  
  [1X2 [33X[0;0YTutorial for the [5XGAP[105X[101X[1X Character Table Library[133X[101X
  
  [33X[0;0YThis  chapter  gives  an overview of the basic functionality provided by the
  [5XGAP[105X  Character  Table Library. The main concepts and interface functions are
  presented  in  the  sections [14X2.1[114X  and  [14X2.2[114X,  Section [14X2.3[114X  shows  a few small
  examples.[133X
  
  
  [1X2.1 [33X[0;0YConcepts used in the [5XGAP[105X[101X[1X Character Table Library[133X[101X
  
  [33X[0;0YThe main idea behind working with the [5XGAP[105X Character Table Library is to deal
  with  character  tables of groups but [13Xwithout[113X having access to these groups.
  This  situation  occurs  for  example  if  one extracts information from the
  printed [5XAtlas[105X of Finite Groups ([CCNPW85]).[133X
  
  [33X[0;0YThis  restriction  means  first  of  all  that  we  need a way to access the
  character  tables,  see  Section [14X2.2[114X for that. Once we have such a character
  table, we can compute all those data about the underlying group [22XG[122X, say, that
  are  determined  by  the character table. Chapter [14X'Reference: Attributes and
  Properties  for  Groups  and  Character  Tables'[114X  lists  such attributes and
  properties.  For  example,  it can be computed from the character table of [22XG[122X
  whether [22XG[122X is solvable or not.[133X
  
  [33X[0;0YQuestions  that  cannot  be answered using only the character table of [22XG[122X can
  perhaps  be treated using additional information. For example, the structure
  of  subgroups of [22XG[122X is in general not determined by the character table of [22XG[122X,
  but  the  character table may yield partial information. Two examples can be
  found in the sections [14X2.3-4[114X and [14X2.3-6[114X.[133X
  
  [33X[0;0YIn the character table context, the role of homomorphisms between two groups
  is  taken  by  [13Xclass  fusions[113X. Monomorphisms correspond to subgroup fusions,
  epimorphisms  correspond  to factor fusions. Given two character tables of a
  group  [22XG[122X  and  a subgroup [22XH[122X of [22XG[122X, one can in general compute only [13Xcandidates[113X
  for  the  class  fusion  of [22XH[122X into [22XG[122X, for example using [2XPossibleClassFusions[102X
  ([14XReference:   PossibleClassFusions[114X).   Note   that  [22XG[122X  may  contain  several
  nonconjugate  subgroups  isomorphic  with  [22XH[122X, which may have different class
  fusions.[133X
  
  [33X[0;0YOne  can  often  reduce  a  question about a group [22XG[122X to a question about its
  maximal subgroups. In the character table context, it is often sufficient to
  know  the  character  table  of  [22XG[122X,  the  character  tables  of  its maximal
  subgroups,  and  their class fusions into [22XG[122X. We are in this situation if the
  attribute [2XMaxes[102X ([14X3.7-1[114X) is set in the character table of [22XG[122X.[133X
  
  [33X[0;0Y[13XSummary:[113X  The  character  theoretic  approach  that  is supported by the [5XGAP[105X
  Character  Table  Library, that is, an approach without explicitly using the
  underlying  groups,  has  the  advantages that it can be used to answer many
  questions,  and  that  these  computations  are  usually  cheap, compared to
  computations with groups. Disadvantages are that this approach is not always
  successful, and that answers are often [21Xnonconstructive[121X in the sense that one
  can show the existence of something without getting one's hands on it.[133X
  
  
  [1X2.2 [33X[0;0YAccessing a Character Table from the Library[133X[101X
  
  [33X[0;0YAs  stated  in Section [14X2.1[114X, we must define how character tables from the [5XGAP[105X
  Character Table Library can be accessed.[133X
  
  
  [1X2.2-1 [33X[0;0YAccessing a Character Table via a name[133X[101X
  
  [33X[0;0YThe most common way to access a character table from the [5XGAP[105X Character Table
  Library  is to call [2XCharacterTable[102X ([14X3.1-2[114X) with argument a string that is an
  [13Xadmissible  name[113X  for  the  character  table.  Typical  admissible names are
  similar to the group names used in the [5XAtlas[105X of Finite Groups [CCNPW85]. One
  of  these  names  is  the  [2XIdentifier[102X  ([14XReference: Identifier (for character
  tables)[114X)  value  of  the  character  table, this name is used by [5XGAP[105X when it
  prints library character tables.[133X
  
  [33X[0;0YFor  example, an admissible name for the character table of an almost simple
  group  is  the  [5XAtlas[105X name, such as [10XA5[110X, [10XM11[110X, or [10XL2(11).2[110X. Other names may be
  admissible,  for  example  [10XS6[110X  is  admissible for the symmetric group on six
  points, which is called [22XA_6.2_1[122X in the [5XAtlas[105X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCharacterTable( "J1" );[127X[104X
    [4X[28XCharacterTable( "J1" )[128X[104X
    [4X[25Xgap>[125X [27XCharacterTable( "L2(11)" );[127X[104X
    [4X[28XCharacterTable( "L2(11)" )[128X[104X
    [4X[25Xgap>[125X [27XCharacterTable( "S5" );[127X[104X
    [4X[28XCharacterTable( "A5.2" )[128X[104X
  [4X[32X[104X
  
  
  [1X2.2-2 [33X[0;0YAccessing a Character Table via properties[133X[101X
  
  [33X[0;0YIf  one  does  not know an admissible name of the character table of a group
  one is interested in, or if one does not know whether ths character table is
  available  at  all,  one can use [2XAllCharacterTableNames[102X ([14X3.1-3[114X) to compute a
  list of identifiers of all available character tables with given properties.
  Analogously,  [2XOneCharacterTableName[102X  ([14X3.1-4[114X) can be used to compute one such
  identifier.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAllCharacterTableNames( Size, 120 );[127X[104X
    [4X[28X[ "2.A5", "2.A6M2", "2xA5", "A5.2", "A6.2_1M3", "D120", "L2(25)M3" ][128X[104X
    [4X[25Xgap>[125X [27XOneCharacterTableName( NrConjugacyClasses, n -> n <= 4 );[127X[104X
    [4X[28X"S3"[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  certain  filters, such as [2XSize[102X ([14XReference: Size[114X) and [2XNrConjugacyClasses[102X
  ([14XReference:  NrConjugacyClasses[114X),  the  computations  are  fast  because the
  values  for  all  library tables are precomputed. See [2XAllCharacterTableNames[102X
  ([14X3.1-3[114X) for an overview of these filters.[133X
  
  [33X[0;0YThe  function  [2XBrowseCTblLibInfo[102X ([14X3.5-2[114X) provides an interactive overview of
  available  character tables, which allows one for example to search also for
  substrings  in  identifiers  of character tables. This function is available
  only if the [5XBrowse[105X package has been loaded.[133X
  
  
  [1X2.2-3 [33X[0;0YAccessing a Character Table via a Table of Marks[133X[101X
  
  [33X[0;0YLet  [22XG[122X  be  a group whose table of marks is available via the [5XTomLib[105X package
  (see  [NMP11]  for how to access tables of marks from this library) then the
  [5XGAP[105X  Character  Table Library contains the character table of [22XG[122X, and one can
  access   this  table  by  using  the  table  of  marks  as  an  argument  of
  [2XCharacterTable[102X ([14X3.2-2[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "M11" );[127X[104X
    [4X[28XTableOfMarks( "M11" )[128X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( tom );[127X[104X
    [4X[28XCharacterTable( "M11" )[128X[104X
  [4X[32X[104X
  
  
  [1X2.2-4 [33X[0;0YAccessing a Character Table relative to another Character Table[133X[101X
  
  [33X[0;0YIf  one  has  already a character table from the [5XGAP[105X Character Table Library
  that  belongs to the group [22XG[122X, say, then names of related tables can be found
  as follows.[133X
  
  [33X[0;0YThe  value  of  the  attribute  [2XMaxes[102X  ([14X3.7-1[114X),  if  known,  is  the list of
  identifiers  of  the character tables of all classes of maximal subgroups of
  [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "M11" );[127X[104X
    [4X[28XCharacterTable( "M11" )[128X[104X
    [4X[25Xgap>[125X [27XHasMaxes( t );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XMaxes( t );[127X[104X
    [4X[28X[ "A6.2_3", "L2(11)", "3^2:Q8.2", "A5.2", "2.S4" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  the  [2XMaxes[102X ([14X3.7-1[114X) value of the character table with identifier [22Xid[122X, say,
  is known then the character table of the groups in the [22Xi[122X-th class of maximal
  subgroups can be accessed via the [21Xrelative name[121X [22Xid[122X[10XM[110X[22Xi[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCharacterTable( "M11M2" );[127X[104X
    [4X[28XCharacterTable( "L2(11)" )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe    value    of    the    attribute    [2XNamesOfFusionSources[102X   ([14XReference:
  NamesOfFusionSources[114X)  is  the list of identifiers of those character tables
  which  store  class  fusions  to  [22XG[122X.  So  these  character  tables belong to
  subgroups of [22XG[122X and groups that have [22XG[122X as a factor group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNamesOfFusionSources( t );[127X[104X
    [4X[28X[ "A5.2", "A6.2_3", "P48/G1/L1/V1/ext2", "P48/G1/L1/V2/ext2", [128X[104X
    [4X[28X  "L2(11)", "2.S4", "3^5:M11", "3^6.M11", "3^2:Q8.2", "M11N2", "5:4", [128X[104X
    [4X[28X  "11:5" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe    value    of    the    attribute    [2XComputedClassFusions[102X   ([14XReference:
  ComputedClassFusions[114X)  is  the list of records whose [10Xname[110X components are the
  identifiers  of those character tables to which class fusions are stored. So
  these character tables belong to overgroups and factor groups of [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList( ComputedClassFusions( t ), r -> r.name );[127X[104X
    [4X[28X[ "A11", "M12", "M23", "HS", "McL", "ON", "3^5:M11", "B" ][128X[104X
  [4X[32X[104X
  
  
  [1X2.2-5 [33X[0;0YDifferent character tables for the same group[133X[101X
  
  [33X[0;0YThe  [5XGAP[105X  Character  Table  Library  may contain several different character
  tables  of  a given group, in the sense that the rows and columns are sorted
  differently.[133X
  
  [33X[0;0YFor  example, the [5XAtlas[105X table of the alternating group [22XA_5[122X is available, and
  since  [22XA_5[122X  is  isomorphic with the groups PSL[22X(2, 4)[122X and PSL[22X(2, 5)[122X, two more
  character  tables  of  [22XA_5[122X  can  be  constructed in a natural way. The three
  tables are of course permutation isomorphic. The first two are sorted in the
  same way, but the rows and columns of the third one are sorted differently.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt1:= CharacterTable( "A5" );;[127X[104X
    [4X[25Xgap>[125X [27Xt2:= CharacterTable( "PSL", 2, 4 );;[127X[104X
    [4X[25Xgap>[125X [27Xt3:= CharacterTable( "PSL", 2, 5 );;[127X[104X
    [4X[25Xgap>[125X [27XTransformingPermutationsCharacterTables( t1, t2 );[127X[104X
    [4X[28Xrec( columns := (), group := Group([ (4,5) ]), rows := () )[128X[104X
    [4X[25Xgap>[125X [27XTransformingPermutationsCharacterTables( t1, t3 );[127X[104X
    [4X[28Xrec( columns := (2,4)(3,5), group := Group([ (2,3) ]), [128X[104X
    [4X[28X  rows := (2,5,3,4) )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAnother  situation  where  several  character  tables for the same group are
  available  is  that  a  group contains several classes of isomorphic maximal
  subgroups such that the class fusions are different.[133X
  
  [33X[0;0YFor  example,  the  Mathieu  group  [22XM_12[122X  contains  two  classes  of maximal
  subgroups of index [22X12[122X, which are isomorphic with [22XM_11[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "M12" );[127X[104X
    [4X[28XCharacterTable( "M12" )[128X[104X
    [4X[25Xgap>[125X [27Xmx:= Maxes( t );[127X[104X
    [4X[28X[ "M11", "M12M2", "A6.2^2", "M12M4", "L2(11)", "3^2.2.S4", "M12M7", [128X[104X
    [4X[28X  "2xS5", "M8.S4", "4^2:D12", "A4xS3" ][128X[104X
    [4X[25Xgap>[125X [27Xs1:= CharacterTable( mx[1] );[127X[104X
    [4X[28XCharacterTable( "M11" )[128X[104X
    [4X[25Xgap>[125X [27Xs2:= CharacterTable( mx[2] );[127X[104X
    [4X[28XCharacterTable( "M12M2" )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  class fusions into [22XM_12[122X are stored on the library tables of the maximal
  subgroups.  The  groups  in  the  first class of [22XM_11[122X type subgroups contain
  elements in the classes [10X4B[110X, [10X6B[110X, and [10X8B[110X of [22XM_12[122X, and the groups in the second
  class contain elements in the classes [10X4A[110X, [10X6A[110X, and [10X8A[110X. Note that according to
  the  [5XAtlas[105X  (see [CCNPW85, p. 33]), the permutation characters of the action
  of [22XM_12[122X on the cosets of [22XM_11[122X type subgroups from the two classes of maximal
  subgroups are [10X1a + 11a[110X and [10X1a + 11b[110X, respectively.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( s1, t );[127X[104X
    [4X[28X[ 1, 3, 4, 7, 8, 10, 12, 12, 15, 14 ][128X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( s2, t );[127X[104X
    [4X[28X[ 1, 3, 4, 6, 8, 10, 11, 11, 14, 15 ][128X[104X
    [4X[25Xgap>[125X [27XDisplay( t );[127X[104X
    [4X[28XM12[128X[104X
    [4X[28X[128X[104X
    [4X[28X      2   6  4  6  1  2  5  5  1  2  1  3  3   1   .   .[128X[104X
    [4X[28X      3   3  1  1  3  2  .  .  .  1  1  .  .   .   .   .[128X[104X
    [4X[28X      5   1  1  .  .  .  .  .  1  .  .  .  .   1   .   .[128X[104X
    [4X[28X     11   1  .  .  .  .  .  .  .  .  .  .  .   .   1   1[128X[104X
    [4X[28X[128X[104X
    [4X[28X         1a 2a 2b 3a 3b 4a 4b 5a 6a 6b 8a 8b 10a 11a 11b[128X[104X
    [4X[28X     2P  1a 1a 1a 3a 3b 2b 2b 5a 3b 3a 4a 4b  5a 11b 11a[128X[104X
    [4X[28X     3P  1a 2a 2b 1a 1a 4a 4b 5a 2a 2b 8a 8b 10a 11a 11b[128X[104X
    [4X[28X     5P  1a 2a 2b 3a 3b 4a 4b 1a 6a 6b 8a 8b  2a 11a 11b[128X[104X
    [4X[28X    11P  1a 2a 2b 3a 3b 4a 4b 5a 6a 6b 8a 8b 10a  1a  1a[128X[104X
    [4X[28X[128X[104X
    [4X[28XX.1       1  1  1  1  1  1  1  1  1  1  1  1   1   1   1[128X[104X
    [4X[28XX.2      11 -1  3  2 -1 -1  3  1 -1  . -1  1  -1   .   .[128X[104X
    [4X[28XX.3      11 -1  3  2 -1  3 -1  1 -1  .  1 -1  -1   .   .[128X[104X
    [4X[28XX.4      16  4  . -2  1  .  .  1  1  .  .  .  -1   A  /A[128X[104X
    [4X[28XX.5      16  4  . -2  1  .  .  1  1  .  .  .  -1  /A   A[128X[104X
    [4X[28XX.6      45  5 -3  .  3  1  1  . -1  . -1 -1   .   1   1[128X[104X
    [4X[28XX.7      54  6  6  .  .  2  2 -1  .  .  .  .   1  -1  -1[128X[104X
    [4X[28XX.8      55 -5  7  1  1 -1 -1  .  1  1 -1 -1   .   .   .[128X[104X
    [4X[28XX.9      55 -5 -1  1  1  3 -1  .  1 -1 -1  1   .   .   .[128X[104X
    [4X[28XX.10     55 -5 -1  1  1 -1  3  .  1 -1  1 -1   .   .   .[128X[104X
    [4X[28XX.11     66  6  2  3  . -2 -2  1  . -1  .  .   1   .   .[128X[104X
    [4X[28XX.12     99 -1  3  .  3 -1 -1 -1 -1  .  1  1  -1   .   .[128X[104X
    [4X[28XX.13    120  . -8  3  .  .  .  .  .  1  .  .   .  -1  -1[128X[104X
    [4X[28XX.14    144  4  .  . -3  .  . -1  1  .  .  .  -1   1   1[128X[104X
    [4X[28XX.15    176 -4  . -4 -1  .  .  1 -1  .  .  .   1   .   .[128X[104X
    [4X[28X[128X[104X
    [4X[28XA = E(11)+E(11)^3+E(11)^4+E(11)^5+E(11)^9[128X[104X
    [4X[28X  = (-1+Sqrt(-11))/2 = b11[128X[104X
  [4X[32X[104X
  
  [33X[0;0YPermutation  equivalent  library  tables  are  related to each other. In the
  above  example,  the  table [10Xs2[110X is a [13Xduplicate[113X of [10Xs1[110X, and there are functions
  for making the relations explicit.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsDuplicateTable( s2 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIdentifierOfMainTable( s2 );[127X[104X
    [4X[28X"M11"[128X[104X
    [4X[25Xgap>[125X [27XIdentifiersOfDuplicateTables( s1 );[127X[104X
    [4X[28X[ "HSM9", "M12M2", "ONM11" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSee Section [14X3.6[114X for details about duplicate character tables.[133X
  
  
  [1X2.3 [33X[0;0YExamples of Using the [5XGAP[105X[101X[1X Character Table Library[133X[101X
  
  [33X[0;0YThe   sections   [14X2.3-1[114X,   [14X2.3-2[114X,   and   [14X2.3-3[114X   show   how   the   function
  [2XAllCharacterTableNames[102X  ([14X3.1-3[114X)  can  be used to search for character tables
  with  certain  properties.  The [5XGAP[105X Character Table Library serves as a tool
  for finding and checking conjectures in these examples.[133X
  
  [33X[0;0YIn Section [14X2.3-6[114X, a question about a subgroup of the sporadic simple Fischer
  group  [22XG  =  Fi_23[122X  is  answered  using  only  character tables from the [5XGAP[105X
  Character Table Library.[133X
  
  [33X[0;0YMore examples can be found in [BGLMN10], [Brea], [Bred], [Brec], [Bree].[133X
  
  
  [1X2.3-1 [33X[0;0YExample: Ambivalent Simple Groups[133X[101X
  
  [33X[0;0YA  group  [22XG[122X  is called [13Xambivalent[113X if each element in [22XG[122X is [22XG[122X-conjugate to its
  inverse.   Equivalently,   [22XG[122X   is  ambivalent  if  all  its  characters  are
  real-valued.   We   are   interested  in  simple  ambivalent  groups.  Since
  ambivalence   is  invariant  under  permutation  equivalence,  we  may  omit
  duplicate character tables.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xisambivalent:= tbl -> PowerMap( tbl, -1 )[127X[104X
    [4X[25X>[125X [27X                           = [ 1 .. NrConjugacyClasses( tbl ) ];;[127X[104X
    [4X[25Xgap>[125X [27XAllCharacterTableNames( IsSimple, true, IsDuplicateTable, false,[127X[104X
    [4X[25X>[125X [27X                           isambivalent, true );[127X[104X
    [4X[28X[ "3D4(2)", "A10", "A14", "A5", "A6", "J1", "J2", "L2(101)", [128X[104X
    [4X[28X  "L2(109)", "L2(113)", "L2(121)", "L2(125)", "L2(13)", "L2(16)", [128X[104X
    [4X[28X  "L2(17)", "L2(25)", "L2(29)", "L2(32)", "L2(37)", "L2(41)", [128X[104X
    [4X[28X  "L2(49)", "L2(53)", "L2(61)", "L2(64)", "L2(73)", "L2(8)", [128X[104X
    [4X[28X  "L2(81)", "L2(89)", "L2(97)", "O7(5)", "O8+(2)", "O8+(3)", [128X[104X
    [4X[28X  "O8+(7)", "O8-(2)", "O8-(3)", "O9(3)", "S10(2)", "S12(2)", "S4(4)", [128X[104X
    [4X[28X  "S4(5)", "S4(8)", "S4(9)", "S6(2)", "S6(4)", "S6(5)", "S8(2)" ][128X[104X
  [4X[32X[104X
  
  
  [1X2.3-2 [33X[0;0YExample: Simple [22Xp[122X[101X[1X-pure Groups[133X[101X
  
  [33X[0;0YA  group  [22XG[122X  is  called [13X[22Xp[122X-pure[113X for a prime integer [22Xp[122X that divides [22X|G|[122X if the
  centralizer   orders   of   nonidentity   [22Xp[122X-elements   in  [22XG[122X  are  [22Xp[122X-powers.
  Equivalently,  [22XG[122X  is  [22Xp[122X-pure if [22Xp[122X divides [22X|G|[122X and each element in [22XG[122X of order
  divisible  by [22Xp[122X is a [22Xp[122X-element. (This property was studied by L. Héthelyi in
  2002.)[133X
  
  [33X[0;0YWe are interested in small nonabelian simple [22Xp[122X-pure groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xisppure:= function( p )[127X[104X
    [4X[25X>[125X [27X     return tbl -> Size( tbl ) mod p = 0 and[127X[104X
    [4X[25X>[125X [27X       ForAll( OrdersClassRepresentatives( tbl ),[127X[104X
    [4X[25X>[125X [27X               n -> n mod p <> 0 or IsPrimePowerInt( n ) );[127X[104X
    [4X[25X>[125X [27X   end;;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in [ 2, 3, 5, 7, 11, 13 ] do[127X[104X
    [4X[25X>[125X [27X     Print( i, "\n",[127X[104X
    [4X[25X>[125X [27X       AllCharacterTableNames( IsSimple, true, IsAbelian, false,[127X[104X
    [4X[25X>[125X [27X           IsDuplicateTable, false, isppure( i ), true ),[127X[104X
    [4X[25X>[125X [27X       "\n" );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[28X2[128X[104X
    [4X[28X[ "A5", "A6", "L2(16)", "L2(17)", "L2(31)", "L2(32)", "L2(64)", [128X[104X
    [4X[28X  "L2(8)", "L3(2)", "L3(4)", "Sz(32)", "Sz(8)" ][128X[104X
    [4X[28X3[128X[104X
    [4X[28X[ "A5", "A6", "L2(17)", "L2(19)", "L2(27)", "L2(53)", "L2(8)", [128X[104X
    [4X[28X  "L2(81)", "L3(2)", "L3(4)" ][128X[104X
    [4X[28X5[128X[104X
    [4X[28X[ "A5", "A6", "A7", "L2(11)", "L2(125)", "L2(25)", "L2(49)", "L3(4)", [128X[104X
    [4X[28X  "M11", "M22", "S4(7)", "Sz(32)", "Sz(8)", "U4(2)", "U4(3)" ][128X[104X
    [4X[28X7[128X[104X
    [4X[28X[ "A7", "A8", "A9", "G2(3)", "HS", "J1", "J2", "L2(13)", "L2(49)", [128X[104X
    [4X[28X  "L2(8)", "L2(97)", "L3(2)", "L3(4)", "M22", "O8+(2)", "S6(2)", [128X[104X
    [4X[28X  "Sz(8)", "U3(3)", "U3(5)", "U4(3)", "U6(2)" ][128X[104X
    [4X[28X11[128X[104X
    [4X[28X[ "A11", "A12", "A13", "Co2", "HS", "J1", "L2(11)", "L2(121)", [128X[104X
    [4X[28X  "L2(23)", "L5(3)", "M11", "M12", "M22", "M23", "M24", "McL", "ON", [128X[104X
    [4X[28X  "Suz", "U5(2)", "U6(2)" ][128X[104X
    [4X[28X13[128X[104X
    [4X[28X[ "2E6(2)", "2F4(2)'", "3D4(2)", "A13", "A14", "A15", "F4(2)", [128X[104X
    [4X[28X  "Fi22", "G2(3)", "G2(4)", "L2(13)", "L2(25)", "L2(27)", "L3(3)", [128X[104X
    [4X[28X  "L4(3)", "O7(3)", "O8+(3)", "S4(5)", "S6(3)", "Suz", "Sz(8)", [128X[104X
    [4X[28X  "U3(4)" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YLooking  at these examples, we may observe that the alternating group [22XA_n[122X of
  degree  [22Xn[122X  is [22X2[122X-pure iff [22Xn ∈ { 4, 5, 6 }[122X, [22X3[122X-pure iff [22Xn ∈ { 3, 4, 5, 6 }[122X, and
  [22Xp[122X-pure, for [22Xp ≥ 5[122X, iff [22Xn ∈ { p, p+1, p+2 }[122X.[133X
  
  [33X[0;0YAlso,  the  Suzuki  groups  [22XSz(q)[122X  are  [22X2[122X-pure  since  the  centralizers  of
  nonidentity [22X2[122X-elements are contained in Sylow [22X2[122X-subgroups.[133X
  
  [33X[0;0YFrom  the  inspection of the generic character table(s) of [22XPSL(2, q)[122X, we see
  that  [22XPSL(2,  p^d)[122X  is  [22Xp[122X-pure  Additionally, exactly the following cases of
  [22Xl[122X-purity occur, for a prime [22Xl[122X.[133X
  
  [30X    [33X[0;6Y[22Xq[122X is even and [22Xq-1[122X or [22Xq+1[122X is a power of [22Xl[122X.[133X
  
  [30X    [33X[0;6YFor [22Xq ≡ 1 mod 4[122X, [22X(q+1)/2[122X is a power of [22Xl[122X or [22Xq-1[122X is a power of [22Xl = 2[122X.[133X
  
  [30X    [33X[0;6YFor [22Xq ≡ 3 mod 4[122X, [22X(q-1)/2[122X is a power of [22Xl[122X or [22Xq+1[122X is a power of [22Xl = 2[122X.[133X
  
  
  [1X2.3-3 [33X[0;0YExample: Simple Groups with only one [22Xp[122X[101X[1X-Block[133X[101X
  
  [33X[0;0YAre there nonabelian simple groups with only one [22Xp[122X-block, for some prime [22Xp[122X?[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfun:= function( tbl )[127X[104X
    [4X[25X>[125X [27X     local result, p, bl;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X     result:= false;[127X[104X
    [4X[25X>[125X [27X     for p in Set( Factors( Size( tbl ) ) ) do[127X[104X
    [4X[25X>[125X [27X       bl:= PrimeBlocks( tbl, p );[127X[104X
    [4X[25X>[125X [27X       if Length( bl.defect ) = 1 then[127X[104X
    [4X[25X>[125X [27X         result:= true;[127X[104X
    [4X[25X>[125X [27X         Print( "only one block: ", Identifier( tbl ), ", p = ", p, "\n" );[127X[104X
    [4X[25X>[125X [27X       fi;[127X[104X
    [4X[25X>[125X [27X     od;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X     return result;[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
    [4X[25Xgap>[125X [27XAllCharacterTableNames( IsSimple, true, IsAbelian, false,[127X[104X
    [4X[25X>[125X [27X                           IsDuplicateTable, false, fun, true );[127X[104X
    [4X[28Xonly one block: M22, p = 2[128X[104X
    [4X[28Xonly one block: M24, p = 2[128X[104X
    [4X[28X[ "M22", "M24" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe see that the sporadic simple groups [22XM_22[122X and [22XM_24[122X have only one [22X2[122X-block.[133X
  
  
  [1X2.3-4 [33X[0;0YExample:The Sylow [22X3[122X[101X[1X subgroup of [22X3.O'N[122X[101X[1X[133X[101X
  
  [33X[0;0YWe  want  to  determine the structure of the Sylow [22X3[122X-subgroups of the triple
  cover [22XG = 3.O'N[122X of the sporadic simple O'Nan group [22XO'N[122X. The Sylow [22X3[122X-subgroup
  of  [22XO'N[122X  is  an  elementary  abelian  group  of order [22X3^4[122X, since the Sylow [22X3[122X
  normalizer in [22XO'N[122X has the structure [22X3^4:2^1+4D_10[122X (see [CCNPW85, p. 132]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCharacterTable( "ONN3" );[127X[104X
    [4X[28XCharacterTable( "3^4:2^(1+4)D10" )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YLet  [22XP[122X  be  a  Sylow  [22X3[122X-subgroup  of  [22XG[122X.  Then  [22XP[122X  is not abelian, since the
  centralizer order of any preimage of an element of order three in the simple
  factor  group  of  [22XG[122X is not divisible by [22X3^5[122X. Moreover, the exponent of [22XP[122X is
  three.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X3t:= CharacterTable( "3.ON" );;[127X[104X
    [4X[25Xgap>[125X [27Xorders:= OrdersClassRepresentatives( 3t );;[127X[104X
    [4X[25Xgap>[125X [27Xord3:= PositionsProperty( orders, x -> x = 3 );[127X[104X
    [4X[28X[ 2, 3, 7 ][128X[104X
    [4X[25Xgap>[125X [27Xsizes:= SizesCentralizers( 3t ){ ord3 };[127X[104X
    [4X[28X[ 1382446517760, 1382446517760, 3240 ][128X[104X
    [4X[25Xgap>[125X [27XSize( 3t );[127X[104X
    [4X[28X1382446517760[128X[104X
    [4X[25Xgap>[125X [27XCollected( Factors( sizes[3] ) );[127X[104X
    [4X[28X[ [ 2, 3 ], [ 3, 4 ], [ 5, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27X9 in orders;[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo both the centre and the Frattini subgroup of [22XP[122X are equal to the centre of
  [22XG[122X, hence [22XP[122X is an extraspecial group [22X3^1+4_+[122X.[133X
  
  
  [1X2.3-5 [33X[0;0YExample: Primitive Permutation Characters of [22X2.A_6[122X[101X[1X[133X[101X
  
  [33X[0;0YIt is often interesting to compute the primitive permutation characters of a
  group  [22XG[122X,  that  is,  the  characters of the permutation actions of [22XG[122X on the
  cosets  of  its  maximal  subgroups.  These  characters  can be computed for
  example  when the character tables of [22XG[122X, the character tables of its maximal
  subgroups,  and the class fusions from these character tables into the table
  of [22XG[122X are known.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "2.A6" );;[127X[104X
    [4X[25Xgap>[125X [27XHasMaxes( tbl );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xmaxes:= Maxes( tbl );[127X[104X
    [4X[28X[ "2.A5", "2.A6M2", "3^2:8", "2.Symm(4)", "2.A6M5" ][128X[104X
    [4X[25Xgap>[125X [27Xmx:= List( maxes, CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xprim1:= List( mx, s -> TrivialCharacter( s )^tbl );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( tbl,[127X[104X
    [4X[25X>[125X [27X     rec( chars:= prim1, centralizers:= false, powermap:= false ) );[127X[104X
    [4X[28X2.A6[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 2a 4a 3a 6a 3b 6b 8a 8b 5a 10a 5b 10b[128X[104X
    [4X[28X[128X[104X
    [4X[28XY.1     6  6  2  3  3  .  .  .  .  1   1  1   1[128X[104X
    [4X[28XY.2     6  6  2  .  .  3  3  .  .  1   1  1   1[128X[104X
    [4X[28XY.3    10 10  2  1  1  1  1  2  2  .   .  .   .[128X[104X
    [4X[28XY.4    15 15  3  3  3  .  .  1  1  .   .  .   .[128X[104X
    [4X[28XY.5    15 15  3  .  .  3  3  1  1  .   .  .   .[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThese permutation characters are the ones listed in [CCNPW85, p. 4].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( tbl, prim1 ).ATLAS;[127X[104X
    [4X[28X[ "1a+5a", "1a+5b", "1a+9a", "1a+5a+9a", "1a+5b+9a" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAlternatively, one can compute the primitive permutation characters from the
  table of marks if this table and the fusion into it are known.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( tbl );[127X[104X
    [4X[28XTableOfMarks( "2.A6" )[128X[104X
    [4X[25Xgap>[125X [27Xallperm:= PermCharsTom( tbl, tom );;[127X[104X
    [4X[25Xgap>[125X [27Xprim2:= allperm{ MaximalSubgroupsTom( tom )[1] };;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( tbl,[127X[104X
    [4X[25X>[125X [27X     rec( chars:= prim2, centralizers:= false, powermap:= false ) );[127X[104X
    [4X[28X2.A6[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 2a 4a 3a 6a 3b 6b 8a 8b 5a 10a 5b 10b[128X[104X
    [4X[28X[128X[104X
    [4X[28XY.1     6  6  2  3  3  .  .  .  .  1   1  1   1[128X[104X
    [4X[28XY.2     6  6  2  .  .  3  3  .  .  1   1  1   1[128X[104X
    [4X[28XY.3    10 10  2  1  1  1  1  2  2  .   .  .   .[128X[104X
    [4X[28XY.4    15 15  3  .  .  3  3  1  1  .   .  .   .[128X[104X
    [4X[28XY.5    15 15  3  3  3  .  .  1  1  .   .  .   .[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see  that  the two approaches yield the same permutation characters, but
  the  two  lists are sorted in a different way. The latter is due to the fact
  that  the  rows  of  the  table  of  marks  are ordered in a way that is not
  compatible  with  the ordering of maximal subgroups for the character table.
  Moreover,  there  is no way to choose the fusion from the character table to
  the  table  of  marks  in  such  a  way  that  the  two lists of permutation
  characters would become equal. The component [10Xperm[110X in the [2XFusionToTom[102X ([14X3.2-4[114X)
  record of the character table describes the incompatibility.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFusionToTom( tbl );[127X[104X
    [4X[28Xrec( map := [ 1, 2, 5, 4, 8, 3, 7, 11, 11, 6, 13, 6, 13 ], [128X[104X
    [4X[28X  name := "2.A6", perm := (4,5), [128X[104X
    [4X[28X  text := "fusion map is unique up to table autom." )[128X[104X
  [4X[32X[104X
  
  
  [1X2.3-6 [33X[0;0YExample: A Permutation Character of [22XFi_23[122X[101X[1X[133X[101X
  
  [33X[0;0YLet  [22Xx[122X  be  a [10X3B[110X element in the sporadic simple Fischer group [22XG = Fi_23[122X. The
  normalizer   [22XM[122X   of   [22Xx[122X   in   [22XG[122X   is   a   maximal  subgroup  of  the  type
  [22X3^{1+8}_+.2^{1+6}_-.3^{1+2}_+.2S_4[122X. We are interested in the distribution of
  the  elements  of  the  normal  subgroup [22XN[122X of the type [22X3^{1+8}_+[122X in [22XM[122X to the
  conjugacy classes of [22XG[122X.[133X
  
  [33X[0;0YThis  information  can be computed from the permutation character [22Xπ = 1_N^G[122X,
  so  we try to compute this permutation character. We have [22Xπ = (1_N^M)^G[122X, and
  [22X1_N^M[122X  can  be  computed  as  the  inflation of the regular character of the
  factor  group  [22XM/N[122X  to  [22XM[122X.  Note  that  the  character tables of [22XG[122X and [22XM[122X are
  available,  as well as the class fusion of [22XM[122X in [22XG[122X, and that [22XN[122X is the largest
  normal [22X3[122X-subgroup of [22XM[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "Fi23" );[127X[104X
    [4X[28XCharacterTable( "Fi23" )[128X[104X
    [4X[25Xgap>[125X [27Xmx:= Maxes( t );[127X[104X
    [4X[28X[ "2.Fi22", "O8+(3).3.2", "2^2.U6(2).2", "S8(2)", "S3xO7(3)", [128X[104X
    [4X[28X  "2..11.m23", "3^(1+8).2^(1+6).3^(1+2).2S4", "Fi23M8", "A12.2", [128X[104X
    [4X[28X  "(2^2x2^(1+8)).(3xU4(2)).2", "2^(6+8):(A7xS3)", "S4xS6(2)", [128X[104X
    [4X[28X  "S4(4).4", "L2(23)" ][128X[104X
    [4X[25Xgap>[125X [27Xm:= CharacterTable( mx[7] );[127X[104X
    [4X[28XCharacterTable( "3^(1+8).2^(1+6).3^(1+2).2S4" )[128X[104X
    [4X[25Xgap>[125X [27Xn:= ClassPositionsOfPCore( m, 3 );[127X[104X
    [4X[28X[ 1 .. 6 ][128X[104X
    [4X[25Xgap>[125X [27Xf:= m / n;[127X[104X
    [4X[28XCharacterTable( "3^(1+8).2^(1+6).3^(1+2).2S4/[ 1, 2, 3, 4, 5, 6 ]" )[128X[104X
    [4X[25Xgap>[125X [27Xreg:= 0 * [ 1 .. NrConjugacyClasses( f ) ];;[127X[104X
    [4X[25Xgap>[125X [27Xreg[1]:= Size( f );;[127X[104X
    [4X[25Xgap>[125X [27Xinfl:= reg{ GetFusionMap( m, f ) };[127X[104X
    [4X[28X[ 165888, 165888, 165888, 165888, 165888, 165888, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xind:= Induced( m, t, [ infl ] );[127X[104X
    [4X[28X[ ClassFunction( CharacterTable( "Fi23" ), [128X[104X
    [4X[28X    [ 207766624665600, 0, 0, 0, 603832320, 127567872, 6635520, [128X[104X
    [4X[28X      663552, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( t, ind ).contained;[127X[104X
    [4X[28X[ [ 1, 0, 0, 0, 864, 1538, 3456, 13824, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XPositionsProperty( OrdersClassRepresentatives( t ), x -> x = 3 );[127X[104X
    [4X[28X[ 5, 6, 7, 8 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThus [22XN[122X contains [22X864[122X elements in the class [10X3A[110X, [22X1538[122X elements in the class [10X3B[110X,
  and so on.[133X
  
