab,pd-bab,\,\text{%
class }3\rangle  \tag{7.4424}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-xbab,pb-bab,pc-ca,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.4425}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-bab,pb,pc-ca,pd,\,\text{class }%
3\rangle  \tag{7.4426}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc,pa-\omega bab,pb,pc-ca,pd,\,\text{%
class }3\rangle  \tag{7.4427}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc-bab,pa,pb,pc-ca,pd-xbab,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.4428}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc-bab,pa,pb,pc-ca-xbab,pd-bab,\,%
\text{class }3\rangle \;(x\neq 0,\,x\sim x^{\prime }\text{ if }%
x^{3}=x^{\prime 3}\func{mod}p)  \tag{7.4429}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc-bab,pa-xbab,pb,pc-ca,pd,\,\text{%
class }3\rangle \;(x\neq 0,\,x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}%
\func{mod}p)  \tag{7.4430}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc-bab,pa,pb-xbab,pc-ca,pd,\,\text{%
class }3\rangle \;(x\neq 0,\,x\sim x^{\prime }\text{ if }x^{3}=x^{\prime 3}%
\func{mod}p)  \tag{7.4431}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc-\omega bab,pa,pb,pc-ca,pd-xbab,\,%
\text{class }3\rangle \;(\text{all }x,\;p=1\func{mod}3)  \tag{7.4432}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc-\omega
bab,pa,pb,pc-ca-xbab,pd-bab,\,\text{class }3\rangle \;(x\neq 0,\,x\sim
x^{\prime }\text{ if }x^{3}=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3) 
\tag{7.4433}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc-\omega bab,pa-xbab,pb,pc-ca,pd,\,%
\text{class }3\rangle \;(x\neq 0,\,x\sim x^{\prime }\text{ if }%
x^{3}=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.4434}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc-\omega bab,pa,pb-xbab,pc-ca,pd,\,%
\text{class }3\rangle \;(x\neq 0,\,x\sim x^{\prime }\text{ if }%
x^{3}=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3)  \tag{7.4435}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc-\omega
^{2}bab,pa,pb,pc-ca,pd-xbab,\,\text{class }3\rangle \;(\text{all }x,\;p=1%
\func{mod}3)  \tag{7.4436}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc-\omega
^{2}bab,pa,pb,pc-ca-xbab,pd-bab,\,\text{class }3\rangle \;(x\neq 0,\,x\sim
x^{\prime }\text{ if }x^{3}=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3) 
\tag{7.4437}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc-\omega
^{2}bab,pa-xbab,pb,pc-ca,pd,\,\text{class }3\rangle \;(x\neq 0,\,x\sim
x^{\prime }\text{ if }x^{3}=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3) 
\tag{7.4438}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,baa,cb,da-bab,db-ca,dc-\omega
^{2}bab,pa,pb-xbab,pc-ca,pd,\,\text{class }3\rangle \;(x\neq 0,\,x\sim
x^{\prime }\text{ if }x^{3}=x^{\prime 3}\func{mod}p,\;p=1\func{mod}3) 
\tag{7.4439}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc,pa,pb,pc-ca,pd-baa,\,\text{class }%
3\rangle  \tag{7.4440}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc,pa,pb,pc-ca-baa,pd-baa,\,\text{class }%
3\rangle  \tag{7.4441}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc,pa,pb-xbaa,pc-ca,pd,\,\text{class }%
3\rangle \;(\text{all }x)  \tag{7.4442}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc,pa,pb-xbaa,pc-ca-baa,pd,\,\text{class 
}3\rangle \;(\text{all }x)  \tag{7.4443}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc,pa-baa,pb-xbaa,pc-ca,pd,\,\text{class 
}3\rangle \;(\text{all }x)  \tag{7.4444}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc,pa-baa,pb-xbaa,pc-ca-baa,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.4445}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db-ca,dc,pa,pb,pc-ca-xbaa,pd-baa,\,\text{%
class }3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.4446}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db-ca,dc,pa,pb,pc-ca-xbaa,pd-\omega baa,\,%
\text{class }3\rangle \;(\text{all }x,\,x\sim -x)  \tag{7.4447}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db-ca,dc,pa-xbaa,pb-ybaa,pc-ca-baa,pd,\,%
\text{class }3\rangle \;(\text{all }x,\,y)  \tag{7.4448}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db-ca,dc,pa,pb-xbaa,pc-ca,pd,\,\text{class 
}3\rangle \;(\text{all }x)  \tag{7.4449}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb-baa,da,db-ca,dc,pa-baa,pb-xbaa,pc-ca,pd,\,\text{%
class }3\rangle \;(\text{all }x)  \tag{7.4450}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc-baa,pa,pb-ybaa,pc-ca-xbaa,pd-baa,\,%
\text{class }3\rangle \;(\text{all }x,\,y,\,x\sim -x)  \tag{7.4451}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc-baa,pa,pb-ybaa,pc-ca-xbaa,pd,\,\text{%
class }3\rangle \;(\text{all }x,\,y,\,x\sim -x)  \tag{7.4452}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc-baa,pa-baa,pb-ybaa,pc-ca-xbaa,pd,\,%
\text{class }3\rangle \;(\text{all }x,\,y,\,x\sim -x)  \tag{7.4453}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc-\omega
baa,pa,pb-ybaa,pc-ca-xbaa,pd-baa,\,\text{class }3\rangle \;(\text{all }%
x,\,y,\,x\sim -x)  \tag{7.4454}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc-\omega baa,pa,pb-ybaa,pc-ca-xbaa,pd,\,%
\text{class }3\rangle \;(\text{all }x,\,y,\,x\sim -x)  \tag{7.4455}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,cb,da,db-ca,dc-\omega
baa,pa-baa,pb-ybaa,pc-ca-xbaa,pd,\,\text{class }3\rangle \;(\text{all }%
x,\,y,x\sim -x)  \tag{7.4456}
\end{equation}

\subsection{Descendants of 6.60}

$7+\gcd (p-1,3)$ algebras

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-\omega ba,pa,pb,pc,pd,\,\text{%
class }3\rangle  \tag{7.4457}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-\omega ba,pa,pb,pc-baa,pd,\,%
\text{class }3\rangle  \tag{7.4458}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-\omega
ba,pa,pb-baa,pc-xbaa,pd,\,\text{class }3\rangle \;(x\text{ FIXED with }%
x^{2}-\omega \text{ not a square }\func{mod}p)  \tag{7.4459}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb,da,db-ca,dc-\omega ba,pa,pb,pc,pd-baa,\,%
\text{class }3\rangle  \tag{7.4460}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb-baa,da,db-ca,dc-\omega ba,pa,pb,pc,pd,\,%
\text{class }3\rangle  \tag{7.4461}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb-baa,da,db-ca,dc-\omega
ba,pa,pb,pc-baa,pd,\,\text{class }3\rangle  \tag{7.4462}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb-baa,da,db-ca,dc-\omega
ba,pa,pb-baa,pc-xbaa,pd,\,\text{class }3\rangle \;(x\text{ FIXED with }%
x^{2}-\omega \text{ not a square }\func{mod}p)  \tag{7.4463}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb-baa,da,db-ca,dc-\omega
ba,pa,pb,pc,pd-baa,\,\text{class }3\rangle  \tag{7.4464}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb-baa,da,db-ca,dc-\omega
ba,pa,pb,pc,pd-\omega baa,\,\text{class }3\rangle \;(p=1\func{mod}3) 
\tag{7.4465}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,bac,bad,cb-baa,da,db-ca,dc-\omega
ba,pa,pb,pc,pd-\omega ^{2}baa,\,\text{class }3\rangle \;(p=1\func{mod}3) 
\tag{7.4466}
\end{equation}

\subsection{Descendants of 6.63}

4 algebras

\begin{equation}
\langle a,b,c,d\,|\,ba,ca,cb,da,db,dc,pb,pc,pd,\,\text{class }4\rangle 
\tag{7.4467}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ba-p^3a,ca,cb,da,db,dc,pb,pc,pd,\,\text{class }4\rangle 
\tag{7.4468}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ba,ca,cb-p^3a,da,db,dc,pb,pc,pd,\,\text{class }4\rangle 
\tag{7.4469}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ba,ca,cb-p^3a,da-p^3a,db,dc,pb,pc,pd,\,\text{class }%
4\rangle  \tag{7.4470}
\end{equation}

\subsection{Descendants of 6.67}

$18+8\gcd (p-1,3)+3\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc,pa,pb,pc-baaa,pd,\,\text{class }%
4\rangle  \tag{7.4471}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc,pa,pb-baaa,pc,pd,\,\text{class }%
4\rangle  \tag{7.4472}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc,pa,pb-\omega baaa,pc,pd,\,\text{class 
}4\rangle \;(p=1\func{mod}3)  \tag{7.4473}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc,pa,pb-\omega ^{2}baaa,pc,pd,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.4474}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc,pa,pb,pc,pd,\,\text{class }4\rangle 
\tag{7.4475}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc,pa-baaa,pb,pc,pd,\,\text{class }%
4\rangle  \tag{7.4476}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc,pa,pb,pc-baaa,pd,\,\text{class }%
4\rangle  \tag{7.4477}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc,pa,pb-baaa,pc,pd,\,\text{class }%
4\rangle  \tag{7.4478}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc,pa,pb-\omega baaa,pc,pd,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.4479}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc,pa,pb-\omega ^{2}baaa,pc,pd,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4480}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc,pa,pb,pc,pd,\,\text{class }%
4\rangle  \tag{7.4481}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc,pa-baaa,pb,pc,pd,\,\text{class }%
4\rangle  \tag{7.4482}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc,pa-\omega baaa,pb,pc,pd,\,\text{%
class }4\rangle  \tag{7.4483}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc,pa-\omega ^{2}baaa,pb,pc,pd,\,%
\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.4484}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc,pa-\omega ^{3}baaa,pb,pc,pd,\,%
\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.4485}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baaa,da,db,dc,pa,pb,pc,pd-baaa,\,\text{class }%
4\rangle  \tag{7.4486}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baaa,da,db,dc,pa,pb,pc-baaa,pd,\,\text{class }%
4\rangle  \tag{7.4487}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baaa,da,db,dc,pa,pb-baaa,pc,pd,\,\text{class }%
4\rangle  \tag{7.4488}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baaa,da,db,dc,pa,pb-\omega baaa,pc,pd,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.4489}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baaa,da,db,dc,pa,pb-\omega ^{2}baaa,pc,pd,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4490}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baaa,da,db,dc,pa,pb,pc,pd,\,\text{class }%
4\rangle  \tag{7.4491}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb-baaa,da,db,dc,pa-baaa,pb,pc,pd,\,\text{class }%
4\rangle  \tag{7.4492}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb-baaa,da,db,dc,pa,pb,pc,pd-baaa,\,\text{%
class }4\rangle  \tag{7.4493}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb-baaa,da,db,dc,pa,pb,pc-baaa,pd,\,\text{%
class }4\rangle  \tag{7.4494}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb-baaa,da,db,dc,pa,pb,pc-\omega baaa,pd,\,%
\text{class }4\rangle  \tag{7.4495}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb-baaa,da,db,dc,pa,pb-baaa,pc,pd,\,\text{%
class }4\rangle  \tag{7.4496}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb-baaa,da,db,dc,pa,pb-\omega baaa,pc,pd,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4497}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb-baaa,da,db,dc,pa,pb-\omega
^{2}baaa,pc,pd,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4498}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb-baaa,da,db,dc,pa,pb,pc,pd,\,\text{class }%
4\rangle  \tag{7.4499}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb-baaa,da,db,dc,pa-baaa,pb,pc,pd,\,\text{%
class }4\rangle  \tag{7.4500}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb-baaa,da,db,dc,pa-\omega baaa,pb,pc,pd,\,%
\text{class }4\rangle  \tag{7.4501}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb-baaa,da,db,dc,pa-\omega
^{2}baaa,pb,pc,pd,\,\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.4502}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb-baaa,da,db,dc,pa-\omega
^{3}baaa,pb,pc,pd,\,\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.4503}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baaa,pa,pb,pc,pd,\,\text{class }%
4\rangle  \tag{7.4504}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baaa,pa,pb,pc-baaa,pd,\,\text{class }%
4\rangle  \tag{7.4505}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baaa,pa,pb-baaa,pc,pd,\,\text{class }%
4\rangle  \tag{7.4506}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baaa,pa,pb-baaa,pc-baaa,pd,\,\text{%
class }4\rangle  \tag{7.4507}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baaa,pa,pb-\omega baaa,pc,pd,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.4508}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baaa,pa,pb-\omega baaa,pc-baaa,pd,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4509}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baaa,pa,pb-\omega ^{2}baaa,pc,pd,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4510}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baaa,pa,pb-\omega
^{2}baaa,pc-baaa,pd,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4511}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab,ca,cb,da,db,dc-baaa,pa-baaa,pb,pc,pd,\,\text{class }%
4\rangle  \tag{7.4512}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc-baaa,pa,pb,pc,pd,\,\text{class }%
4\rangle  \tag{7.4513}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc-baaa,pa,pb,pc-baaa,pd,\,\text{%
class }4\rangle  \tag{7.4514}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc-baaa,pa,pb-baaa,pc,pd,\,\text{%
class }4\rangle  \tag{7.4515}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc-baaa,pa,pb-baaa,pc-baaa,pd,\,%
\text{class }4\rangle  \tag{7.4516}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc-baaa,pa,pb-\omega baaa,pc,pd,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4517}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc-baaa,pa,pb-\omega
baaa,pc-baaa,pd,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4518}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc-baaa,pa,pb-\omega
^{2}baaa,pc,pd,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4519}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc-baaa,pa,pb-\omega
^{2}baaa,pc-baaa,pd,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4520}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc-baaa,pa-baaa,pb,pc,pd,\,\text{%
class }4\rangle  \tag{7.4521}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc-baaa,pa-\omega baaa,pb,pc,pd,\,%
\text{class }4\rangle  \tag{7.4522}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc-baaa,pa-\omega
^{2}baaa,pb,pc,pd,\,\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.4523}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,bab-baaa,ca,cb,da,db,dc-baaa,pa-\omega
^{3}baaa,pb,pc,pd,\,\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.4524}
\end{equation}

\subsection{Descendants of 6.72}

$17+(p+7)\gcd (p-1,3)+\gcd (p-1,4)$ algebras

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc,pa,pb,pc,pd-baaa,\,\text{class }%
4\rangle  \tag{7.4525}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc,pa,pb,pc,pd,\,\text{class }%
4\rangle  \tag{7.4526}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc,pa,pb,pc-baaa,pd,\,\text{class }%
4\rangle  \tag{7.4527}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc,pa,pb-baaa,pc,pd,\,\text{class }%
4\rangle  \tag{7.4528}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc,pa,pb-baaa,pc-baaa,pd,\,\text{%
class }4\rangle  \tag{7.4529}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc,pa,pb-\omega baaa,pc,pd,\,\text{%
class }4\rangle \;(p=1\func{mod}3)  \tag{7.4530}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc,pa,pb-\omega baaa,pc-baaa,pd,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4531}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc,pa,pb-\omega ^{2}baaa,pc,pd,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4532}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc,pa,pb-\omega
^{2}baaa,pc-baaa,pd,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4533}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc,pa-baaa,pb,pc,pd,\,\text{class }%
4\rangle  \tag{7.4534}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc,pa-baaa,pb,pc-baaa,pd,\,\text{%
class }4\rangle  \tag{7.4535}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc,pa,pb,pc,pd-baaa,\,\text{%
class }4\rangle  \tag{7.4536}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc,pa,pb,pc,pd,\,\text{class }%
4\rangle  \tag{7.4537}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc,pa,pb,pc-baaa,pd,\,\text{%
class }4\rangle  \tag{7.4538}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc,pa,pb-baaa,pc,pd,\,\text{%
class }4\rangle  \tag{7.4539}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc,pa,pb-baaa,pc-baaa,pd,\,%
\text{class }4\rangle  \tag{7.4540}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc,pa,pb-\omega baaa,pc,pd,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4541}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc,pa,pb-\omega
baaa,pc-baaa,pd,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4542}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc,pa,pb-\omega
^{2}baaa,pc,pd,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4543}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc,pa,pb-\omega
^{2}baaa,pc-baaa,pd,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4544}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc,pa-baaa,pb,pc,pd,\,\text{%
class }4\rangle  \tag{7.4545}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc-baaa,pa,pb,pc,pd-baaa,\,\text{%
class }4\rangle  \tag{7.4546}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc-baaa,pa,pb,pc,pd-\omega baaa,\,%
\text{class }4\rangle  \tag{7.4547}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc-baaa,pa,pb,pc,pd,\,\text{class }%
4\rangle  \tag{7.4548}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc-baaa,pa,pb,pc-baaa,pd,\,\text{%
class }4\rangle  \tag{7.4549}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc-baaa,pa,pb-baaa,pc,pd,\,\text{%
class }4\rangle  \tag{7.4550}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc-baaa,pa,pb-baaa,pc-baaa,pd,\,%
\text{class }4\rangle  \tag{7.4551}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc-baaa,pa,pb-\omega baaa,pc,pd,\,%
\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4552}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc-baaa,pa,pb-\omega
baaa,pc-baaa,pd,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4553}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc-baaa,pa,pb-\omega
^{2}baaa,pc,pd,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4554}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc-baaa,pa,pb-\omega
^{2}baaa,pc-baaa,pd,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4555}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db,dc-baaa,pa-baaa,pb,pc,pd,\,\text{%
class }4\rangle  \tag{7.4556}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc-baaa,pa,pb,pc,pd-baaa,\,%
\text{class }4\rangle  \tag{7.4557}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc-baaa,pa,pb,pc,pd-\omega
baaa,\,\text{class }4\rangle  \tag{7.4558}
\end{equation}

\begin{equation}
\langle
a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc-baaa,pa,pb-xbaaa,pc-baaa,pd,\,\text{%
class }4\rangle \;(\text{all }x)  \tag{7.4559}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc-baaa,pa,pb-xbaaa,pc-\omega
baaa,pd,\,\text{class }4\rangle \;(\text{all }x,\;p=1\func{mod}3) 
\tag{7.4560}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc-baaa,pa,pb-xbaaa,pc-\omega
^{2}baaa,pd,\,\text{class }4\rangle \;(\text{all }x,\;p=1\func{mod}3) 
\tag{7.4561}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc-baaa,pa,pb-baaa,pc,pd,\,%
\text{class }4\rangle  \tag{7.4562}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc-baaa,pa,pb-\omega
baaa,pc,pd,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4563}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc-baaa,pa,pb-\omega
^{2}baaa,pc,pd,\,\text{class }4\rangle \;(p=1\func{mod}3)  \tag{7.4564}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc-baaa,pa,pb,pc,pd,\,\text{%
class }4\rangle  \tag{7.4565}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc-baaa,pa-baaa,pb,pc,pd,\,%
\text{class }4\rangle  \tag{7.4566}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc-baaa,pa-\omega
baaa,pb,pc,pd,\,\text{class }4\rangle  \tag{7.4567}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc-baaa,pa-\omega
^{2}baaa,pb,pc,pd,\,\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.4568}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ca,cb-baa,bab,da,db-baaa,dc-baaa,pa-\omega
^{3}baaa,pb,pc,pd,\,\text{class }4\rangle \;(p=1\func{mod}4)  \tag{7.4569}
\end{equation}

\subsection{Descendants of 6.9}

7 algebras

\begin{equation}
\langle a,b,c,d\,|\,ba,ca,cb,da,db,dc,p^2b,pc,pd,\,\text{class }3\rangle 
\tag{7.4570}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ba-p^2a,ca,cb,da,db,dc,p^2b,pc,pd,\,\text{class }3\rangle
\tag{7.4571}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ba,ca-p^2a,cb,da,db,dc,p^2b,pc,pd,\,\text{class }3\rangle
\tag{7.4572}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ba,ca,cb-p^2a,da,db,dc,p^2b,pc,pd,\,\text{class }3\rangle
\tag{7.4573}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ba,ca,cb-p^2a,da-p^2a,db,dc,p^2b,pc,pd,\,\text{class }%
3\rangle  \tag{7.4574}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ba,ca,cb,da,db,dc-p^2a,p^2b,pc,pd,\,\text{class }3\rangle
\tag{7.4575}
\end{equation}

\begin{equation}
\langle a,b,c,d\,|\,ba-p^2a,ca,cb,da,db,dc-p^2a,p^2b,pc,pd,\,\text{class }%
3\rangle  \tag{7.4576}
\end{equation}

\subsection{Descendants of 5.1}

178 algebras if $p=3$, and $p^{2}+15p+125$ otherwise

\begin{equation}
\langle a,b,c,d,e\,|\,ba,ca,da,ea,cb,db,eb,dc,ec,ed,pc,pd,pe,\,\text{class }%
2\rangle  \tag{7.4577}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,da,ea,cb,db,eb,dc,ec,ed,pb,pc,pd,pe,\,\text{class }%
2\rangle  \tag{7.4578}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,da,ea,cb,db,eb,dc,ec,ed,pa,pb,pd,pe,\,\text{class }%
2\rangle  \tag{7.4579}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,da,ea,cb,db,eb,dc,ec,ed,pa,pb,pd-ba,pe,\,\text{%
class }2\rangle  \tag{7.4580}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,da,ea,cb,db,eb,dc,ec,ed,pa-ba,pb,pd,pe,\,\text{%
class }2\rangle  \tag{7.4581}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,da,ea,cb,db,eb,dc,ec,ed,pb,pc-ba,pd,pe,\,\text{%
class }2\rangle  \tag{7.4582}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,da,ea,cb,db,eb,dc,ec,ed,pb-ba,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4583}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,da,ea,cb,db,eb,dc-ba,ec,ed,pb,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4584}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,da,ea,cb,db,eb,dc-ba,ec,ed,pa,pb,pc,pd,\,\text{%
class }2\rangle  \tag{7.4585}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,da,ea,cb,db,eb,dc-ba,ec,ed,pa-ba,pb,pc,pd,\,\text{%
class }2\rangle  \tag{7.4586}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,da,ea,cb,db,eb,dc-ba,ec,ed,pb,pc,pd,pe-ba,\,\text{%
class }2\rangle  \tag{7.4587}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,da,ea,cb,db,eb,dc-ba,ec,ed,pb-ba,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4588}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,da,ea,cb,db,eb,dc-ba,ec,ed,pb,pc-ba,pd,pe,\,\text{%
class }2\rangle  \tag{7.4589}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed,pa,pb,pc,pd,pe,\,\text{class }%
2\rangle  \tag{7.4590}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ba,eb,dc,ec,ed,pa,pb,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4591}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed,pa,pb,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4592}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc-\omega ba,ec,ed,pa,pb,pc,pd,pe,\,%
\text{class }2\rangle  \tag{7.4593}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4594}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4595}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db,dc,ea,eb,ec,ed,pa-ba,pb,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4596}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db,dc,ea,eb,ec,ed,pa,pb-ba,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4597}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db,dc,ea,eb,ec,ed,pa,pb-ca,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4598}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db,dc,ea,eb,ec,ed,pa,pb,pc,pd-ca,pe,\,\text{%
class }2\rangle  \tag{7.4599}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa-ba,pb,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4600}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa,pb-ba,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4601}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa-ba,pb-ba,pc,pd,pe,\,%
\text{class }2\rangle  \tag{7.4602}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa-ba-ca,pb,pc,pd,pe,\,%
\text{class }2\rangle  \tag{7.4603}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa-ba-ca,pb-ba-ca,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4604}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa,pb,pc,pd,pe-ba,\,\text{%
class }2\rangle  \tag{7.4605}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa,pb,pc,pd,pe-ba-ca,\,%
\text{class }2\rangle  \tag{7.4606}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc,ea,eb,ec,ed,pa-ba,pb,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4607}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc,ea,eb,ec,ed,pa,pb,pc-ba,pd,pe,\,\text{%
class }2\rangle  \tag{7.4608}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc,ea,eb,ec,ed,pa-ca,pb,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4609}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc,ea,eb,ec,ed,pa,pb,pc-ca,pd,pe,\,\text{%
class }2\rangle  \tag{7.4610}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc,ea,eb,ec,ed,pa,pb,pc,pd,pe-ba,\,\text{%
class }2\rangle  \tag{7.4611}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc,ea,eb,ec,ed,pa,pb,pc,pd,pe-ca,\,\text{%
class }2\rangle  \tag{7.4612}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc-\omega
ba,ea,eb,ec,ed,pa-ba,pb,pc,pd,pe,\,\text{class }2\rangle  \tag{7.4613}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc-\omega
ba,ea,eb,ec,ed,pa,pb,pc,pd,pe-ba,\,\text{class }2\rangle  \tag{7.4614}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb-ba,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4615}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb-ca,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4616}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb-ca,pc-ca,pd,pe,\,%
\text{class }2\rangle  \tag{7.4617}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb,pc,pd-ba,pe,\,\text{%
class }2\rangle  \tag{7.4618}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb,pc-ba,pd-ba,pe,\,%
\text{class }2\rangle  \tag{7.4619}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb,pc,pd-ca,pe,\,\text{%
class }2\rangle  \tag{7.4620}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb,pc-ca,pd-ca,pe,\,%
\text{class }2\rangle  \tag{7.4621}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb,pc-ba,pd,pe,\,\text{%
class }2\rangle  \tag{7.4622}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb,pc-ca,pd,pe,\,\text{%
class }2\rangle  \tag{7.4623}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa-ba,pb,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4624}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa-ca,pb,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4625}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb,pc,pd,pe-ba,\,%
\text{class }2\rangle  \tag{7.4626}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb,pc-ba,pd,pe-ba,\,%
\text{class }2\rangle  \tag{7.4627}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb,pc-\omega
ba,pd,pe-ba,\,\text{class }2\rangle  \tag{7.4628}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb-ba,pc,pd,pe,\,%
\text{class }2\rangle  \tag{7.4629}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb,pc-ba,pd,pe,\,%
\text{class }2\rangle  \tag{7.4630}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa-ba,pb,pc,pd,pe,\,%
\text{class }2\rangle  \tag{7.4631}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb,pc,pd-ba,pe,\,%
\text{class }2\rangle  \tag{7.4632}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed,pa,pb,pc,pd-ba,pe-ca,\,\text{%
class }2\rangle  \tag{7.4633}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db,dc,ea,eb,ec,ed,pa-ca,pb-ba,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4634}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db,dc,ea,eb,ec,ed,pa-ba,pb-ca,pc,pd,pe,\,\text{%
class }2\rangle  \tag{7.4635}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db,dc,ea,eb,ec,ed,pa,pb-ba,pc-xca,pd,pe,\,\text{%
class }2\rangle \;(x \neq 0, x \sim x^{-1})  \tag{7.4636}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db,dc,ea,eb,ec,ed,pa,pb-ba-ca,pc-ca,pd,pe,\,%
\text{class }2\rangle  \tag{7.4637}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db,dc,ea,eb,ec,ed,pa,pb-\omega ca,pc-ba,pd,pe,\,%
\text{class }2\rangle  \tag{7.4638}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db,dc,ea,eb,ec,ed,pa,pb-xca,pc-ba-ca,pd,pe,\,%
\text{class }2\rangle \;(\text{all }x\text{ with }1+4x\text{ not a square }%
\func{\func{mod}}p)  \tag{7.4639}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db,dc,ea,eb,ec,ed,pa-ba,pb,pc,pd-ca,pe,\,\text{%
class }2\rangle  \tag{7.4640}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db,dc,ea,eb,ec,ed,pa,pb-ba,pc,pd-ca,pe,\,\text{%
class }2\rangle  \tag{7.4641}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db,dc,ea,eb,ec,ed,pa,pb,pc-ba,pd-ca,pe,\,\text{%
class }2\rangle  \tag{7.4642}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa-ba,pb-ba,pc-ca,pd-ca,pe,\,\text{%
class }2\rangle  \tag{7.4643}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa-ba,pb-ba,pc,pd-ca,pe,\,%
\text{class }2\rangle  \tag{7.4644}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa-ba+ca,pb-ba,pc,pd-ca,pe,\,\text{%
class }2\rangle  \tag{7.4645}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa+ca,pb-ba,pc,pd-ca,pe,\,%
\text{class }2\rangle  \tag{7.4646}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa,pb-ba,pc,pd-ca,pe,\,%
\text{class }2\rangle  \tag{7.4647}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa-ba-ca,pb-ba-xca,pc,pd,pe,\,\text{%
class }2\rangle \;(x \neq 0,1)  \tag{7.4648}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa-ba-ca,pb-ba,pc,pd,pe,\,%
\text{class }2\rangle  \tag{7.4649}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa-ba-ca,pb-ca,pc,pd,pe,\,%
\text{class }2\rangle  \tag{7.4650}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa-ba,pb-ca,pc,pd,pe,\,%
\text{class }2\rangle  \tag{7.4651}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa-ca,pb-ba,pc,pd,pe,\,%
\text{class }2\rangle  \tag{7.4652}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa-ca,pb,pc,pd,pe-ba,\,%
\text{class }2\rangle  \tag{7.4653}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa+ca,pb,pc,pd-ca,pe-ba,\,%
\text{class }2\rangle  \tag{7.4654}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa,pb,pc,pd-ca,pe-ba,\,%
\text{class }2\rangle  \tag{7.4655}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa-ca,pb,pc,pd,pe-ba-ca,\,%
\text{class }2\rangle  \tag{7.4656}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ba,dc,ea,eb,ec,ed,pa,pb,pc,pd-ca,pe-ba-ca,\,%
\text{class }2\rangle  \tag{7.4657}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc,ea,eb,ec,ed,pa-ba,pb-ca,pc,pd,pe,\,%
\text{class }2\rangle  \tag{7.4658}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc,ea,eb,ec,ed,pa-ca,pb,pc-ba,pd,pe,\,%
\text{class }2\rangle  \tag{7.4659}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc,ea,eb,ec,ed,pa,pb-ca,pc-ba,pd,pe,\,%
\text{class }2\rangle  \tag{7.4660}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc,ea,eb,ec,ed,pa,pb-\omega
ca,pc-ba,pd,pe,\,\text{class }2\rangle  \tag{7.4661}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc,ea,eb,ec,ed,pa-ba,pb,pc-ca,pd,pe,\,%
\text{class }2\rangle  \tag{7.4662}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc,ea,eb,ec,ed,pa,pb-xba,pc-ca,pd,pe,\,%
\text{class }2\rangle \;(x \neq 0)  \tag{7.4663}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc,ea,eb,ec,ed,pa,pb,pc-ba,pd-ca,pe,\,%
\text{class }2\rangle  \tag{7.4664}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc,ea,eb,ec,ed,pa-ca,pb,pc,pd,pe-ba,\,%
\text{class }2\rangle  \tag{7.4665}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc,ea,eb,ec,ed,pa,pb,pc-ca,pd,pe-ba,\,%
\text{class }2\rangle  \tag{7.4666}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc,ea,eb,ec,ed,pa-ba,pb,pc,pd,pe-ca,\,%
\text{class }2\rangle  \tag{7.4667}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc,ea,eb,ec,ed,pa,pb,pc-ba,pd,pe-ca,\,%
\text{class }2\rangle  \tag{7.4668}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc-\omega
ba,ea,eb,ec,ed,pa,pb,pc-ba,pd-ca,pe,\,\text{class }2\rangle  \tag{7.4669}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc-\omega
ba,ea,eb,ec,ed,pa,pb-ba,pc-xba-yca,pd,pe,\,\text{class }2\rangle 
\tag{7.4670}
\end{equation}%
In presentation 7.4670 we have two parameters $x,y$ with $y\neq 0$.
Parameter pairs $(x,y)$ and $(z,t)$ give isomorphic algebras if and only if%
\[
\left( 
\begin{array}{ll}
1 & 0 \\ 
z & t%
\end{array}%
\right) =\left( 
\begin{array}{ll}
\mu & \nu \\ 
\omega \nu & \mu%
\end{array}%
\right) \left( 
\begin{array}{ll}
1 & 0 \\ 
x & y%
\end{array}%
\right) \left( 
\begin{array}{ll}
\mu +\nu x & \nu y \\ 
\omega \nu y & \mu +\nu x%
\end{array}%
\right) ^{-1} 
\]%
for some non-singular matrix $\left( 
\begin{array}{ll}
\mu & \nu \\ 
\omega \nu & \mu%
\end{array}%
\right) $.

\begin{equation}
\langle a,b,c,d,e\,|\,cb,da,db-ca,dc-\omega
ba,ea,eb,ec,ed,pa-ca,pb,pc,pd,pe-ba,\,\text{class }2\rangle  \tag{7.4671}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb,pc,pd-ba,pe-ca,\,%
\text{class }2\rangle  \tag{7.4672}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb,pc-ba,pd-ba,pe-ca,\,%
\text{class }2\rangle  \tag{7.4673}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb,pc-ca,pd-ba,pe-ca,\,%
\text{class }2\rangle  \tag{7.4674}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa-ca,pb,pc,pd-ba,pe,\,%
\text{class }2\rangle  \tag{7.4675}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa-ba,pb,pc,pd-ca,pe,\,%
\text{class }2\rangle  \tag{7.4676}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa-ca,pb,pc-ba,pd-ba,pe,\,%
\text{class }2\rangle  \tag{7.4677}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb,pc-ba,pd-ca,pe,\,%
\text{class }2\rangle  \tag{7.4678}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa-ba,pb,pc-ca,pd-ca,pe,\,%
\text{class }2\rangle  \tag{7.4679}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb,pc-ca,pd-ba,pe,\,%
\text{class }2\rangle  \tag{7.4680}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb,pc-ca,pd-ba-ca,pe,\,%
\text{class }2\rangle  \tag{7.4681}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa-ca,pb,pc-ba,pd,pe,\,%
\text{class }2\rangle  \tag{7.4682}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa-ba,pb,pc-ca,pd,pe,\,%
\text{class }2\rangle  \tag{7.4683}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb-ba,pc-xca,pd,pe,\,%
\text{class }2\rangle \;(x \neq 0)  \tag{7.4684}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb-ba,pc-xca,pd-ca,pe,\,%
\text{class }2\rangle \;(x \neq 0)  \tag{7.4685}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb-ba,pc,pd-ca,pe,\,%
\text{class }2\rangle  \tag{7.4686}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa-ca,pb-ba,pc,pd,pe,\,%
\text{class }2\rangle  \tag{7.4687}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa-ca,pb-ba,pc,pd-ca,pe,\,%
\text{class }2\rangle  \tag{7.4688}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb-ba,pc-ba-ca,pd,pe,\,%
\text{class }2\rangle  \tag{7.4689}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb-ba,pc-ba-ca,pd-ca,pe,\,\text{%
class }2\rangle  \tag{7.4690}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa-ba,pb-ca,pc,pd,pe,\,%
\text{class }2\rangle  \tag{7.4691}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb-ca,pc,pd-ba,pe,\,%
\text{class }2\rangle  \tag{7.4692}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa-ba,pb-ca,pc,pd-ba,pe,\,%
\text{class }2\rangle  \tag{7.4693}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa-ba,pb-ca,pc-ca,pd,pe,\,%
\text{class }2\rangle  \tag{7.4694}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb-ca,pc-ca,pd-ba,pe,\,%
\text{class }2\rangle  \tag{7.4695}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa-ba,pb-ca,pc-ca,pd-ba,pe,\,\text{%
class }2\rangle  \tag{7.4696}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb-ca,pc-ba,pd,pe,\,%
\text{class }2\rangle  \tag{7.4697}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb-ca,pc-ba,pd-ba,pe,\,%
\text{class }2\rangle  \tag{7.4698}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb-ca,pc-\omega
ba,pd,pe,\,\text{class }2\rangle  \tag{7.4699}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb-ca,pc-\omega
ba,pd-ba,pe,\,\text{class }2\rangle  \tag{7.4700}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb-ca,pc-xba-ca,pd,pe,\,%
\text{class }2\rangle \;(x \neq 0)  \tag{7.4701}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,da,ea,cb,db,eb,dc,ec,ed-ba,pa,pb-ca,pc-xba-ca,pd-ba,pe,\,\text{%
class }2\rangle \;(x \neq 0)  \tag{7.4702}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa-ba,pb,pc,pd-xca,pe,\,%
\text{class }2\rangle \;(x \neq 0, x \sim x^{-1})  \tag{7.4703}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa-ba-ca,pb,pc,pd-ca,pe,\,\text{%
class }2\rangle  \tag{7.4704}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa-\omega
ca,pb,pc,pd-ba,pe,\,\text{class }2\rangle  \tag{7.4705}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa-xca,pb,pc,pd-ba-ca,pe,\,\text{%
class }2\rangle \;(\text{all }x\text{ with }1+4x\text{ not a square }\func{%
\func{mod}}p)  \tag{7.4706}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa-ca,pb,pc,pd,pe-ba,\,%
\text{class }2\rangle  \tag{7.4707}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb,pc,pd-ca,pe-ba,\,%
\text{class }2\rangle  \tag{7.4708}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa-ca,pb,pc,pd-ca,pe-ba,\,\text{%
class }2\rangle  \tag{7.4709}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa-ca,pb,pc-ba,pd,pe-ba,\,\text{%
class }2\rangle  \tag{7.4710}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa-xca,pb,pc-ba,pd-ca,pe-ba,\,%
\text{class }2\rangle \;(x \sim -x)  \tag{7.4711}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa-ca,pb,pc-\omega
ba,pd,pe-ba,\,\text{class }2\rangle  \tag{7.4712}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa-xca,pb,pc-\omega
ba,pd-ca,pe-ba,\,\text{class }2\rangle \;(x\sim -x)  \tag{7.4713}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb-ba,pc,pd-ca,pe,\,%
\text{class }2\rangle  \tag{7.4714}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa-ca,pb-ba,pc,pd,pe,\,%
\text{class }2\rangle  \tag{7.4715}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa-ca,pb-ba,pc,pd-ca,pe,\,\text{%
class }2\rangle  \tag{7.4716}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb,pc-ba,pd-ca,pe,\,%
\text{class }2\rangle  \tag{7.4717}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa-ca,pb,pc-ba,pd,pe,\,%
\text{class }2\rangle  \tag{7.4718}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb,pc-xba-ca,pd,pe-ba-yca,\,%
\text{class }2\rangle \;(\text{all }x,\,y\text{ with }xy\neq 1,(x,\,y)\sim
(y,x))  \tag{7.4719}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb,pc-xba,pd,pe-ba-ca,\,\text{%
class }2\rangle \;(x \neq 0)  \tag{7.4720}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb,pc-ba,pd,pe-ca,\,%
\text{class }2\rangle  \tag{7.4721}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb,pc-ba,pd,pe-\omega
ca,\,\text{class }2\rangle \;(p=1\func{mod}3)  \tag{7.4722}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|%
\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb-ca,pc-tba-xca,pd,pe-yba-zca,\,\text{%
class }2\rangle  \tag{7.4723}
\end{equation}%
We consider the orbits of matrices 
\[
\left( 
\begin{array}{ll}
u & v \\ 
t & x \\ 
y & z%
\end{array}%
\right) 
\]%
where $(tz-xy)^{2}-(ux-vt)(uz-vy)$ is not a square, under the action of
non-singular matrices $\left( 
\begin{array}{ll}
\alpha & \varepsilon \\ 
\gamma & \eta%
\end{array}%
\right) $ given by 
\[
\left( 
\begin{array}{ll}
u & v \\ 
t & x \\ 
y & z%
\end{array}%
\right) \rightarrow (\alpha \eta -\gamma \varepsilon )^{-2}\left( 
\begin{array}{lll}
(\alpha \eta +\gamma \varepsilon ) & 2\gamma \eta & -2\alpha \varepsilon \\ 
\varepsilon \eta & \eta ^{2} & -\varepsilon ^{2} \\ 
-\alpha \gamma & -\gamma ^{2} & \alpha ^{2}%
\end{array}%
\right) \left( 
\begin{array}{ll}
u & v \\ 
t & x \\ 
y & z%
\end{array}%
\right) \left( 
\begin{array}{ll}
\eta & -\gamma \\ 
-\varepsilon & \alpha%
\end{array}%
\right) . 
\]%
Each orbit contains a matrix with $u=0$ and $v=1$, and we pick one matrix of
this form to obtain a representative set of parameters $t,x,y,z$ for
presentation 7.4723. The number of orbits is $4$ when $p=3$, $(p^{2}-1)/2$
when $p=1\func{mod}3$, and $(p^{2}+1)/2$ when $p=2\func{mod}3$.

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb-ca,pc,pd,pe-xba,\,%
\text{class }2\rangle \;(x \neq 0)  \tag{7.4724}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb-ca,pc,pd-ca,pe-xba,\,\text{%
class }2\rangle \;(x \neq 0)  \tag{7.4725}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb-ca,pc,pd,pe+ba-ca,\,\text{%
class }2\rangle  \tag{7.4726}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb-ca,pc,pd-ca,pe+ba-ca,\,%
\text{class }2\rangle  \tag{7.4727}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb-ca,pc-ba,pd,pe,\,%
\text{class }2\rangle  \tag{7.4728}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb-ca,pc-ba,pd-ba,pe,\,\text{%
class }2\rangle  \tag{7.4729}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb-ca,pc-\omega
ba,pd,pe,\,\text{class }2\rangle  \tag{7.4730}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb-ca,pc-\omega
ba,pd-ba,pe,\,\text{class }2\rangle  \tag{7.4731}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb-ca,pc-xba-ca,pd,pe,\,\text{%
class }2\rangle \;(x \neq 0)  \tag{7.4732}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,da,ea,cb,db-ca,eb,dc,ec,ed-ba,pa,pb-ca,pc-xba-ca,pd-ba,pe,\,%
\text{class }2\rangle \;(x \neq 0)  \tag{7.4733}
\end{equation}

\subsection{Descendants of 6.2}

5 algebras

\begin{equation}
\langle a,b,c,d,e\,|\,ba,ca,da,ea,cb,db,eb,dc,ec,ed,pb,pc,pd,pe,\,\text{%
class }3\rangle  \tag{7.4734}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ba-p^2a,ca,da,ea,cb,db,eb,dc,ec,ed,pb,pc,pd,pe,\,\text{%
class }3\rangle  \tag{7.4735}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ba,ca,da,ea,cb-p^2a,db,eb,dc,ec,ed,pb,pc,pd,pe,\,\text{%
class }3\rangle  \tag{7.4736}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ba,ca,da-p^2a,ea,cb-p^2a,db,eb,dc,ec,ed,pb,pc,pd,pe,\,%
\text{class }3\rangle  \tag{7.4737}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ba,ca,da,ea,cb-p^2a,db,eb,dc,ec,ed-p^2a,pb,pc,pd,pe,\,%
\text{class }3\rangle  \tag{7.4738}
\end{equation}

\subsection{Descendants of 6.3}

25 algebras

\begin{equation}
\langle a,b,c,d,e\,|\,ca,cb,bab,da,db,dc,ea,eb,ec,ed,pa,pb,pc,pd,pe,\,\text{%
class }3\rangle  \tag{7.4739}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,cb,bab,da,db,dc,ea,eb,ec,ed,pa-baa,pb,pc,pd,pe,\,%
\text{class }3\rangle  \tag{7.4740}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,cb,bab,da,db,dc,ea,eb,ec,ed,pa,pb-baa,pc,pd,pe,\,%
\text{class }3\rangle  \tag{7.4741}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,cb,bab,da,db,dc,ea,eb,ec,ed,pa,pb-\omega
baa,pc,pd,pe,\,\text{class }3\rangle  \tag{7.4742}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,cb,bab,da,db,dc,ea,eb,ec,ed,pa,pb,pc-baa,pd,pe,\,%
\text{class }3\rangle  \tag{7.4743}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,cb-baa,bab,da,db,dc,ea,eb,ec,ed,pa,pb,pc,pd,pe,\,%
\text{class }3\rangle  \tag{7.4744}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,ca,cb-baa,bab,da,db,dc,ea,eb,ec,ed,pa-baa,pb,pc,pd,pe,\,\text{%
class }3\rangle  \tag{7.4745}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,ca,cb-baa,bab,da,db,dc,ea,eb,ec,ed,pa,pb-baa,pc,pd,pe,\,\text{%
class }3\rangle  \tag{7.4746}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,ca,cb-baa,bab,da,db,dc,ea,eb,ec,ed,pa,pb-\omega
baa,pc,pd,pe,\,\text{class }3\rangle  \tag{7.4747}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,ca,cb-baa,bab,da,db,dc,ea,eb,ec,ed,pa,pb,pc-baa,pd,pe,\,\text{%
class }3\rangle  \tag{7.4748}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,ca,cb-baa,bab,da,db,dc,ea,eb,ec,ed,pa,pb,pc,pd-baa,pe,\,\text{%
class }3\rangle  \tag{7.4749}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,bab,ca,cb,da,db,dc-baa,ea,eb,ec,ed,pa,pb,pc,pd,pe,\,%
\text{class }3\rangle  \tag{7.4750}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,bab,ca,cb,da,db,dc-baa,ea,eb,ec,ed,pa-baa,pb,pc,pd,pe,\,\text{%
class }3\rangle  \tag{7.4751}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,bab,ca,cb,da,db,dc-baa,ea,eb,ec,ed,pa,pb-baa,pc,pd,pe,\,\text{%
class }3\rangle  \tag{7.4752}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,bab,ca,cb,da,db,dc-baa,ea,eb,ec,ed,pa,pb-\omega
baa,pc,pd,pe,\,\text{class }3\rangle  \tag{7.4753}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,bab,ca,cb,da,db,dc-baa,ea,eb,ec,ed,pa,pb,pc-baa,pd,pe,\,\text{%
class }3\rangle  \tag{7.4754}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,bab,ca,cb,da,db,dc-baa,ea,eb,ec,ed,pa,pb-baa,pc-baa,pd,pe,\,%
\text{class }3\rangle  \tag{7.4755}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,bab,ca,cb,da,db,dc-baa,ea,eb,ec,ed,pa,pb-\omega
baa,pc-baa,pd,pe,\,\text{class }3\rangle  \tag{7.4756}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,bab,ca,cb,da,db,dc-baa,ea,eb,ec,ed,pa,pb,pc,pd,pe-baa,\,\text{%
class }3\rangle  \tag{7.4757}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,bab,ca,cb,da,db,dc-baa,ea,eb-baa,ec,ed,pa,pb,pc,pd,pe-baa,\,%
\text{class }3\rangle  \tag{7.4758}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,bab,ca,cb,da,db,dc-baa,ea,eb-baa,ec,ed,pa,pb,pc-baa,pd,pe,\,%
\text{class }3\rangle  \tag{7.4759}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,bab,ca,cb,da,db,dc-baa,ea,eb-baa,ec,ed,pa,pb-baa,pc,pd,pe,\,%
\text{class }3\rangle  \tag{7.4760}
\end{equation}

\begin{equation}
\langle a,b,c,d,e\,|\,bab,ca,cb,da,db,dc-baa,ea,eb-baa,ec,ed,pa,pb-\omega
baa,pc,pd,pe,\,\text{class }3\rangle  \tag{7.4761}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,bab,ca,cb,da,db,dc-baa,ea,eb-baa,ec,ed,pa,pb,pc,pd,pe,\,\text{%
class }3\rangle  \tag{7.4762}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e\,|\,bab,ca,cb,da,db,dc-baa,ea,eb-baa,ec,ed,pa-baa,pb,pc,pd,pe,\,%
\text{class }3\rangle  \tag{7.4763}
\end{equation}

\subsection{Descendants of 6.1}

9 algebras

\begin{equation}
\langle
a,b,c,d,e,f\,|\,ba,ca,da,ea,fa,cb,db,eb,fb,dc,ec,fc,ed,fd,fe,pb,pc,pd,pe,pf,%
\,\text{class }2\rangle  \tag{7.4764}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e,f\,|\,ca,da,ea,fa,cb,db,eb,fb,dc,ec,fc,ed,fd,fe,pa,pb,pc,pd,pe,pf,%
\,\text{class }2\rangle  \tag{7.4765}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e,f\,|%
\,ca,da,ea,fa,cb,db,eb,fb,dc,ec,fc,ed,fd,fe,pa-ba,pb,pc,pd,pe,pf,\,\text{%
class }2\rangle  \tag{7.4766}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e,f\,|%
\,ca,da,ea,fa,cb,db,eb,fb,dc,ec,fc,ed,fd,fe,pa,pb,pc-ba,pd,pe,pf,\,\text{%
class }2\rangle  \tag{7.4767}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e,f\,|%
\,ca,da,ea,fa,cb,db,eb,fb,dc-ba,ec,fc,ed,fd,fe,pa,pb,pc,pd,pe,pf,\,\text{%
class }2\rangle  \tag{7.4768}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e,f\,|%
\,ca,da,ea,fa,cb,db,eb,fb,dc-ba,ec,fc,ed,fd,fe,pa-ba,pb,pc,pd,pe,pf,\,\text{%
class }2\rangle  \tag{7.4769}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e,f\,|%
\,ca,da,ea,fa,cb,db,eb,fb,dc-ba,ec,fc,ed,fd,fe,pa,pb,pc,pd,pe-ba,pf,\,\text{%
class }2\rangle  \tag{7.4770}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e,f\,|%
\,ca,da,ea,fa,cb,db,eb,fb,dc-ba,ec,fc,ed,fd,fe-ba,pa,pb,pc,pd,pe,pf,\,\text{%
class }2\rangle  \tag{7.4771}
\end{equation}

\begin{equation}
\langle
a,b,c,d,e,f\,|%
\,ca,da,ea,fa,cb,db,eb,fb,dc-ba,ec,fc,ed,fd,fe-ba,pa-ba,pb,pc,pd,pe,pf,\,%
\text{class }2\rangle  \tag{7.4772}
\end{equation}

\bibliographystyle{plain}
\bibliography{acompat,mike}

\end{document}
