  
  [1X4 [33X[0;0YContents of the [5XGAP[105X[101X[1X Character Table Library[133X[101X
  
  [33X[0;0YThis chapter informs you about[133X
  
  [30X    [33X[0;6Ythe currently available character tables (see Section [14X4.1[114X),[133X
  
  [30X    [33X[0;6Ygeneric character tables (see Section [14X4.2[114X),[133X
  
  [30X    [33X[0;6Ythe  subsets  of  [5XAtlas[105X  tables  (see Section [14X4.3[114X) and [5XCAS[105X tables (see
        Section [14X4.4[114X),[133X
  
  [30X    [33X[0;6Yinstalling   the   library,   and   related   user   preferences  (see
        Section [14X4.5[114X).[133X
  
  [33X[0;0YThe  following  rather  technical sections are thought for those who want to
  maintain or extend the Character Table Library.[133X
  
  [30X    [33X[0;6Ythe technicalities of the access to library tables (see Section [14X4.6[114X),[133X
  
  [30X    [33X[0;6Yhow to extend the library (see Section [14X4.7[114X), and[133X
  
  [30X    [33X[0;6Ysanity checks (see Section [14X4.8[114X).[133X
  
  
  [1X4.1 [33X[0;0YOrdinary and Brauer Tables in the [5XGAP[105X[101X[1X Character Table Library[133X[101X
  
  [33X[0;0YThis  section  gives  a  brief overview of the contents of the [5XGAP[105X character
  table  library.  For  the details about, e. g., the structure of data files,
  see Section [14X4.6[114X.[133X
  
  [33X[0;0YThe  changes in the character table library since the first release of [5XGAP[105X 4
  are listed in a file that can be fetched from[133X
  
  [33X[0;0Y[7Xhttps://www.math.rwth-aachen.de/~Thomas.Breuer/ctbllib/htm/ctbldiff.htm[107X.[133X
  
  [33X[0;0YThere  are  three  different  kinds  of character tables in the [5XGAP[105X library,
  namely  [13Xordinary  character  tables[113X,  [13XBrauer  tables[113X,  and [13Xgeneric character
  tables[113X. Note that the Brauer table and the corresponding ordinary table of a
  group determine the [13Xdecomposition matrix[113X of the group (and the decomposition
  matrices  of  its blocks). These decomposition matrices can be computed from
  the  ordinary  and  modular  irreducibles  with [5XGAP[105X, see Section [14X'Reference:
  Operations  Concerning Blocks'[114X for details. A collection of PDF files of the
  known  decomposition  matrices  of  [5XAtlas[105X  tables in the [5XGAP[105X Character Table
  Library can also be found at[133X
  
  [33X[0;0Y[7Xhttps://www.math.rwth-aachen.de/~MOC/decomposition/[107X.[133X
  
  
  [1X4.1-1 [33X[0;0YOrdinary Character Tables[133X[101X
  
  [33X[0;0YTwo  different  aspects  are  useful  to  list the ordinary character tables
  available  in [5XGAP[105X, namely the aspect of the [13Xsource[113X of the tables and that of
  [13Xrelations[113X between the tables.[133X
  
  [33X[0;0YAs  for the source, there are first of all two big sources, namely the [5XAtlas[105X
  of  Finite  Groups (see Section [14X4.3[114X) and the [5XCAS[105X library of character tables
  (see [NPP84]).  Many  [5XAtlas[105X  tables  are  contained  in the [5XCAS[105X library, and
  difficulties  may  arise because the succession of characters and classes in
  [5XCAS[105X tables and [5XAtlas[105X tables are in general different, so see Section [14X4.4[114X for
  the  relations  between  these  two variants of character tables of the same
  group.  A subset of the [5XCAS[105X tables is the set of tables of Sylow normalizers
  of  sporadic  simple  groups  as published in [Ost86]  this may be viewed as
  another  source of character tables. The library also contains the character
  tables   of   factor   groups  of  space  groups  (computed  by  W. Hanrath,
  see [Han88])  that are part of [HP89], in the form of two microfiches; these
  tables  are  given  in  [5XCAS[105X format (see Section [14X4.4[114X) on the microfiches, but
  they had not been part of the [21Xofficial[121X [5XCAS[105X library.[133X
  
  [33X[0;0YTo  avoid  confusion about the ordering of classes and characters in a given
  table, authorship and so on, the [2XInfoText[102X ([14XReference: InfoText[114X) value of the
  table contains the information[133X
  
  [8X[10Xorigin: ATLAS of finite groups[110X[8X[108X
        [33X[0;6Yfor [5XAtlas[105X tables (see Section [14X4.3[114X),[133X
  
  [8X[10Xorigin: Ostermann[110X[8X[108X
        [33X[0;6Yfor tables contained in [Ost86],[133X
  
  [8X[10Xorigin: CAS library[110X[8X[108X
        [33X[0;6Yfor  any  table  of the [5XCAS[105X table library that is contained neither in
        the [5XAtlas[105X nor in [Ost86], and[133X
  
  [8X[10Xorigin: Hanrath library[110X[8X[108X
        [33X[0;6Yfor tables contained in the microfiches in [HP89].[133X
  
  [33X[0;0YThe  [2XInfoText[102X  ([14XReference:  InfoText[114X)  value  usually contains more detailed
  information,  for  example that the table in question is the character table
  of  a maximal subgroup of an almost simple group. If the table was contained
  in  the  [5XCAS[105X  library  then  additional information may be available via the
  [2XCASInfo[102X ([14X4.4-1[114X) value.[133X
  
  [33X[0;0YIf  one  is interested in the aspect of relations between the tables, i. e.,
  the  internal  structure of the library of ordinary tables, the contents can
  be listed up the following way.[133X
  
  [33X[0;0YWe have[133X
  
  [30X    [33X[0;6Yall  [5XAtlas[105X  tables  (see Section [14X4.3[114X), i. e., the tables of the simple
        groups  which  are  contained  in  the [5XAtlas[105X of Finite Groups, and the
        tables of cyclic and bicyclic extensions of these groups,[133X
  
  [30X    [33X[0;6Ymost  tables  of  maximal subgroups of sporadic simple groups ([13Xnot all[113X
        for the Monster group),[133X
  
  [30X    [33X[0;6Ymany  tables  of  maximal  subgroups  of other [5XAtlas[105X tables; the [2XMaxes[102X
        ([14X3.7-1[114X)  value for the table is set if all tables of maximal subgroups
        are available,[133X
  
  [30X    [33X[0;6Ythe tables of many Sylow [22Xp[122X-normalizers of sporadic simple groups; this
        includes  the tables printed in [Ost86] except [22XJ_4N2[122X, [22XCo_1N2[122X, [22XFi_22N2[122X,
        but also other tables are available; more generally, several tables of
        normalizers  of  other  radical  [22Xp[122X-subgroups  are  available,  such as
        normalizers of defect groups of [22Xp[122X-blocks,[133X
  
  [30X    [33X[0;6Ysome tables of element centralizers,[133X
  
  [30X    [33X[0;6Ysome tables of Sylow [22Xp[122X-subgroups,[133X
  
  [30X    [33X[0;6Yand a few other tables, e. g. [10XW(F4)[110X[133X
  
  [33X[0;0Y[13XNote[113X  that  class  fusions stored on library tables are not guaranteed to be
  compatible for any two subgroups of a group and their intersection, and they
  are not guaranteed to be consistent w. r. t. the composition of maps.[133X
  
  
  [1X4.1-2 [33X[0;0YBrauer Tables[133X[101X
  
  [33X[0;0YThe  library  contains  all tables of the [5XAtlas[105X of Brauer Tables ([JLPW95]),
  and  many  other Brauer tables of bicyclic extensions of simple groups which
  are known yet. The Brauer tables in the library contain the information[133X
  
  origin: modular ATLAS of finite groups
  
  [33X[0;0Yin their [2XInfoText[102X ([14XReference: InfoText[114X) string.[133X
  
  
  [1X4.2 [33X[0;0YGeneric Character Tables[133X[101X
  
  [33X[0;0YGeneric  character  tables  provide  a  means for writing down the character
  tables  of  all  groups  in  a  (usually infinite) series of similar groups,
  e. g.,  cyclic  groups,  or  symmetric  groups, or the general linear groups
  GL[22X(2,q)[122X where [22Xq[122X ranges over certain prime powers.[133X
  
  [33X[0;0YLet [22X{ G_q | q ∈ I }[122X be such a series, where [22XI[122X is an index set. The character
  table  of one fixed member [22XG_q[122X could be computed using a function that takes
  [22Xq[122X  as  only  argument and constructs the table of [22XG_q[122X. It is, however, often
  desirable  to  compute  not  only  the  whole  table  but to access just one
  specific  character,  or  to  compute  just  one  character  value,  without
  computing the whole character table.[133X
  
  [33X[0;0YFor  example,  both  the conjugacy classes and the irreducible characters of
  the  symmetric group [22XS_n[122X are in bijection with the partitions of [22Xn[122X. Thus for
  given  [22Xn[122X  it  makes  sense  to  ask  for  the  character  corresponding to a
  particular partition, or just for its character value at another partition.[133X
  
  [33X[0;0YA  generic character table in [5XGAP[105X allows one such local evaluations. In this
  sense,  [5XGAP[105X  can  deal  also  with  character  tables that are too big to be
  computed and stored as a whole.[133X
  
  [33X[0;0YCurrently  the  only  operations for generic tables supported by [5XGAP[105X are the
  specialisation  of  the  parameter [22Xq[122X in order to compute the whole character
  table   of   [22XG_q[122X,  and  local  evaluation  (see [2XClassParameters[102X  ([14XReference:
  ClassParameters[114X)  for  an example). [5XGAP[105X does [13Xnot[113X support the computation of,
  e. g., generic scalar products.[133X
  
  [33X[0;0YWhile the numbers of conjugacy classes for the members of a series of groups
  are  usually  not  bounded,  there  is always a fixed finite number of [13Xtypes[113X
  (equivalence  classes)  of  conjugacy  classes;  very  often the equivalence
  relation is isomorphism of the centralizers of the representatives.[133X
  
  [33X[0;0YFor  each  type  [22Xt[122X  of  classes  and a fixed [22Xq ∈ I[122X, a [13Xparametrisation[113X of the
  classes  in [22Xt[122X is a function that assigns to each conjugacy class of [22XG_q[122X in [22Xt[122X
  a [13Xparameter[113X by which it is uniquely determined. Thus the classes are indexed
  by pairs [22X[t,p_t][122X consisting of a type [22Xt[122X and a parameter [22Xp_t[122X for that type.[133X
  
  [33X[0;0YFor  any  generic  table,  there  has  to  be  a  fixed  number  of types of
  irreducible characters of [22XG_q[122X, too. Like the classes, the characters of each
  type are parametrised.[133X
  
  [33X[0;0YIn  [5XGAP[105X,  the parametrisations of classes and characters for tables computed
  from   generic   tables  is  stored  using  the  attributes  [2XClassParameters[102X
  ([14XReference:    ClassParameters[114X)    and    [2XCharacterParameters[102X    ([14XReference:
  CharacterParameters[114X).[133X
  
  
  [1X4.2-1 [33X[0;0YAvailable generic character tables[133X[101X
  
  [33X[0;0YCurrently,  generic  tables  of the following groups –in alphabetical order–
  are available in [5XGAP[105X. (A list of the names of generic tables known to [5XGAP[105X is
  [10XLIBTABLE.GENERIC.firstnames[110X.)  We  list  the  function calls needed to get a
  specialized  table,  the  generic  table  itself  can be accessed by calling
  [2XCharacterTable[102X ([14XReference: CharacterTable[114X) with the first argument only; for
  example,  [10XCharacterTable(  "Cyclic"  )[110X  yields  the  generic table of cyclic
  groups.[133X
  
  [8X[10XCharacterTable( "Alternating", [110X[8X[22Xn[122X[10X )[110X[8X[108X
        [33X[0;6Ythe table of the [13Xalternating[113X group on [22Xn[122X letters,[133X
  
  [8X[10XCharacterTable( "Cyclic", [110X[8X[22Xn[122X[10X )[110X[8X[108X
        [33X[0;6Ythe table of the [13Xcyclic[113X group of order [22Xn[122X,[133X
  
  [8X[10XCharacterTable( "Dihedral", [110X[8X[22X2n[122X[10X )[110X[8X[108X
        [33X[0;6Ythe table of the [13Xdihedral[113X group of order [22X2n[122X,[133X
  
  [8X[10XCharacterTable( "DoubleCoverAlternating", [110X[8X[22Xn[122X[10X )[110X[8X[108X
        [33X[0;6Ythe  table  of  the  [13XSchur  double cover of the alternating[113X group on [22Xn[122X
        letters (see [Noe02]),[133X
  
  [8X[10XCharacterTable( "DoubleCoverSymmetric", [110X[8X[22Xn[122X[10X )[110X[8X[108X
        [33X[0;6Ythe table of the [13Xstandard Schur double cover of the symmetric[113X group on
        [22Xn[122X letters (see [Noe02]),[133X
  
  [8X[10XCharacterTable( "GL", 2, [110X[8X[22Xq[122X[10X )[110X[8X[108X
        [33X[0;6Ythe table of the [13Xgeneral linear[113X group [10XGL(2,[110X[22Xq[122X[10X)[110X, for a prime power [22Xq[122X,[133X
  
  [8X[10XCharacterTable( "GU", 3, [110X[8X[22Xq[122X[10X )[110X[8X[108X
        [33X[0;6Ythe table of the [13Xgeneral unitary[113X group [10XGU(3,[110X[22Xq[122X[10X)[110X, for a prime power [22Xq[122X,[133X
  
  [8X[10XCharacterTable( "P:Q", [110X[8X[22X[ p, q ][122X[10X )[110X[8X and [10XCharacterTable( "P:Q", [110X[8X[22X[ p, q, k ][122X[10X )[110X[8X[108X
        [33X[0;6Ythe table of the [13XFrobenius extension[113X of the nontrivial cyclic group of
        odd  order [22Xp[122X by the nontrivial cyclic group of order [22Xq[122X where [22Xq[122X divides
        [22Xp_i-1[122X  for  all  prime divisors [22Xp_i[122X of [22Xp[122X; if [22Xp[122X is a prime power then [22Xq[122X
        determines  the group uniquely and thus the first version can be used,
        otherwise  the  action of the residue class of [22Xk[122X modulo [22Xp[122X is taken for
        forming  orbits  of  length  [22Xq[122X each on the nonidentity elements of the
        group of order [22Xp[122X,[133X
  
  [8X[10XCharacterTable( "PSL", 2, [110X[8X[22Xq[122X[10X )[110X[8X[108X
        [33X[0;6Ythe table of the [13Xprojective special linear[113X group [10XPSL(2,[110X[22Xq[122X[10X)[110X, for a prime
        power [22Xq[122X,[133X
  
  [8X[10XCharacterTable( "SL", 2, [110X[8X[22Xq[122X[10X )[110X[8X[108X
        [33X[0;6Ythe table of the [13Xspecial linear[113X group [10XSL(2,[110X[22Xq[122X[10X)[110X, for a prime power [22Xq[122X,[133X
  
  [8X[10XCharacterTable( "SU", 3, [110X[8X[22Xq[122X[10X )[110X[8X[108X
        [33X[0;6Ythe table of the [13Xspecial unitary[113X group [10XSU(3,[110X[22Xq[122X[10X)[110X, for a prime power [22Xq[122X,[133X
  
  [8X[10XCharacterTable( "Suzuki", [110X[8X[22Xq[122X[10X )[110X[8X[108X
        [33X[0;6Ythe  table of the [13XSuzuki[113X group [10XSz([110X[22Xq[122X[10X)[110X [22X= ^2B_2(q)[122X, for [22Xq[122X an odd power of
        [22X2[122X,[133X
  
  [8X[10XCharacterTable( "Symmetric", [110X[8X[22Xn[122X[10X )[110X[8X[108X
        [33X[0;6Ythe table of the [13Xsymmetric[113X group on [22Xn[122X letters,[133X
  
  [8X[10XCharacterTable( "WeylB", [110X[8X[22Xn[122X[10X )[110X[8X[108X
        [33X[0;6Ythe table of the [13XWeyl[113X group of type [22XB_n[122X,[133X
  
  [8X[10XCharacterTable( "WeylD", [110X[8X[22Xn[122X[10X )[110X[8X[108X
        [33X[0;6Ythe table of the [13XWeyl[113X group of type [22XD_n[122X.[133X
  
  [33X[0;0YIn addition to the above calls that really use generic tables, the following
  calls  to  [2XCharacterTable[102X  ([14XReference:  CharacterTable[114X)  are  to some extent
  [21Xgeneric[121X  constructions.  But  note  that  no local evaluation is possible in
  these  cases, as no generic table object exists in [5XGAP[105X that can be asked for
  local information.[133X
  
  [8X[10XCharacterTable( "Quaternionic", [110X[8X[22X4n[122X[10X )[110X[8X[108X
        [33X[0;6Ythe table of the [13Xgeneralized quaternionic[113X group of order [22X4n[122X,[133X
  
  [8X[10XCharacterTableWreathSymmetric( tbl, [110X[8X[22Xn[122X[10X )[110X[8X[108X
        [33X[0;6Ythe  character table of the wreath product of the group whose table is
        [10Xtbl[110X     with     the     symmetric     group     on     [22Xn[122X     letters,
        see [2XCharacterTableWreathSymmetric[102X                          ([14XReference:
        CharacterTableWreathSymmetric[114X).[133X
  
  [1X4.2-2 CharacterTableSpecialized[101X
  
  [33X[1;0Y[29X[2XCharacterTableSpecialized[102X( [3Xgentbl[103X, [3Xq[103X ) [32X function[133X
  
  [33X[0;0YFor  a record [3Xgentbl[103X representing a generic character table, and a parameter
  value [3Xq[103X, [2XCharacterTableSpecialized[102X returns a character table object computed
  by evaluating [3Xgentbl[103X at [3Xq[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xc5:= CharacterTableSpecialized( CharacterTable( "Cyclic" ), 5 );[127X[104X
    [4X[28XCharacterTable( "C5" )[128X[104X
    [4X[25Xgap>[125X [27XDisplay( c5 );[127X[104X
    [4X[28XC5[128X[104X
    [4X[28X[128X[104X
    [4X[28X     5  1  1  1  1  1[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 5a 5b 5c 5d[128X[104X
    [4X[28X    5P 1a 1a 1a 1a 1a[128X[104X
    [4X[28X[128X[104X
    [4X[28XX.1     1  1  1  1  1[128X[104X
    [4X[28XX.2     1  A  B /B /A[128X[104X
    [4X[28XX.3     1  B /A  A /B[128X[104X
    [4X[28XX.4     1 /B  A /A  B[128X[104X
    [4X[28XX.5     1 /A /B  B  A[128X[104X
    [4X[28X[128X[104X
    [4X[28XA = E(5)[128X[104X
    [4X[28XB = E(5)^2[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y(Also  [10XCharacterTable(  "Cyclic",  5 )[110X could have been used to construct the
  above table.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XHasClassParameters( c5 );  HasCharacterParameters( c5 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XClassParameters( c5 );  CharacterParameters( c5 );[127X[104X
    [4X[28X[ [ 1, 0 ], [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ] ][128X[104X
    [4X[28X[ [ 1, 0 ], [ 1, 1 ], [ 1, 2 ], [ 1, 3 ], [ 1, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27XClassParameters( CharacterTable( "Symmetric", 3 ) );[127X[104X
    [4X[28X[ [ 1, [ 1, 1, 1 ] ], [ 1, [ 2, 1 ] ], [ 1, [ 3 ] ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YHere  are  examples  for  the  [21Xlocal evaluation[121X of generic character tables,
  first  a  character  value of the cyclic group shown above, then a character
  value and a representative order of a symmetric group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCharacterTable( "Cyclic" ).irreducibles[1][1]( 5, 2, 3 );[127X[104X
    [4X[28XE(5)[128X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "Symmetric" );;[127X[104X
    [4X[25Xgap>[125X [27Xtbl.irreducibles[1][1]( 5, [ 3, 2 ], [ 2, 2, 1 ] );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xtbl.orders[1]( 5, [ 2, 1, 1, 1 ] );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  
  [1X4.2-3 [33X[0;0YComponents of generic character tables[133X[101X
  
  [33X[0;0YAny  generic  table  in  [5XGAP[105X  is  represented  by  a  record.  The following
  components are supported for generic character table records.[133X
  
  [8X[10Xcentralizers[110X[8X[108X
        [33X[0;6Ylist  of  functions,  one  for each class type [22Xt[122X, with arguments [22Xq[122X and
        [22Xp_t[122X, returning the centralizer order of the class [22X[t,p_t][122X,[133X
  
  [8X[10Xcharparam[110X[8X[108X
        [33X[0;6Ylist  of  functions,  one  for each character type [22Xt[122X, with argument [22Xq[122X,
        returning the list of character parameters of type [22Xt[122X,[133X
  
  [8X[10Xclassparam[110X[8X[108X
        [33X[0;6Ylist  of  functions,  one  for  each  class  type  [22Xt[122X, with argument [22Xq[122X,
        returning the list of class parameters of type [22Xt[122X,[133X
  
  [8X[10Xclasstext[110X[8X[108X
        [33X[0;6Ylist  of  functions,  one  for each class type [22Xt[122X, with arguments [22Xq[122X and
        [22Xp_t[122X,  returning  a  representative of the class with parameter [22X[t,p_t][122X
        (note  that  this  element  need  [13Xnot[113X  actually  lie  in  the group in
        question,   for   example   it  may  be  a  diagonal  matrix  but  the
        characteristic polynomial in the group s irreducible),[133X
  
  [8X[10Xdomain[110X[8X[108X
        [33X[0;6Yfunction  of  [22Xq[122X  returning  [9Xtrue[109X  if [22Xq[122X is a valid parameter, and [9Xfalse[109X
        otherwise,[133X
  
  [8X[10Xidentifier[110X[8X[108X
        [33X[0;6Yidentifier string of the generic table,[133X
  
  [8X[10Xirreducibles[110X[8X[108X
        [33X[0;6Ylist of list of functions, in row [22Xi[122X and column [22Xj[122X the function of three
        arguments, namely [22Xq[122X and the parameters [22Xp_t[122X and [22Xp_s[122X of the class type [22Xt[122X
        and the character type [22Xs[122X,[133X
  
  [8X[10XisGenericTable[110X[8X[108X
        [33X[0;6Yalways [9Xtrue[109X[133X
  
  [8X[10Xlibinfo[110X[8X[108X
        [33X[0;6Yrecord  with  components  [10Xfirstname[110X ([2XIdentifier[102X ([14XReference: Identifier
        for  character  tables[114X)  value  of  the table) and [10Xothernames[110X (list of
        other admissible names)[133X
  
  [8X[10Xmatrix[110X[8X[108X
        [33X[0;6Yfunction of [22Xq[122X returning the matrix of irreducibles of [22XG_q[122X,[133X
  
  [8X[10Xorders[110X[8X[108X
        [33X[0;6Ylist  of  functions,  one  for each class type [22Xt[122X, with arguments [22Xq[122X and
        [22Xp_t[122X,  returning  the  representative  order  of elements of type [22Xt[122X and
        parameter [22Xp_t[122X,[133X
  
  [8X[10Xpowermap[110X[8X[108X
        [33X[0;6Ylist  of  functions,  one  for  each  class  type  [22Xt[122X,  each with three
        arguments  [22Xq[122X,  [22Xp_t[122X,  and  [22Xk[122X,  returning  the  pair [22X[s,p_s][122X of type and
        parameter for the [22Xk[122X-th power of the class with parameter [22X[t,p_t][122X,[133X
  
  [8X[10Xsize[110X[8X[108X
        [33X[0;6Yfunction of [22Xq[122X returning the order of [22XG_q[122X,[133X
  
  [8X[10Xspecializedname[110X[8X[108X
        [33X[0;6Yfunction  of  [22Xq[122X  returning  the  [2XIdentifier[102X ([14XReference: Identifier for
        character tables[114X) value of the table of [22XG_q[122X,[133X
  
  [8X[10Xtext[110X[8X[108X
        [33X[0;6Ystring informing about the generic table[133X
  
  [33X[0;0YIn  the  specialized table, the [2XClassParameters[102X ([14XReference: ClassParameters[114X)
  and  [2XCharacterParameters[102X  ([14XReference:  CharacterParameters[114X)  values  are the
  lists of parameters [22X[t,p_t][122X of classes and characters, respectively.[133X
  
  [33X[0;0YIf  the  [10Xmatrix[110X  component  is present then its value implements a method to
  compute  the  complete  table of small members [22XG_q[122X more efficiently than via
  local  evaluation; this method will be called when the generic table is used
  to    compute    the    whole    character    table    for    a    given   [22Xq[122X
  (see [2XCharacterTableSpecialized[102X ([14X4.2-2[114X)).[133X
  
  
  [1X4.2-4 [33X[0;0YExample: The generic table of cyclic groups[133X[101X
  
  [33X[0;0YFor  the  cyclic group [22XC_q = ⟨ x ⟩[122X of order [22Xq[122X, there is one type of classes.
  The  class  parameters  are  integers  [22Xk  ∈  { 0, ..., q-1 }[122X, the class with
  parameter  [22Xk[122X  consists of the group element [22Xx^k[122X. Group order and centralizer
  orders  are the identity function [22Xq ↦ q[122X, independent of the parameter [22Xk[122X. The
  representative order function maps the parameter pair [22X[q,k][122X to [22Xq / gcd(q,k)[122X,
  which is the order of [22Xx^k[122X in [22XC_q[122X; the [22Xp[122X-th power map is the function mapping
  the triple [22X(q,k,p)[122X to the parameter [22X[1,(kp mod q)][122X.[133X
  
  [33X[0;0YThere  is  one  type of characters, with parameters [22Xl ∈ { 0, ..., q-1 }[122X; for
  [22Xe_q[122X  a  primitive  complex  [22Xq[122X-th  root  of  unity,  the character values are
  [22Xχ_l(x^k) = e_q^kl[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPrint( CharacterTable( "Cyclic" ), "\n" );[127X[104X
    [4X[28Xrec([128X[104X
    [4X[28X  centralizers := [ function ( n, k )[128X[104X
    [4X[28X            return n;[128X[104X
    [4X[28X        end ],[128X[104X
    [4X[28X  charparam := [ function ( n )[128X[104X
    [4X[28X            return [ 0 .. n - 1 ];[128X[104X
    [4X[28X        end ],[128X[104X
    [4X[28X  classparam := [ function ( n )[128X[104X
    [4X[28X            return [ 0 .. n - 1 ];[128X[104X
    [4X[28X        end ],[128X[104X
    [4X[28X  domain := <Category "(IsInt and IsPosRat)">,[128X[104X
    [4X[28X  identifier := "Cyclic",[128X[104X
    [4X[28X  irreducibles := [ [ function ( n, k, l )[128X[104X
    [4X[28X                return E( n ) ^ (k * l);[128X[104X
    [4X[28X            end ] ],[128X[104X
    [4X[28X  isGenericTable := true,[128X[104X
    [4X[28X  libinfo := rec([128X[104X
    [4X[28X      firstname := "Cyclic",[128X[104X
    [4X[28X      othernames := [  ] ),[128X[104X
    [4X[28X  orders := [ function ( n, k )[128X[104X
    [4X[28X            return n / Gcd( n, k );[128X[104X
    [4X[28X        end ],[128X[104X
    [4X[28X  powermap := [ function ( n, k, pow )[128X[104X
    [4X[28X            return [ 1, k * pow mod n ];[128X[104X
    [4X[28X        end ],[128X[104X
    [4X[28X  size := function ( n )[128X[104X
    [4X[28X        return n;[128X[104X
    [4X[28X    end,[128X[104X
    [4X[28X  specializedname := function ( q )[128X[104X
    [4X[28X        return Concatenation( "C", String( q ) );[128X[104X
    [4X[28X    end,[128X[104X
    [4X[28X  text := "generic character table for cyclic groups" )[128X[104X
  [4X[32X[104X
  
  
  [1X4.2-5 [33X[0;0YExample: The generic table of the general linear group GL[22X(2,q)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe  have four types [22Xt_1, t_2, t_3, t_4[122X of classes, according to the rational
  canonical form of the elements. [22Xt_1[122X describes scalar matrices, [22Xt_2[122X nonscalar
  diagonal  matrices, [22Xt_3[122X companion matrices of [22X(X - ρ)^2[122X for nonzero elements
  [22Xρ  ∈  F_q[122X, and [22Xt_4[122X companion matrices of irreducible polynomials of degree [22X2[122X
  over [22XF_q[122X.[133X
  
  [33X[0;0YThe  sets  of  class  parameters  of the types are in bijection with nonzero
  elements in [22XF_q[122X for [22Xt_1[122X and [22Xt_3[122X, with the set[133X
  
  
  [24X[33X[0;6Y{ { ρ, τ }; ρ, τ ∈ F_q, ρ ‡ 0, τ ‡ 0, ρ ‡ τ }[133X[124X
  
  [33X[0;0Yfor [22Xt_2[122X, and with the set [22X{ { ϵ, ϵ^q }; ϵ ∈ F_{q^2} ∖ F_q }[122X for [22Xt_4[122X.[133X
  
  [33X[0;0YThe  centralizer  order  functions  are [22Xq ↦ (q^2-1)(q^2-q)[122X for type [22Xt_1[122X, [22Xq ↦
  (q-1)^2[122X for type [22Xt_2[122X, [22Xq ↦ q(q-1)[122X for type [22Xt_3[122X, and [22Xq ↦ q^2-1[122X for type [22Xt_4[122X.[133X
  
  [33X[0;0YThe  representative  order  function of [22Xt_1[122X maps [22X(q, ρ)[122X to the order of [22Xρ[122X in
  [22XF_q[122X,  that  of  [22Xt_2[122X  maps  [22X(q, { ρ, τ })[122X to the least common multiple of the
  orders of [22Xρ[122X and [22Xτ[122X.[133X
  
  [33X[0;0YThe file contains something similar to the following table.[133X
  
  [4X[32X[104X
    [4Xrec([104X
    [4Xidentifier := "GL2",[104X
    [4Xspecializedname := ( q -> Concatenation( "GL(2,", String(q), ")" ) ),[104X
    [4Xsize := ( q -> (q^2-1)*(q^2-q) ),[104X
    [4Xtext := "generic character table of GL(2,q), see Robert Steinberg: ...",[104X
    [4Xcentralizers := [ function( q, k ) return (q^2-1) * (q^2-q); end,[104X
    [4X                  ..., ..., ... ],[104X
    [4Xclassparam := [ ( q -> [ 0 .. q-2 ] ), ..., ..., ... ],[104X
    [4Xcharparam := [ ( q -> [ 0 .. q-2 ] ), ..., ..., ... ],[104X
    [4Xpowermap := [ function( q, k, pow ) return [ 1, (k*pow) mod (q-1) ]; end,[104X
    [4X              ..., ..., ... ],[104X
    [4Xorders:= [ function( q, k ) return (q-1)/Gcd( q-1, k ); end,[104X
    [4X           ..., ..., ... ],[104X
    [4Xirreducibles := [ [ function( q, k, l ) return E(q-1)^(2*k*l); end,[104X
    [4X                    ..., ..., ... ],[104X
    [4X                  [ ..., ..., ..., ... ],[104X
    [4X                  [ ..., ..., ..., ... ],[104X
    [4X                  [ ..., ..., ..., ... ] ],[104X
    [4Xclasstext := [ ..., ..., ..., ... ],[104X
    [4Xdomain := IsPrimePowerInt,[104X
    [4XisGenericTable := true )[104X
  [4X[32X[104X
  
  
  [1X4.3 [33X[0;0Y[5XAtlas[105X[101X[1X Tables[133X[101X
  
  [33X[0;0YThe  [5XGAP[105X  character  table library contains all character tables of bicyclic
  extensions  of simple groups that are included in the [5XAtlas[105X of Finite Groups
  ([CCN+85], from now on called [5XAtlas[105X), and the Brauer tables contained in the
  [5XAtlas[105X of Brauer Characters ([JLPW95]).[133X
  
  [33X[0;0YThese tables have the information[133X
  
  origin: ATLAS of finite groups
  
  [33X[0;0Yor[133X
  
  origin: modular ATLAS of finite groups
  
  [33X[0;0Yin  their [2XInfoText[102X ([14XReference: InfoText[114X) value, they are simply called [5XAtlas[105X
  tables further on.[133X
  
  [33X[0;0YFor  displaying  [5XAtlas[105X  tables with the row labels used in the [5XAtlas[105X, or for
  displaying    decomposition   matrices,   see [2XLaTeXStringDecompositionMatrix[102X
  ([14XReference:   LaTeXStringDecompositionMatrix[114X)  and [2XAtlasLabelsOfIrreducibles[102X
  ([14X4.3-6[114X).[133X
  
  [33X[0;0YIn  addition  to the information given in Chapters 6 to 8 of the [5XAtlas[105X which
  tell you how to read the printed tables, there are some rules relating these
  to the corresponding [5XGAP[105X tables.[133X
  
  
  [1X4.3-1 [33X[0;0YImprovements to the [5XAtlas[105X[101X[1X[133X[101X
  
  [33X[0;0YFor the [5XGAP[105X Character Table Library not the printed versions of the [5XAtlas[105X of
  Finite  Groups  and  the  [5XAtlas[105X  of  Brauer  Characters are relevant but the
  revised versions given by the currently three lists of improvements that are
  maintained  by Simon Norton. The first such list is contained in [BN95], and
  is  printed  in  the Appendix of [JLPW95]; it contains the improvements that
  had  been  known  until  the  [21X[5XAtlas[105X  of Brauer Characters[121X was published. The
  second  list  contains  the  improvements to the [5XAtlas[105X of Finite Groups that
  were  found  since  the  publication  of [JLPW95].  It  can  be found in the
  internet, an HTML version at[133X
  
  [33X[0;0Y[7Xhttp://web.mat.bham.ac.uk/atlas/html/atlasmods.html[107X[133X
  
  [33X[0;0Yand a DVI version at[133X
  
  [33X[0;0Y[7Xhttp://web.mat.bham.ac.uk/atlas/html/atlasmods.dvi[107X.[133X
  
  [33X[0;0YThe  third list contains the improvements to the [5XAtlas[105X of Brauer Characters,
  HTML and PDF versions can be found in the internet at[133X
  
  [33X[0;0Y[7Xhttps://www.math.rwth-aachen.de/~MOC/ABCerr.html[107X[133X
  
  [33X[0;0Yand[133X
  
  [33X[0;0Y[7Xhttps://www.math.rwth-aachen.de/~MOC/ABCerr.pdf[107X,[133X
  
  [33X[0;0Yrespectively.[133X
  
  [33X[0;0YAlso  some  tables  are regarded as [5XAtlas[105X tables that are not printed in the
  [5XAtlas[105X  but available in [5XAtlas[105X format, according to the lists of improvements
  mentioned above. Currently these are the tables related to [22XL_2(49)[122X, [22XL_2(81)[122X,
  [22XL_6(2)[122X, [22XO_8^-(3)[122X, [22XO_8^+(3)[122X, [22XS_10(2)[122X, and [22X^2E_6(2).3[122X.[133X
  
  
  [1X4.3-2 [33X[0;0YPower Maps[133X[101X
  
  [33X[0;0YFor  the  tables of [22X3.McL[122X, [22X3_2.U_4(3)[122X and its covers, and [22X3_2.U_4(3).2_3[122X and
  its  covers,  the  power maps are not uniquely determined by the information
  from   the   [5XAtlas[105X   but   determined   only   up  to  matrix  automorphisms
  (see [2XMatrixAutomorphisms[102X    ([14XReference:    MatrixAutomorphisms[114X))    of   the
  irreducible  characters. In these cases, the first possible map according to
  lexicographical ordering was chosen, and the automorphisms are listed in the
  [2XInfoText[102X ([14XReference: InfoText[114X) strings of the tables.[133X
  
  
  [1X4.3-3 [33X[0;0YProjective Characters and Projections[133X[101X
  
  [33X[0;0YIf  [22XG[122X  (or  [22XG.a[122X)  has  a  nontrivial  Schur  multiplier  then  the attribute
  [2XProjectivesInfo[102X  ([14X3.7-2[114X)  of  the [5XGAP[105X table object of [22XG[122X (or [22XG.a[122X) is set; the
  [10Xchars[110X  component  of  the  record in question is the list of values lists of
  those  faithful  projective  irreducibles  that  are  printed  in  the [5XAtlas[105X
  (so-called  [13Xproxy  character[113X),  and the [10Xmap[110X component lists the positions of
  columns  in  the  covering  for  which the column is printed in the [5XAtlas[105X (a
  so-called  [13Xproxy  class[113X,  this  preimage  is  denoted  by  [22Xg_0[122X in Chapter 7,
  Section 14 of the [5XAtlas[105X).[133X
  
  
  [1X4.3-4 [33X[0;0YTables of Isoclinic Groups[133X[101X
  
  [33X[0;0YAs  described  in  Chapter 6,  Section 7 and in Chapter 7, Section 18 of the
  [5XAtlas[105X, there exist two (often nonisomorphic) groups of structure [22X2.G.2[122X for a
  simple  group  [22XG[122X,  which are isoclinic. The table in the [5XGAP[105X Character Table
  Library  is the one printed in the [5XAtlas[105X, the table of the isoclinic variant
  can     be    constructed    using    [2XCharacterTableIsoclinic[102X    ([14XReference:
  CharacterTableIsoclinic[114X).[133X
  
  
  [1X4.3-5 [33X[0;0YOrdering of Characters and Classes[133X[101X
  
  [33X[0;0Y(Throughout this section, [22XG[122X always means the simple group involved.)[133X
  
  [31X1[131X   [33X[0;6YFor  [22XG[122X itself, the ordering of classes and characters in the [5XGAP[105X table
        coincides with the one in the [5XAtlas[105X.[133X
  
  [31X2[131X   [33X[0;6YFor an automorphic extension [22XG.a[122X, there are three types of characters.[133X
  
        [30X    [33X[0;12YIf  a  character  [22Xχ[122X  of  [22XG[122X  extends  to  [22XG.a[122X  then the different
              extensions  [22Xχ^0,  χ^1, ..., χ^{a-1}[122X are consecutive in the table
              of [22XG.a[122X (see [CCN+85, Chapter 7, Section 16]).[133X
  
        [30X    [33X[0;12YIf  some  characters of [22XG[122X fuse to give a single character of [22XG.a[122X
              then the position of that character in the table of [22XG.a[122X is given
              by the position of the first involved character of [22XG[122X.[133X
  
        [30X    [33X[0;12YIf  both  extension  and  fusion  occur for a character then the
              resulting  characters  are  consecutive in the table of [22XG.a[122X, and
              each replaces the first involved character of [22XG[122X.[133X
  
  [31X3[131X   [33X[0;6YSimilarly,  there  are  different  types of classes for an automorphic
        extension [22XG.a[122X, as follows.[133X
  
        [30X    [33X[0;12YIf  some  classes collapse then the resulting class replaces the
              first involved class of [22XG[122X.[133X
  
        [30X    [33X[0;12YFor [22Xa > 2[122X, any proxy class and its algebraic conjugates that are
              not printed in the [5XAtlas[105X are consecutive in the table of [22XG.a[122X; if
              more than two classes of [22XG.a[122X have the same proxy class (the only
              case  that  actually  occurs  is for [22Xa = 5[122X) then the ordering of
              non-printed  classes  is the natural one of corresponding Galois
              conjugacy operators [22X*k[122X (see [CCN+85, Chapter 7, Section 19]).[133X
  
        [30X    [33X[0;12YFor  [22Xa_1[122X,  [22Xa_2[122X  dividing  [22Xa[122X  such that [22Xa_1 ≤ a_2[122X, the classes of
              [22XG.a_1[122X  in  [22XG.a[122X  precede  the  classes  of [22XG.a_2[122X not contained in
              [22XG.a_1[122X.  This ordering is the same as in the [5XAtlas[105X, with the only
              exception [22XU_3(8).6[122X.[133X
  
  [31X4[131X   [33X[0;6YFor  a  central  extension  [22XM.G[122X,  there  are  two  different  types of
        characters, as follows.[133X
  
        [30X    [33X[0;12YEach  character  can  be  regarded  as a faithful character of a
              factor  group  [22Xm.G[122X,  where [22Xm[122X divides [22XM[122X. Characters with the same
              kernel  are  consecutive  as  in  the  [5XAtlas[105X,  the  ordering  of
              characters  with  different  kernels  is  given  by the order of
              precedence [22X1, 2, 4, 3, 6, 12[122X for the different values of [22Xm[122X.[133X
  
        [30X    [33X[0;12YIf  [22Xm  >  2[122X,  a faithful character of [22Xm.G[122X that is printed in the
              [5XAtlas[105X  (a  so-called  [13Xproxy  character[113X)  represents  two or more
              Galois conjugates. In each [5XAtlas[105X table in [5XGAP[105X, a proxy character
              always  precedes the non-printed characters with this proxy. The
              case [22Xm = 12[122X is the only one that actually occurs where more than
              one  character  for  a  proxy  is not printed. In this case, the
              non-printed    characters   are   ordered   according   to   the
              corresponding  Galois  conjugacy  operators [22X*5[122X, [22X*7[122X, [22X*11[122X (in this
              order).[133X
  
  [31X5[131X   [33X[0;6YFor the classes of a central extension we have the following.[133X
  
        [30X    [33X[0;12YThe  preimages  of a [22XG[122X-class in [22XM.G[122X are subsequent, the ordering
              is  the  same  as  that  of  the  lifting order rows in [CCN+85,
              Chapter 7, Section 7].[133X
  
        [30X    [33X[0;12YThe  primitive roots of unity chosen to represent the generating
              central  element  (i. e., the element in the second class of the
              [5XGAP[105X  table) are [10XE(3)[110X, [10XE(4)[110X, [10XE(6)^5[110X ([10X= E(2)*E(3)[110X), and [10XE(12)^7[110X ([10X=
              E(3)*E(4)[110X), for [22Xm = 3[122X, [22X4[122X, [22X6[122X, and [22X12[122X, respectively.[133X
  
  [31X6[131X   [33X[0;6YFor   tables   of  bicyclic  extensions  [22Xm.G.a[122X,  both  the  rules  for
        automorphic  and  central  extensions  hold.  Additionally we have the
        following three rules.[133X
  
        [30X    [33X[0;12YWhenever  classes of the subgroup [22Xm.G[122X collapse in [22Xm.G.a[122X then the
              resulting class replaces the first involved class.[133X
  
        [30X    [33X[0;12YWhenever  characters  of the subgroup [22Xm.G[122X collapse fuse in [22Xm.G.a[122X
              then the result character replaces the first involved character.[133X
  
        [30X    [33X[0;12YExtensions  of a character are subsequent, and the extensions of
              a proxy character precede the extensions of characters with this
              proxy that are not printed.[133X
  
        [30X    [33X[0;12YPreimages  of  a  class  of [22XG.a[122X in [22Xm.G.a[122X are subsequent, and the
              preimages  of a proxy class precede the preimages of non-printed
              classes with this proxy.[133X
  
  [1X4.3-6 AtlasLabelsOfIrreducibles[101X
  
  [33X[1;0Y[29X[2XAtlasLabelsOfIrreducibles[102X( [3Xtbl[103X[, [3Xshort[103X] ) [32X function[133X
  
  [33X[0;0YLet  [3Xtbl[103X be the (ordinary or Brauer) character table of a bicyclic extension
  of  a simple group that occurs in the [5XAtlas[105X of Finite Groups [CCN+85] or the
  [5XAtlas[105X  of  Brauer  Characters [JLPW95].  [2XAtlasLabelsOfIrreducibles[102X returns a
  list  of  strings,  the  [22Xi[122X-th  entry  being a label for the [22Xi[122X-th irreducible
  character of [3Xtbl[103X.[133X
  
  [33X[0;0YThe  labels  have  the  following form. We state the rules only for ordinary
  characters,  the  rules for Brauer characters are obtained by replacing [22Xχ[122X by
  [22Xφ[122X.[133X
  
  [33X[0;0YFirst  consider  only  downward extensions [22Xm.G[122X of a simple group [22XG[122X. If [22Xm ≤ 2[122X
  then  only  labels  of  the  form [22Xχ_i[122X occur, which denotes the [22Xi[122X-th ordinary
  character shown in the [5XAtlas[105X.[133X
  
  [33X[0;0YThe  labels  of faithful ordinary characters of groups [22Xm.G[122X with [22Xm ≥ 3[122X are of
  the  form [22Xχ_i[122X, [22Xχ_i^*[122X, or [22Xχ_i^{*k}[122X, which means the [22Xi[122X-th character printed in
  the  [5XAtlas[105X,  the unique character that is not printed and for which [22Xχ_i[122X acts
  as  proxy (see [CCN+85, Chapter 7, Sections 8 and 19]), and the image of the
  printed   character   [22Xχ_i[122X   under   the  algebraic  conjugacy  operator  [22X*k[122X,
  respectively.[133X
  
  [33X[0;0YFor  groups  [22Xm.G.a[122X  with [22Xa > 1[122X, the labels of the irreducible characters are
  derived   from   the   labels  of  the  irreducible  constituents  of  their
  restrictions to [22Xm.G[122X, as follows.[133X
  
  [31X1[131X   [33X[0;6YIf the ordinary irreducible character [22Xχ_i[122X of [22Xm.G[122X extends to [22Xm.G.a[122X then
        the  [22Xa^'[122X  extensions  are denoted by [22Xχ_{i,0}, χ_{i,1}, ..., χ_{i,a^'}[122X,
        where [22Xχ_{i,0}[122X is the character whose values are printed in the [5XAtlas[105X.[133X
  
  [31X2[131X   [33X[0;6YThe  label  [22Xχ_{i_1  + i_2 + ⋯ + i_a}[122X means that [22Xa[122X different characters
        [22Xχ_{i_1},  χ_{i_2},  ...,  χ_{i_a}[122X  of  [22Xm.G[122X  induce  to  an irreducible
        character of [22Xm.G.a[122X with this label.[133X
  
        [33X[0;6YIf either [9Xtrue[109X or the string [10X"short"[110X is entered as the second argument
        then  the  label has the short form [22Xχ_{i_1+}[122X. Note that [22Xi_2, i_3, ...,
        i_a[122X can be read off from the fusion signs in the [5XAtlas[105X.[133X
  
  [31X3[131X   [33X[0;6YFinally,  the  label [22Xχ_{i_1,j_1 + i_2,j_2 + ⋯ + i_{a^'},j_{a^'}}[122X means
        that  the  characters [22Xχ_{i_1}, χ_{i_2}, ..., χ_{i_{a^'}}[122X of [22Xm.G[122X extend
        to  a  group  that  lies  properly  between  [22Xm.G[122X  and  [22Xm.G.a[122X,  and the
        extensions  [22Xχ_{i_1, j_1}, χ_{i_2, j_2}, ... χ_{i_{a^'},j_{a^'}}[122X induce
        to an irreducible character of [22Xm.G.a[122X with this label.[133X
  
        [33X[0;6YIf  [9Xtrue[109X or the string [10X"short"[110X was entered as the second argument then
        the label has the short form [22Xχ_{i,j+}[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAtlasLabelsOfIrreducibles( CharacterTable( "3.A7.2" ) );[127X[104X
    [4X[28X[ "\\chi_{1,0}", "\\chi_{1,1}", "\\chi_{2,0}", "\\chi_{2,1}", [128X[104X
    [4X[28X  "\\chi_{3+4}", "\\chi_{5,0}", "\\chi_{5,1}", "\\chi_{6,0}", [128X[104X
    [4X[28X  "\\chi_{6,1}", "\\chi_{7,0}", "\\chi_{7,1}", "\\chi_{8,0}", [128X[104X
    [4X[28X  "\\chi_{8,1}", "\\chi_{9,0}", "\\chi_{9,1}", "\\chi_{17+17\\ast 2}",[128X[104X
    [4X[28X  "\\chi_{18+18\\ast 2}", "\\chi_{19+19\\ast 2}", [128X[104X
    [4X[28X  "\\chi_{20+20\\ast 2}", "\\chi_{21+21\\ast 2}", [128X[104X
    [4X[28X  "\\chi_{22+23\\ast 8}", "\\chi_{22\\ast 8+23}" ][128X[104X
    [4X[25Xgap>[125X [27XAtlasLabelsOfIrreducibles( CharacterTable( "3.A7.2" ), "short" );[127X[104X
    [4X[28X[ "\\chi_{1,0}", "\\chi_{1,1}", "\\chi_{2,0}", "\\chi_{2,1}", [128X[104X
    [4X[28X  "\\chi_{3+}", "\\chi_{5,0}", "\\chi_{5,1}", "\\chi_{6,0}", [128X[104X
    [4X[28X  "\\chi_{6,1}", "\\chi_{7,0}", "\\chi_{7,1}", "\\chi_{8,0}", [128X[104X
    [4X[28X  "\\chi_{8,1}", "\\chi_{9,0}", "\\chi_{9,1}", "\\chi_{17+}", [128X[104X
    [4X[28X  "\\chi_{18+}", "\\chi_{19+}", "\\chi_{20+}", "\\chi_{21+}", [128X[104X
    [4X[28X  "\\chi_{22+}", "\\chi_{23+}" ][128X[104X
  [4X[32X[104X
  
  
  [1X4.3-7 [33X[0;0YExamples of the [5XAtlas[105X[101X[1X Format for [5XGAP[105X[101X[1X Tables[133X[101X
  
  [33X[0;0YWe  give  three  little  examples for the conventions stated in Section [14X4.3[114X,
  listing both the [5XAtlas[105X format and the table displayed by [5XGAP[105X.[133X
  
  [33X[0;0YFirst, let [22XG[122X be the trivial group. We consider the cyclic group [22XC_6[122X of order
  [22X6[122X. It can be viewed in several ways, namely[133X
  
  [30X    [33X[0;6Yas  a downward extension of the factor group [22XC_2[122X which contains [22XG[122X as a
        subgroup,  or equivalently, as an upward extension of the subgroup [22XC_3[122X
        which has a factor group isomorphic to [22XG[122X:[133X
  
  [4X[32X[104X
    [4X┌───────┐ ┌───────┐       ;   @   ;   ;   @      2   1   1   1   1   1   1[104X
    [4X│       │ │       │           1           1      3   1   1   1   1   1   1[104X
    [4X│   G   │ │  G.2  │     p power           A[104X
    [4X│       │ │       │     p' part           A         1a  3a  3b  2a  6a  6b[104X
    [4X└───────┘ └───────┘     ind  1A fus ind  2A     2P  1a  3b  3a  1a  3b  3a[104X
    [4X┌───────┐ ┌───────┐                             3P  1a  1a  1a  2a  2a  2a[104X
    [4X│       │ │       │  χ_1  +   1   :  ++   1[104X
    [4X│  3.G  │ │ 3.G.2 │                           X.1    1   1   1   1   1   1[104X
    [4X│       │ │       │     ind   1 fus ind   2   X.2    1   1   1  -1  -1  -1[104X
    [4X└───────┘ └───────┘           3           6   X.3    1   A  /A   1   A  /A[104X
    [4X                              3           6   X.4    1   A  /A  -1  -A -/A[104X
    [4X                                              X.5    1  /A   A   1  /A   A[104X
    [4X                     χ_2 o2   1   : oo2   1   X.6    1  /A   A  -1 -/A  -A[104X
    [4X[104X
    [4X                                              A = E(3)[104X
    [4X                                                = (-1+ER(-3))/2 = b3[104X
  [4X[32X[104X
  
  [8X[108X
        [33X[0;6Y[10XX.1[110X, [10XX.2[110X extend [22Xχ_1[122X. [10XX.3[110X, [10XX.4[110X extend the proxy character [22Xχ_2[122X. [10XX.5[110X, [10XX.6[110X
        extend  the  not printed character with proxy [22Xχ_2[122X. The classes [10X1a[110X, [10X3a[110X,
        [10X3b[110X are preimages of [10X1A[110X, and [10X2a[110X, [10X6a[110X, [10X6b[110X are preimages of [10X2A[110X.[133X
  
  [30X    [33X[0;6Yas  a downward extension of the factor group [22XC_3[122X which contains [22XG[122X as a
        subgroup,  or equivalently, as an upward extension of the subgroup [22XC_2[122X
        which has a factor group isomorphic to [22XG[122X:[133X
  
  [4X[32X[104X
    [4X┌───────┐ ┌───────┐       ;   @   ;   ;   @      2   1   1   1   1   1   1[104X
    [4X│       │ │       │           1           1      3   1   1   1   1   1   1[104X
    [4X│   G   │ │  G.3  │     p power           A[104X
    [4X│       │ │       │     p' part           A         1a  2a  3a  6a  3b  6b[104X
    [4X└───────┘ └───────┘     ind  1A fus ind  3A     2P  1a  1a  3b  3b  3a  3a[104X
    [4X┌───────┐ ┌───────┐                             3P  1a  2a  1a  2a  1a  2a[104X
    [4X│       │ │       │  χ_1  +   1   : +oo   1[104X
    [4X│  2.G  │ │ 2.G.3 │                           X.1    1   1   1   1   1   1[104X
    [4X│       │ │       │     ind   1 fus ind   3   X.2    1   1   A   A  /A  /A[104X
    [4X└───────┘ └───────┘           2           6   X.3    1   1  /A  /A   A   A[104X
    [4X                                              X.4    1  -1   1  -1   1  -1[104X
    [4X                     χ_2  +   1   : +oo   1   X.5    1  -1   A  -A  /A -/A[104X
    [4X                                              X.6    1  -1  /A -/A   A  -A[104X
    [4X[104X
    [4X                                              A = E(3)[104X
    [4X                                                = (-1+ER(-3))/2 = b3[104X
  [4X[32X[104X
  
  [8X[108X
        [33X[0;6Y[10XX.1[110X  to  [10XX.3[110X  extend [22Xχ_1[122X, [10XX.4[110X to [10XX.6[110X extend [22Xχ_2[122X. The classes [10X1a[110X and [10X2a[110X
        are  preimages  of  [10X1A[110X, [10X3a[110X and [10X6a[110X are preimages of the proxy class [10X3A[110X,
        and [10X3b[110X and [10X6b[110X are preimages of the not printed class with proxy [10X3A[110X.[133X
  
  [30X    [33X[0;6Yas  a downward extension of the factor groups [22XC_3[122X and [22XC_2[122X which have [22XG[122X
        as a factor group:[133X
  
  [4X[32X[104X
    [4X┌───────┐           ;   @        2   1   1   1   1   1   1[104X
    [4X│       │               1        3   1   1   1   1   1   1[104X
    [4X│   G   │         p power[104X
    [4X│       │         p' part           1a  6a  3a  2a  3b  6b[104X
    [4X└───────┘         ind  1A       2P  1a  3a  3b  1a  3a  3b[104X
    [4X┌───────┐                       3P  1a  2a  1a  2a  1a  2a[104X
    [4X│       │      χ_1  +   1[104X
    [4X│  2.G  │                     X.1    1   1   1   1   1   1[104X
    [4X│       │         ind   1     X.2    1  ─1   1  ─1   1  ─1[104X
    [4X└───────┘               2     X.3    1   A  /A   1   A  /A[104X
    [4X┌───────┐                     X.4    1  /A   A   1  /A   A[104X
    [4X│       │      χ_2  +   1     X.5    1  ─A  /A  ─1   A ─/A[104X
    [4X│  3.G  │                     X.6    1 ─/A   A  ─1  /A  ─A[104X
    [4X│       │         ind   1[104X
    [4X└───────┘               3     A = E(3)[104X
    [4X┌───────┐               3       = (─1+ER(─3))/2 = b3[104X
    [4X│       │[104X
    [4X│  6.G  │      χ_3 o2   1[104X
    [4X│       │[104X
    [4X└───────┘         ind   1[104X
    [4X                        6[104X
    [4X                        3[104X
    [4X                        2[104X
    [4X                        3[104X
    [4X                        6[104X
    [4X[104X
    [4X               χ_4 o2   1[104X
  [4X[32X[104X
  
  [8X[108X
        [33X[0;6Y[10XX.1[110X,  [10XX.2[110X correspond to [22Xχ_1, χ_2[122X, respectively; [10XX.3[110X, [10XX.5[110X correspond to
        the  proxies [22Xχ_3[122X, [22Xχ_4[122X, and [10XX.4[110X, [10XX.6[110X to the not printed characters with
        these proxies. The factor fusion onto [22X3.G[122X is given by [10X[ 1, 2, 3, 1, 2,
        3 ][110X, that onto [22XG.2[122X by [10X[ 1, 2, 1, 2, 1, 2 ][110X.[133X
  
  [30X    [33X[0;6Yas  an upward extension of the subgroups [22XC_3[122X or [22XC_2[122X which both contain
        a subgroup isomorphic to [22XG[122X:[133X
  
  [4X[32X[104X
    [4X┌───────┐ ┌───────┐ ┌───────┐ ┌───────┐[104X
    [4X│       │ │       │ │       │ │       │[104X
    [4X│   G   │ │  G.2  │ │  G.3  │ │  G.6  │[104X
    [4X│       │ │       │ │       │ │       │[104X
    [4X└───────┘ └───────┘ └───────┘ └───────┘[104X
    [4X[104X
    [4X     ;   @   ;   ;   @   ;   ;   @   ;     ;   @[104X
    [4X[104X
    [4X         1           1           1             1[104X
    [4X   p power           A           A            AA[104X
    [4X   p' part           A           A            AA[104X
    [4X   ind  1A fus ind  2A fus ind  3A fus   ind  6A[104X
    [4X[104X
    [4Xχ_1  +   1   :  ++   1   : +oo   1   :+oo+oo   1[104X
    [4X[104X
    [4X[104X
    [4X    2   1   1   1   1   1   1[104X
    [4X    3   1   1   1   1   1   1[104X
    [4X[104X
    [4X       1a  2a  3a  3b  6a  6b[104X
    [4X   2P  1a  1a  3b  3a  3b  3a[104X
    [4X   3P  1a  2a  1a  1a  2a  2a[104X
    [4X X.1    1   1   1   1   1   1[104X
    [4X X.2    1  -1   A  /A  -A -/A[104X
    [4X X.3    1   1  /A   A  /A   A[104X
    [4X X.4    1  -1   1   1  -1  -1[104X
    [4X X.5    1   1   A  /A   A  /A[104X
    [4X X.6    1  -1  /A   A -/A  -A[104X
    [4X[104X
    [4X A = E(3)[104X
    [4X   = (-1+ER(-3))/2 = b3[104X
  [4X[32X[104X
  
  [8X[108X
        [33X[0;6YThe  classes  [10X1a[110X,  [10X2a[110X  correspond  to  [22X1A[122X,  [22X2A[122X,  respectively.  [10X3a[110X, [10X6a[110X
        correspond  to  the  proxies  [22X3A[122X,  [22X6A[122X,  and  [10X3b[110X, [10X6b[110X to the not printed
        classes with these proxies.[133X
  
  [33X[0;0YThe second example explains the fusion case. Again, [22XG[122X is the trivial group.[133X
  
  [4X[32X[104X
    [4X┌───────┐ ┌───────┐        ;   @   ;   ;  @      3.G.2[104X
    [4X│       │ │       │            1          1[104X
    [4X│   G   │ │  G.2  │      p power          A         2   1   .   1[104X
    [4X│       │ │       │      p' part          A         3   1   1   .[104X
    [4X└───────┘ └───────┘      ind  1A fus ind 2A[104X
    [4X┌───────┐ ┌───────┐                                    1a 3a 2a[104X
    [4X│       │ │       │   χ_1  +   1   :  ++  1        2P  1a 3a 1a[104X
    [4X│  2.G  │ │ 2.G.2 │                                3P  1a 1a 2a[104X
    [4X│       │ │       │      ind   1 fus ind  2[104X
    [4X└───────┘ └───────┘            2          2      X.1    1  1  1[104X
    [4X┌───────┐ ┌───────                               X.2    1  1 ─1[104X
    [4X│       │ │           χ_2  +   1   :  ++  1      X.3    2 ─1  .[104X
    [4X│  3.G  │ │ 3.G.2[104X
    [4X│       │ │              ind   1 fus ind  2[104X
    [4X└───────┘                      3                 6.G.2[104X
    [4X┌───────┐ ┌───────             3[104X
    [4X│       │ │                                         2   2  1  1  2  2  2[104X
    [4X│  6.G  │ │ 6.G.2     χ_3 o2   1   *   +            3   1  1  1  1  .  .[104X
    [4X│       │ │[104X
    [4X└───────┘                ind   1 fus ind  2            1a 6a 3a 2a 2b 2c[104X
    [4X                               6          2        2P  1a 3a 3a 1a 1a 1a[104X
    [4X                               3                   3P  1a 2a 1a 2a 2b 2c[104X
    [4X                               2[104X
    [4X                               3                 Y.1    1  1  1  1  1  1[104X
    [4X                               6                 Y.2    1  1  1  1 -1 -1[104X
    [4X                                                 Y.3    1 -1  1 -1  1 -1[104X
    [4X                      χ_4 o2   1   *   +         Y.4    1 -1  1 -1 -1  1[104X
    [4X                                                 Y.5    2 -1 -1  2  .  .[104X
    [4X                                                 Y.6    2  1 -1 -2  .  .[104X
  [4X[32X[104X
  
  [33X[0;0YThe  tables  of  [22XG[122X,  [22X2.G[122X, [22X3.G[122X, [22X6.G[122X and [22XG.2[122X are known from the first example,
  that  of  [22X2.G.2[122X will be given in the next one. So here we print only the [5XGAP[105X
  tables of [22X3.G.2 ≅ D_6[122X and [22X6.G.2 ≅ D_12[122X.[133X
  
  [33X[0;0YIn  [22X3.G.2[122X,  the  characters  [10XX.1[110X,  [10XX.2[110X  extend  [22Xχ_1[122X; [22Xχ_3[122X and its non-printed
  partner fuse to give [10XX.3[110X, and the two preimages of [10X1A[110X of order [22X3[122X collapse.[133X
  
  [33X[0;0YIn [22X6.G.2[122X, [10XY.1[110X to [10XY.4[110X are extensions of [22Xχ_1[122X, [22Xχ_2[122X, so these characters are the
  inflated characters from [22X2.G.2[122X (with respect to the factor fusion [10X[ 1, 2, 1,
  2,  3, 4 ][110X). [10XY.5[110X is inflated from [22X3.G.2[122X (with respect to the factor fusion [10X[
  1,  2,  2,  1,  3,  3 ][110X), and [10XY.6[110X is the result of the fusion of [22Xχ_4[122X and its
  non-printed partner.[133X
  
  [33X[0;0YFor  the last example, let [22XG[122X be the elementary abelian group [22X2^2[122X of order [22X4[122X.
  Consider the following tables.[133X
  
  [4X[32X[104X
    [4X┌───────┐ ┌───────┐          ;   @   @   @   @   ;   ;   @[104X
    [4X│       │ │       │              4   4   4   4           1[104X
    [4X│   G   │ │  G.3  │        p power   A   A   A           A[104X
    [4X│       │ │       │        p' part   A   A   A           A[104X
    [4X└───────┘ └───────┘        ind  1A  2A  2B  2C fus ind  3A[104X
    [4X┌───────┐ ┌───────┐[104X
    [4X│       │ │       │     χ_1  +   1   1   1   1   : +oo   1[104X
    [4X│  2.G  │ │ 2.G.3 │     χ_2  +   1   1  ─1  ─1   .   +   0[104X
    [4X│       │ │       │     χ_3  +   1  ─1   1  ─1   |[104X
    [4X└───────┘ └───────┘     χ_4  +   1  ─1  ─1   1   |[104X
    [4X[104X
    [4X                           ind   1   4   4   4 fus ind   3[104X
    [4X                                 2                       6[104X
    [4X[104X
    [4X                        χ_5  -   2   0   0   0   : -oo   1[104X
    [4X[104X
    [4X  G.3[104X
    [4X[104X
    [4X     2   2   2   .   .[104X
    [4X     3   1   .   1   1[104X
    [4X[104X
    [4X        1a  2a  3a  3b[104X
    [4X    2P  1a  1a  3b  3a[104X
    [4X    3P  1a  2a  1a  1a[104X
    [4X[104X
    [4X  X.1    1   1   1   1[104X
    [4X  X.2    1   1   A  /A[104X
    [4X  X.3    1   1  /A   A         2.G.3[104X
    [4X  X.4    3  -1   .   .[104X
    [4X                                  2   3   3   2   1   1   1   1[104X
    [4X  A = E(3)                        3   1   1   .   1   1   1   1[104X
    [4X    = (-1+ER(-3))/2 = b3[104X
    [4X                                     1a  2a  4a  3a  6a  3b  6b[104X
    [4X  2.G                            2P  1a  1a  2a  3b  3b  3a  3a[104X
    [4X                                 3P  1a  2a  4a  1a  2a  1a  2a[104X
    [4X     2   3   3   2   2   2[104X
    [4X                               X.1    1   1   1   1   1   1   1[104X
    [4X        1a  2a  4a  4b  4c     X.2    1   1   1   A   A  /A  /A[104X
    [4X    2P  1a  1a  2a  1a  1a     X.3    1   1   1  /A  /A   A   A[104X
    [4X    3P  1a  2a  4a  4b  4c     X.4    3   3  -1   .   .   .   .[104X
    [4X                               X.5    2  -2   .   1   1   1   1[104X
    [4X  X.1    1   1   1   1   1     X.6    2  -2   .   A  -A  /A -/A[104X
    [4X  X.2    1   1   1  -1  -1     X.7    2  -2   .  /A -/A   A  -A[104X
    [4X  X.3    1   1  -1   1  -1[104X
    [4X  X.4    1   1  -1  -1   1     A = E(3)[104X
    [4X  X.5    2  -2   .   .   .       = (-1+ER(-3))/2 = b3[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the  table  of [22XG.3 ≅ A_4[122X, the characters [22Xχ_2[122X, [22Xχ_3[122X, and [22Xχ_4[122X fuse, and the
  classes [10X2A[110X, [10X2B[110X and [10X2C[110X collapse. For getting the table of [22X2.G ≅ Q_8[122X, one just
  has to split the class [10X2A[110X and adjust the representative orders. Finally, the
  table  of  [22X2.G.3  ≅  SL_2(3)[122X is given; the class fusion corresponding to the
  injection  [22X2.G  ↪  2.G.3[122X  is  [10X[  1,  2,  3,  3,  3  ][110X, and the factor fusion
  corresponding to the epimorphism [22X2.G.3 → G.3[122X is [10X[ 1, 1, 2, 3, 3, 4, 4 ][110X.[133X
  
  
  [1X4.4 [33X[0;0Y[5XCAS[105X[101X[1X Tables[133X[101X
  
  [33X[0;0YOne  of  the  predecessors  of  [5XGAP[105X  was  [5XCAS[105X  ([13XC[113Xharacter  [13XA[113Xlgorithm [13XS[113Xystem,
  see [NPP84]),  which  had  also  a  library  of  character tables. All these
  character tables are available in [5XGAP[105X except if stated otherwise in the file
  [11Xdoc/ctbldiff.pdf[111X.  This sublibrary has been completely revised before it was
  included in [5XGAP[105X, for example, errors have been corrected and power maps have
  been completed.[133X
  
  [33X[0;0YAny  [5XCAS[105X  table  is  accessible  by  each of its [5XCAS[105X names (except if stated
  otherwise in [11Xdoc/ctbldiff.pdf[111X), that is, the table name or the filename used
  in [5XCAS[105X.[133X
  
  [1X4.4-1 CASInfo[101X
  
  [33X[1;0Y[29X[2XCASInfo[102X( [3Xtbl[103X ) [32X attribute[133X
  
  [33X[0;0YLet  [3Xtbl[103X  be  an ordinary character table in the [5XGAP[105X library that was (up to
  permutations  of  classes and characters) contained already in the [5XCAS[105X table
  library.  When one fetches [3Xtbl[103X from the library, one does in general not get
  the  original  [5XCAS[105X  table.  Namely,  in many cases (mostly [5XAtlas[105X tables, see
  Section [14X4.3[114X),  the  identifier  of  the  table  (see [2XIdentifier[102X  ([14XReference:
  Identifier  for  character  tables[114X))  as well as the ordering of classes and
  characters are different for the [5XCAS[105X table and its [5XGAP[105X version.[133X
  
  [33X[0;0YNote that in several cases, the [5XCAS[105X library contains different tables of the
  same  group,  in  particular  these  tables  may  have  different  names and
  orderings of classes and characters.[133X
  
  [33X[0;0YThe  [2XCASInfo[102X  value of [3Xtbl[103X, if stored, is a list of records, each describing
  the  relation  between  [3Xtbl[103X  and  a  character table in the [5XCAS[105X library. The
  records have the components[133X
  
  [8X[10Xname[110X[8X[108X
        [33X[0;6Ythe name of the [5XCAS[105X table,[133X
  
  [8X[10Xpermchars[110X[8X and [10Xpermclasses[110X[8X[108X
        [33X[0;6Ypermutations  of  the  [2XIrr[102X  ([14XReference: Irr[114X) values and the classes of
        [3Xtbl[103X,  respectively, that must be applied in order to get the orderings
        in the original [5XCAS[105X table, and[133X
  
  [8X[10Xtext[110X[8X[108X
        [33X[0;6Ythe text that was stored on the [5XCAS[105X table (which may contain incorrect
        statements).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "m10" );[127X[104X
    [4X[28XCharacterTable( "A6.2_3" )[128X[104X
    [4X[25Xgap>[125X [27XHasCASInfo( tbl );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XCASInfo( tbl );[127X[104X
    [4X[28X[ rec( name := "m10", permchars := (3,5)(4,8,7,6), permclasses := (), [128X[104X
    [4X[28X      text := "names:     m10\norder:     2^4.3^2.5 = 720\nnumber of c\[128X[104X
    [4X[28Xlasses: 8\nsource:    cambridge atlas\ncomments:  point stabilizer of \[128X[104X
    [4X[28Xmathieu-group m11\ntest:      orth, min, sym[3]\n" ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  class  fusions stored on tables from the [5XCAS[105X library have been computed
  anew  with  [5XGAP[105X;  the  [10Xtext[110X  component  of such a fusion record tells if the
  fusion  map  is  equal  to that in the [5XCAS[105X library, up to the permutation of
  classes between the table in [5XCAS[105X and its [5XGAP[105X version.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFirst( ComputedClassFusions( tbl ), x -> x.name = "M11" );[127X[104X
    [4X[28Xrec( map := [ 1, 2, 3, 4, 5, 4, 7, 8 ], name := "M11", [128X[104X
    [4X[28X  text := "fusion is unique up to table automorphisms,\nthe representa\[128X[104X
    [4X[28Xtive is equal to the fusion map on the CAS table" )[128X[104X
  [4X[32X[104X
  
  
  [1X4.5 [33X[0;0YCustomizations of the [5XGAP[105X[101X[1X Character Table Library[133X[101X
  
  
  [1X4.5-1 [33X[0;0YInstalling the [5XGAP[105X[101X[1X Character Table Library[133X[101X
  
  [33X[0;0YTo  install  the  package  unpack the archive file in a directory in the [11Xpkg[111X
  directory  of  your  local copy of [5XGAP[105X 4. This might be the [11Xpkg[111X directory of
  the  [5XGAP[105X 4 home directory, see Section [14X'Reference: Installing a GAP Package'[114X
  for details. It is however also possible to keep an additional [11Xpkg[111X directory
  in  your  private  directories,  see [14X'Reference:  GAP Root Directories'[114X. The
  latter possibility [13Xmust[113X be chosen if you do not have write access to the [5XGAP[105X
  root directory.[133X
  
  [33X[0;0YThe  package  consists entirely of [5XGAP[105X code, no external binaries need to be
  compiled.[133X
  
  [33X[0;0YFor checking the installation of the package, you should start [5XGAP[105X and call[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XReadPackage( "ctbllib", "tst/testinst.g" );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  the  installation  is  o. k.  then  [9Xtrue[109X  is printed, and the [5XGAP[105X prompt
  appears again; otherwise the output lines tell you what should be changed.[133X
  
  [33X[0;0YMore testfiles are available in the [11Xtst[111X directory of the package.[133X
  
  [33X[0;0YPDF  and  HTML  versions  of  the  package  manual  are available in the [11Xdoc[111X
  directory of the package.[133X
  
  
  [1X4.5-2 [33X[0;0YUnloading Character Table Data[133X[101X
  
  [33X[0;0YData files from the [5XGAP[105X Character Table Library may be read only once during
  a [5XGAP[105X session –this is efficient but requires memory– or the cached data may
  be  erased  as  soon as a second data file is to be read –this requires less
  memory but is usually less efficient.[133X
  
  [33X[0;0YOne  can  choose  between  these  two  possibilities via the user preference
  [10X"UnloadCTblLibFiles"[110X  of the [5XCTblLib[105X package, see [2XUserPreference[102X ([14XReference:
  UserPreference[114X).  The default value of this preference is [9Xtrue[109X, that is, the
  contents  of  only  one data file is kept in memory. Call [10XSetUserPreference(
  "CTblLib",  "UnloadCTblLibFiles",  false  );[110X  if  you  want  to  change this
  behaviour.[133X
  
  
  [1X4.5-3 [33X[0;0YChanging the display format of several functions[133X[101X
  
  [33X[0;0YThe  way  how  the  functions [2XDisplayAtlasContents[102X ([14X3.5-6[114X), [2XDisplayAtlasInfo[102X
  ([14XAtlasRep:      DisplayAtlasInfo[114X),      [2XDisplayAtlasMap[102X     ([14X3.5-8[114X),     and
  [2XDisplayCTblLibInfo[102X  ([14X3.5-1[114X)  show  tabular information can be customized via
  the  user  preference [10X"DisplayFunction"[110X of the [5XAtlasRep[105X package, see Section
  [14X'AtlasRep: User preference DisplayFunction'[114X.[133X
  
  
  [1X4.5-4 [33X[0;0YThe path for calling [5XMAGMA[105X[101X[1X[133X[101X
  
  [33X[0;0YThis  preference  describes the path for calling [5XMAGMA[105X. An empty string (the
  default) means that Magma is not available.[133X
  
  [33X[0;0YThis preference is used by [2XCharacterTableComputedByMagma[102X ([14X6.5-3[114X).[133X
  
  
  [1X4.6 [33X[0;0YTechnicalities of the Access to Character Tables from the Library[133X[101X
  
  
  [1X4.6-1 [33X[0;0YData Files of the [5XGAP[105X[101X[1X Character Table Library[133X[101X
  
  [33X[0;0YThe  data  files  of  the  [5XGAP[105X  Character  Table  Library reside in the [11Xdata[111X
  directory of the package [5XCTblLib[105X.[133X
  
  [33X[0;0YThe filenames start with [10Xct[110X (for [21Xcharacter table[121X), followed by either [10Xo[110X (for
  [21Xordinary[121X),  [10Xb[110X  (for  [21XBrauer[121X),  or [10Xg[110X (for [21Xgeneric[121X), then a description of the
  contents (up to [22X5[122X characters, e. g., [10Xalter[110X for the tables of alternating and
  related groups), and the suffix [10X.tbl[110X.[133X
  
  [33X[0;0YThe  file [10Xctb[110X[22Xdescr[122X[10X.tbl[110X contains the known Brauer tables corresponding to the
  ordinary tables in the file [10Xcto[110X[22Xdescr[122X[10X.tbl[110X.[133X
  
  [33X[0;0YEach data file of the table library is supposed to consist of[133X
  
  [31X1[131X   [33X[0;6Ycomment lines, starting with a hash character [10X#[110X in the first column,[133X
  
  [31X2[131X   [33X[0;6Yan assignment to a component of [10XLIBTABLE.LOADSTATUS[110X, at the end of the
        file, and[133X
  
  [31X3[131X   [33X[0;6Yfunction calls of the form[133X
  
        [30X    [33X[0;12Y[10XMBT( [110X[22Xname, data[122X[10X )[110X ([21Xmake Brauer table[121X),[133X
  
        [30X    [33X[0;12Y[10XMOT( [110X[22Xname, data[122X[10X )[110X ([21Xmake ordinary table[121X),[133X
  
        [30X    [33X[0;12Y[10XALF(  [110X[22Xfrom,  to,  map[122X[10X  )[110X,  [10XALF(  [110X[22Xfrom, to, map, textlines[122X[10X )[110X ([21Xadd
              library fusion[121X),[133X
  
        [30X    [33X[0;12Y[10XALN( [110X[22Xname, listofnames[122X[10X )[110X ([21Xadd library name[121X), and[133X
  
        [30X    [33X[0;12Y[10XARC( [110X[22Xname, component, compdata[122X[10X )[110X ([21Xadd record component[121X).[133X
  
        [33X[0;6YHere [22Xname[122X must be the identifier value of the ordinary character table
        corresponding to the table to which the command refers; [22Xdata[122X must be a
        comma  separated  sequence  of  [5XGAP[105X  objects;  [22Xfrom[122X  and  [22Xto[122X  must  be
        identifier values of ordinary character tables, [22Xmap[122X a list of positive
        integers, [22Xtextlines[122X and [22Xlistofnames[122X lists list of strings, [22Xcomponent[122X a
        string, and [22Xcompdata[122X any [5XGAP[105X object.[133X
  
        [33X[0;6Y[10XMOT[110X,  [10XALF[110X,  [10XALN[110X,  and  [10XARC[110X  occur  only  in  files containing ordinary
        character  tables,  and  [10XMBT[110X  occurs  only  in files containing Brauer
        tables.[133X
  
  [33X[0;0YBesides  the  above  calls, the data in files containing ordinary and Brauer
  tables  may  contain  only  the  following  [5XGAP[105X functions. (Files containing
  generic  character  tables  may  contain  calls  to  arbitrary  [5XGAP[105X  library
  functions.)[133X
  
  [33X[0;0Y[10XACM[110X,  [2XConcatenation[102X  ([14XReference:  concatenation of lists[114X), [2XE[102X ([14XReference: E[114X),
  [10XEvalChars[110X,  [10XGALOIS[110X,  [2XLength[102X  ([14XReference:  Length[114X),  [2XShallowCopy[102X  ([14XReference:
  ShallowCopy[114X), [10XTENSOR[110X, and [2XTransposedMat[102X ([14XReference: TransposedMat[114X).[133X
  
  [33X[0;0YThe  function [10XCTblLib.RecomputeTOC[110X in the file [11Xgap4/maketbl.g[111X of the [5XCTblLib[105X
  package  expects  the  file  format  described  above, and to some extent it
  checks this format.[133X
  
  [33X[0;0YThe  function  calls  may  be  continued  over  several  lines  of a file. A
  semicolon  is assumed to be the last character in its line if and only if it
  terminates a function call.[133X
  
  [33X[0;0YNames  of  character tables are strings (see Chapter [14X'Reference: Strings and
  Characters'[114X),  i. e.,  they  are enclosed in double quotes; [13Xstrings in table
  library  files  must  not be split over several lines[113X, because otherwise the
  function  [10XCTblLib.RecomputeTOC[110X  may get confused. Additionally, no character
  table name is allowed to contain double quotes.[133X
  
  [33X[0;0YThere are three different ways how the table data can be stored in the file.[133X
  
  [8XFull ordinary tables[108X
        [33X[0;6Yare  encoded  by  a  call  to  the  function  [10XMOT[110X, where the arguments
        correspond  to the relevant attribute values; each fusion into another
        library  table  is  added  by  a  call  to [10XALF[110X, values to be stored in
        components  of  the  table  object  are added with [10XARC[110X, and admissible
        names  are  notified  with  [10XALN[110X.  The argument of [10XMOT[110X that encodes the
        irreducible  characters  is abbreviated as follows. For each subset of
        characters  that differ just by multiplication with a linear character
        or by Galois conjugacy, only the first one is given by its values, the
        others  are replaced by [10X[TENSOR,[i,j]][110X (which means that the character
        is  the tensor product of the [10Xi[110X-th and the [10Xj[110X-th character in the list)
        or [10X[GALOIS,[i,j]][110X (which means that the character is obtained from the
        [10Xi[110X-th character by applying [10XGaloisCyc( ., j )[110X to it).[133X
  
  [8XBrauer tables[108X
        [33X[0;6Yare  stored  relative  to the corresponding ordinary tables; attribute
        values  that can be computed by restricting from the ordinary table to
        [22Xp[122X-regular  classes  are  not  stored,  and  instead of the irreducible
        characters  the  files contain (inverses of) decomposition matrices or
        Brauer trees for the blocks of nonzero defect.[133X
  
  [8XOrdinary construction tables[108X
        [33X[0;6Yhave  the  attribute  [2XConstructionInfoCharacterTable[102X ([14X3.7-4[114X) set, with
        value  a list that contains the name of the construction function used
        and  the  arguments  for a call to this function; the function call is
        performed by [2XCharacterTable[102X ([14XReference: CharacterTable[114X) when the table
        is  constructed  ([13Xnot[113X when the file containing the table is read). One
        aim  of this mechanism is to store structured character tables such as
        tables  of  direct  products and tables of central extensions of other
        tables in a compact way, see Chapter [14X5[114X.[133X
  
  [1X4.6-2 LIBLIST[101X
  
  [33X[1;0Y[29X[2XLIBLIST[102X [32X global variable[133X
  
  [33X[0;0Y[5XGAP[105X's  knowledge  about  the  ordinary character tables in the [5XGAP[105X Character
  Table  Library is given by several JSON format files that get evaluated when
  the  file  [11Xgap4/ctprimar.g[111X (the [21Xprimary file[121X of the character table library)
  is read. These files can be produced from the data files, see Section [14X4.6-1[114X.[133X
  
  [33X[0;0YThe  information is stored in the global variable [2XLIBLIST[102X, which is a record
  with the following components.[133X
  
  [8X[10Xfirstnames[110X[8X[108X
        [33X[0;6Ythe  list  of  [2XIdentifier[102X ([14XReference: Identifier for character tables[114X)
        values of the ordinary tables,[133X
  
  [8X[10Xfiles[110X[8X[108X
        [33X[0;6Ythe list of filenames containing the data of ordinary tables,[133X
  
  [8X[10Xfilenames[110X[8X[108X
        [33X[0;6Ya  list  of  positive  integers,  value [22Xj[122X at position [22Xi[122X means that the
        table whose identifier is the [22Xi[122X-th in the [10Xfirstnames[110X list is contained
        in the [22Xj[122X-th file of the [10Xfiles[110X component,[133X
  
  [8X[10Xfusionsource[110X[8X[108X
        [33X[0;6Ya list containing at position [22Xi[122X the list of names of tables that store
        a fusion into the table whose identifier is the [22Xi[122X-th in the [10Xfirstnames[110X
        list,[133X
  
  [8X[10Xallnames[110X[8X[108X
        [33X[0;6Ya list of all admissible names of ordinary library tables,[133X
  
  [8X[10Xposition[110X[8X[108X
        [33X[0;6Ya  list  that  stores  at position [22Xi[122X the position in [10Xfirstnames[110X of the
        identifier of the table with the [22Xi[122X-th admissible name in [10Xallnames[110X,[133X
  
  [8X[10Xsimpleinfo[110X[8X[108X
        [33X[0;6Ya  list  of  triples  [22X[  m,  name, a ][122X describing the tables of simple
        groups in the library; [22Xname[122X is the identifier of the table, [22Xm[122X[10X.[110X[22Xname[122X and
        [22Xname[122X[10X.[110X[22Xa[122X  are admissible names for its Schur multiplier and automorphism
        group, respectively, if these tables are available at all,[133X
  
  [8X[10XsporadicSimple[110X[8X[108X
        [33X[0;6Ya  list of identifiers of the tables of the [22X26[122X sporadic simple groups,
        and[133X
  
  [8X[10XGENERIC[110X[8X[108X
        [33X[0;6Ya record with information about generic tables (see Section [14X4.2[114X).[133X
  
  [1X4.6-3 LibInfoCharacterTable[101X
  
  [33X[1;0Y[29X[2XLibInfoCharacterTable[102X( [3Xtblname[103X ) [32X function[133X
  
  [33X[0;0Yis a record with the components[133X
  
  [8X[10XfirstName[110X[8X[108X
        [33X[0;6Ythe  [2XIdentifier[102X  ([14XReference: Identifier for character tables[114X) value of
        the library table for which [3Xtblname[103X is an admissible name, and[133X
  
  [8X[10XfileName[110X[8X[108X
        [33X[0;6Ythe name of the file in which the table data is stored.[133X
  
  [33X[0;0YIf no such table exists in the [5XGAP[105X library then [9Xfail[109X is returned.[133X
  
  [33X[0;0YIf [3Xtblname[103X contains the substring [10X"mod"[110X then it is regarded as the name of a
  Brauer  table.  In  this  case  the  result  is  computed  from that for the
  corresponding  ordinary  table  and  the  characteristic. So if the ordinary
  table  exists  then  the  result  is  a  record although the Brauer table in
  question need not be contained in the [5XGAP[105X library.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLibInfoCharacterTable( "S5" );[127X[104X
    [4X[28Xrec( fileName := "ctoalter", firstName := "A5.2" )[128X[104X
    [4X[25Xgap>[125X [27XLibInfoCharacterTable( "S5mod2" );[127X[104X
    [4X[28Xrec( fileName := "ctbalter", firstName := "A5.2mod2" )[128X[104X
    [4X[25Xgap>[125X [27XLibInfoCharacterTable( "J5" );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  
  [1X4.7 [33X[0;0YHow to Extend the [5XGAP[105X[101X[1X Character Table Library[133X[101X
  
  [33X[0;0Y[5XGAP[105X  users  may  want  to  extend  the  character table library in different
  respects.[133X
  
  [30X    [33X[0;6YProbably  the easiest change is to [13Xadd new admissible names[113X to library
        tables,  in  order  to  use  these  names  in  calls of [2XCharacterTable[102X
        ([14X3.1-2[114X). This can be done using [2XNotifyNameOfCharacterTable[102X ([14X4.7-1[114X).[133X
  
  [30X    [33X[0;6YThe  next  kind  of  changes  is  the  [13Xaddition of new fusions[113X between
        library  tables.  Once  a  fusion map is known, it can be added to the
        library  file  containing  the table of the subgroup, using the format
        produced by [2XLibraryFusion[102X ([14X4.7-2[114X).[133X
  
  [30X    [33X[0;6YThe  last  kind  of changes is the [13Xaddition of new character tables[113X to
        the  [5XGAP[105X  character  table  library.  Data  files containing tables in
        library  format  (i. e.,  in  the  form of calls to [10XMOT[110X or [10XMBT[110X) can be
        produced using [2XPrintToLib[102X ([14X4.7-4[114X).[133X
  
        [33X[0;6YIf  you  have  an ordinary character table in library format which you
        want  to  add  to  the  table library, for example because it shall be
        accessible  via  [2XCharacterTable[102X  ([14X3.1-2[114X),  you must notify this table,
        i. e.,  tell  [5XGAP[105X in which file it can be found, and which names shall
        be  admissible  for  it.  This  can be done using [2XNotifyCharacterTable[102X
        ([14X4.7-5[114X).[133X
  
  [1X4.7-1 NotifyNameOfCharacterTable[101X
  
  [33X[1;0Y[29X[2XNotifyNameOfCharacterTable[102X( [3Xfirstname[103X, [3Xnewnames[103X ) [32X function[133X
  
  [33X[0;0Ynotifies  the  strings  in the list [3Xnewnames[103X as new admissible names for the
  library  table  with [2XIdentifier[102X ([14XReference: Identifier for character tables[114X)
  value [3Xfirstname[103X. If there is already another library table for which some of
  these names are admissible then an error is signaled.[133X
  
  [33X[0;0Y[2XNotifyNameOfCharacterTable[102X modifies the global variable [2XLIBLIST[102X ([14X4.6-2[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCharacterTable( "private" );[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27XNotifyNameOfCharacterTable( "A5", [ "private" ] );[127X[104X
    [4X[25Xgap>[125X [27Xa5:= CharacterTable( "private" );[127X[104X
    [4X[28XCharacterTable( "A5" )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YOne  can  notify  alternative  names for character tables inside data files,
  using  the  function  [10XALN[110X instead of [2XNotifyNameOfCharacterTable[102X. The idea is
  that  the  additional  names of tables from those files can be ignored which
  are  controlled  by  [10XCTblLib.RecomputeTOC[110X.  Therefore,  [10XALN[110X is set to [10XIgnore[110X
  before  the  file  is  read  with  [10XCTblLib.ReadTbl[110X,  otherwise [10XALN[110X is set to
  [2XNotifyNameOfCharacterTable[102X.[133X
  
  [1X4.7-2 LibraryFusion[101X
  
  [33X[1;0Y[29X[2XLibraryFusion[102X( [3Xname[103X, [3Xfus[103X ) [32X function[133X
  
  [33X[0;0YFor a string [3Xname[103X that is an [2XIdentifier[102X ([14XReference: Identifier for character
  tables[114X)  value  of  an  ordinary  character  table in the [5XGAP[105X library, and a
  record [3Xfus[103X with the components[133X
  
  [8X[10Xname[110X[8X[108X
        [33X[0;6Ythe identifier of the destination table, or this table itself,[133X
  
  [8X[10Xmap[110X[8X[108X
        [33X[0;6Ythe fusion map, a list of image positions,[133X
  
  [8X[10Xtext[110X[8X (optional)[108X
        [33X[0;6Ya string describing properties of the fusion, and[133X
  
  [8X[10Xspecification[110X[8X (optional)[108X
        [33X[0;6Ya string or an integer,[133X
  
  [33X[0;0Y[2XLibraryFusion[102X  returns  a  string whose printed value can be used to add the
  fusion  in  question  to  the library file containing the data for the table
  with identifier [3Xname[103X.[133X
  
  [33X[0;0YIf  [3Xname[103X is a character table then its [2XIdentifier[102X ([14XReference: Identifier for
  character tables[114X) value is used as the corresponding string.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs5:= CharacterTable( "S5" );[127X[104X
    [4X[28XCharacterTable( "A5.2" )[128X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( a5, s5 );[127X[104X
    [4X[28X[ [ 1, 2, 3, 4, 4 ] ][128X[104X
    [4X[25Xgap>[125X [27Xfusion:= rec( name:= s5, map:= fus[1], text:= "unique" );;[127X[104X
    [4X[25Xgap>[125X [27XPrint( LibraryFusion( "A5", fusion ) );[127X[104X
    [4X[28XALF("A5","A5.2",[1,2,3,4,4],[[128X[104X
    [4X[28X"unique"[128X[104X
    [4X[28X]);[128X[104X
  [4X[32X[104X
  
  [1X4.7-3 LibraryFusionTblToTom[101X
  
  [33X[1;0Y[29X[2XLibraryFusionTblToTom[102X( [3Xname[103X, [3Xfus[103X ) [32X function[133X
  
  [33X[0;0YFor a string [3Xname[103X that is an [2XIdentifier[102X ([14XReference: Identifier for character
  tables[114X)  value  of  an  ordinary  character  table in the [5XGAP[105X library, and a
  record [3Xfus[103X with the components[133X
  
  [8X[10Xname[110X[8X[108X
        [33X[0;6Ythe  identifier  of  the  destination  table  of  marks, or this table
        itself,[133X
  
  [8X[10Xmap[110X[8X[108X
        [33X[0;6Ythe fusion map, a list of image positions,[133X
  
  [8X[10Xtext[110X[8X (optional)[108X
        [33X[0;6Ya string describing properties of the fusion, and[133X
  
  [8X[10Xperm[110X[8X (optional)[108X
        [33X[0;6Ya permutation,[133X
  
  [33X[0;0Y[2XLibraryFusionTblToTom[102X  returns  a  string whose printed value can be used to
  add  the  fusion in question to the library file containing the data for the
  table with identifier [3Xname[103X.[133X
  
  [33X[0;0YThe  meaning  of the component [10Xperm[110X is as follows. Let [3Xprim[103X be the primitive
  permutation  characters  obtained  by computing the [2XPermCharsTom[102X ([14XReference:
  PermCharsTom  via  fusion  map[114X)  value  of  the  tables of marks, taking the
  sublist  at  the positions in the first component of the [2XMaximalSubgroupsTom[102X
  ([14XReference:   MaximalSubgroupsTom[114X)   value  of  the  tables  of  marks,  and
  restricting  these lists via the [10Xmap[110X component. Permuting [3Xprim[103X with the [10Xperm[110X
  component  via [2XPermuted[102X ([14XReference: Permuted[114X) yields the list of permutation
  characters  obtained  by  inducing  the  trivial characters of the subgroups
  given  by  the  [2XMaxes[102X ([14X3.7-1[114X) value of the character table. If the component
  [10Xperm[110X  is  not  present  and  if  the character table has the attribute [2XMaxes[102X
  ([14X3.7-1[114X)  set  then  the  two  ways  of  computing  the primitive permutation
  characters yield the same list.[133X
  
  [33X[0;0YIf  [3Xname[103X is a character table then its [2XIdentifier[102X ([14XReference: Identifier for
  character tables[114X) value is used as the corresponding string.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "A5" );     [127X[104X
    [4X[28XCharacterTable( "A5" )[128X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "A5" );[127X[104X
    [4X[28XTableOfMarks( "A5" )[128X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleFusionsCharTableTom( tbl, tom );[127X[104X
    [4X[28X[ [ 1, 2, 3, 5, 5 ] ][128X[104X
    [4X[25Xgap>[125X [27Xfusion:= rec( name:= tom, map:= fus[1], text:= "unique" );;[127X[104X
    [4X[25Xgap>[125X [27XPrint( LibraryFusionTblToTom( "A5", fusion ) );[127X[104X
    [4X[28XARC("A5","tomfusion",rec(name:="A5",map:=[1,2,3,5,5],text:=[[128X[104X
    [4X[28X"unique"[128X[104X
    [4X[28X]));[128X[104X
  [4X[32X[104X
  
  [1X4.7-4 PrintToLib[101X
  
  [33X[1;0Y[29X[2XPrintToLib[102X( [3Xfile[103X, [3Xtbl[103X ) [32X function[133X
  
  [33X[0;0Yprints the (ordinary or Brauer) character table [3Xtbl[103X in library format to the
  file [3Xfile[103X[10X.tbl[110X (or to [3Xfile[103X if this has already the suffix [10X.tbl[110X).[133X
  
  [33X[0;0YIf   [3Xtbl[103X   is   an   ordinary   table   then  the  value  of  the  attribute
  [2XNamesOfFusionSources[102X   ([14XReference:   NamesOfFusionSources[114X)   is  ignored  by
  [2XPrintToLib[102X,  since for library tables this information is extracted from the
  source files (see Section [14X4.6-1[114X).[133X
  
  [33X[0;0YThe  names  of  data files in the [5XGAP[105X Character Table Library begin with [10Xcto[110X
  (for  ordinary  tables)  or  [10Xctb[110X  (for  corresponding  Brauer  tables),  see
  Section [14X4.6[114X.  This  is supported also for private extensions of the library,
  that is, if the filenames are chosen this way and the ordinary tables in the
  [10Xcto[110X  files  are  notified  via  [2XNotifyCharacterTable[102X ([14X4.7-5[114X) then the Brauer
  tables  will  be  found in the [10Xctb[110X files. Alternatively, if the filenames of
  the  files  with  the ordinary tables do not start with [10Xcto[110X then [5XGAP[105X expects
  the corresponding Brauer tables in the same file as the ordinary tables.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPrintToLib( "private", a5 );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  above command appends the data of the table [10Xa5[110X to the file [11Xprivate.tbl[111X;
  the first lines printed to this file are[133X
  
  [4X[32X[104X
    [4XMOT("A5",[104X
    [4X[[104X
    [4X"origin: ATLAS of finite groups, tests: 1.o.r., pow[2,3,5]"[104X
    [4X],[104X
    [4X[60,4,3,5,5],[104X
    [4X[,[1,1,3,5,4],[1,2,1,5,4],,[1,2,3,1,1]],[104X
    [4X[[1,1,1,1,1],[3,-1,0,-E(5)-E(5)^4,-E(5)^2-E(5)^3],[104X
    [4X[GALOIS,[2,2]],[4,0,1,-1,-1],[5,1,-1,0,0]],[104X
    [4X[(4,5)]);[104X
    [4XARC("A5","projectives",["2.A5",[[2,0,-1,E(5)+E(5)^4,E(5)^2+E(5)^3],[104X
    [4X[GALOIS,[1,2]],[4,0,1,-1,-1],[6,0,0,1,1]],]);[104X
    [4XARC("A5","extInfo",["2","2"]);[104X
  [4X[32X[104X
  
  [1X4.7-5 NotifyCharacterTable[101X
  
  [33X[1;0Y[29X[2XNotifyCharacterTable[102X( [3Xfirstname[103X, [3Xfilename[103X, [3Xothernames[103X ) [32X function[133X
  
  [33X[0;0Ynotifies  a  new  ordinary  table  to the library. This table has [2XIdentifier[102X
  ([14XReference:   Identifier  for  character  tables[114X)  value  [3Xfirstname[103X,  it  is
  contained  (in library format, see [2XPrintToLib[102X ([14X4.7-4[114X)) in the file with name
  [3Xfilename[103X  (without  suffix  [10X.tbl[110X),  and  the  names  contained  in  the list
  [3Xothernames[103X are admissible for it.[133X
  
  [33X[0;0YIf the initial part of [3Xfilename[103X is one of [10X~/[110X, [10X/[110X or [10X./[110X then it is interpreted
  as  an  [13Xabsolute[113X  path.  Otherwise  it  is  interpreted [13Xrelative[113X to the [11Xdata[111X
  directory of the [5XCTblLib[105X package.[133X
  
  [33X[0;0Y[2XNotifyCharacterTable[102X  modifies  the  global variable [2XLIBLIST[102X ([14X4.6-2[114X) for the
  current  [5XGAP[105X  session,  after  having checked that there is no other library
  table  yet  with  an  admissible  name  equal  to  [3Xfirstname[103X or contained in
  [3Xothernames[103X.[133X
  
  [33X[0;0YFor  example,  let  us  change the name [10XA5[110X to [10Xicos[110X wherever it occurs in the
  file  [11Xprivate.tbl[111X  that was produced above, and then notify the [21Xnew[121X table in
  this file as follows. (The name change is needed because [5XGAP[105X knows already a
  table  with  name  [10XA5[110X  and  would  not accept to add another table with this
  name.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNotifyCharacterTable( "icos", "private", [] );[127X[104X
    [4X[25Xgap>[125X [27Xicos:= CharacterTable( "icos" );[127X[104X
    [4X[28XCharacterTable( "icos" )[128X[104X
    [4X[25Xgap>[125X [27XDisplay( icos );[127X[104X
    [4X[28Xicos[128X[104X
    [4X[28X[128X[104X
    [4X[28X     2  2  2  .  .  .[128X[104X
    [4X[28X     3  1  .  1  .  .[128X[104X
    [4X[28X     5  1  .  .  1  1[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 2a 3a 5a 5b[128X[104X
    [4X[28X    2P 1a 1a 3a 5b 5a[128X[104X
    [4X[28X    3P 1a 2a 1a 5b 5a[128X[104X
    [4X[28X    5P 1a 2a 3a 1a 1a[128X[104X
    [4X[28X[128X[104X
    [4X[28XX.1     1  1  1  1  1[128X[104X
    [4X[28XX.2     3 -1  .  A *A[128X[104X
    [4X[28XX.3     3 -1  . *A  A[128X[104X
    [4X[28XX.4     4  .  1 -1 -1[128X[104X
    [4X[28XX.5     5  1 -1  .  .[128X[104X
    [4X[28X[128X[104X
    [4X[28XA = -E(5)-E(5)^4[128X[104X
    [4X[28X  = (1-ER(5))/2 = -b5[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  the private table is treated as a library table. Note that the table can
  be  accessed  only  if  it has been notified in the current [5XGAP[105X session. For
  frequently   used   private   tables,  it  may  be  reasonable  to  put  the
  [2XNotifyCharacterTable[102X  statements  into  your [11Xgaprc[111X file (see [14X'Reference: The
  gap.ini and gaprc files'[114X), or into a file that is read via the [11Xgaprc[111X file.[133X
  
  [1X4.7-6 NotifyCharacterTables[101X
  
  [33X[1;0Y[29X[2XNotifyCharacterTables[102X( [3Xlist[103X ) [32X function[133X
  
  [33X[0;0Ynotifies  several new ordinary tables to the library. The argument [3Xlist[103X must
  be  a  dense  list  in  which each entry is a lists of the form [10X[ firstname,
  filename,   othernames  ][110X,  with  the  same  meaning  as  the  arguments  of
  [2XNotifyCharacterTable[102X ([14X4.7-5[114X).[133X
  
  
  [1X4.8 [33X[0;0YSanity Checks for the [5XGAP[105X[101X[1X Character Table Library[133X[101X
  
  [33X[0;0YThe  fact  that  the  [5XGAP[105X  Character  Table  Library  is designed as an open
  database  (see  Chapter [14X1[114X) makes it especially desirable to have consistency
  checks available which can be run automatically whenever new data get added.[133X
  
  [33X[0;0YThe  file  [11Xtst/testall.g[111X  of  the  package  contains  [2XTest[102X ([14XReference: Test[114X)
  statements  for  executing  a  collection of such sanity checks; one can run
  them  by  calling  [10XReadPackage(  "CTblLib", "tst/testall.g" )[110X. If no problem
  occurs then [5XGAP[105X prints only lines starting with one of the following.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X+ Input file:[128X[104X
    [4X[28X+ GAP4stones:[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  examples  in  the  package  manual  form  a part of the tests, they are
  collected in the file [11Xtst/docxpl.tst[111X of the package.[133X
  
  [33X[0;0YThe  following  tests  concern only [13Xordinary[113X character tables. In all cases,
  let  [22Xtbl[122X be the ordinary character table of a group [22XG[122X, say. The return value
  is [9Xfalse[109X if an error occurred, and [9Xtrue[109X otherwise.[133X
  
  [8X[10XCTblLib.Test.InfoText( [110X[8X[22Xtbl[122X[10X )[110X[8X[108X
        [33X[0;6Ychecks  some properties of the [2XInfoText[102X ([14XReference: InfoText[114X) value of
        [22Xtbl[122X,  if  available. Currently it is not recommended to use this value
        programmatically.  However, one can rely on the following structure of
        this value for tables in the [5XGAP[105X Character Table Library.[133X
  
        [30X    [33X[0;12YThe value is a string that consists of [10X\n[110X separated lines.[133X
  
        [30X    [33X[0;12YIf  a line of the form [21Xmaximal subgroup of [22Xgrpname[122X[121X occurs, where
              [22Xgrpname[122X  is  the  name of a character table, then a class fusion
              from the table in question to that with name [22Xgrpname[122X is stored.[133X
  
        [30X    [33X[0;12YIf  a  line  of  the form [21X[22Xn[122Xth maximal subgroup of [22Xgrpname[122X[121X occurs
              then  additionally  the  name  [22Xgrpname[122X[10XM[110X[22Xn[122X  is admissible for [22Xtbl[122X.
              Furthermore,  if the table with name [22Xgrpname[122X has a [2XMaxes[102X ([14X3.7-1[114X)
              value then [22Xtbl[122X is referenced in position [22Xn[122X of this list.[133X
  
  [8X[10XCTblLib.Test.RelativeNames( [110X[8X[22Xtbl[122X[10X[, [110X[8X[22Xtblname[122X[10X] )[110X[8X[108X
        [33X[0;6Ychecks some properties of those admissible names for [22Xtbl[122X that refer to
        a  related  group  [22XH[122X,  say.  Let  [22Xname[122X  be  an admissible name for the
        character  table  of  [22XH[122X. (In particular, [22Xname[122X is not an empty string.)
        Then the following relative names are considered.[133X
  
        [8X[22Xname[122X[10XM[110X[8X[22Xn[122X[108X
              [33X[0;12Y[22XG[122X  is  isomorphic  with  the groups in the [22Xn[122X-th class of maximal
              subgroups  of  [22XH[122X.  An  example  is [10X"M12M1"[110X for the Mathieu group
              [22XM_11[122X.  We  consider  only  cases where [22Xname[122X does [13Xnot[113X contain the
              letter  [10Xx[110X.  For  example,  [10X2xM12[110X denotes the direct product of a
              cyclic  group  of order two and the Mathieu group [22XM_12[122X but [13Xnot[113X a
              maximal  subgroup of [21X[10X2x[110X[121X. Similarly, [10X3x2.M22M5[110X denotes the direct
              product  of  a  cyclic  group  of order three and a group in the
              fifth  class  of  maximal  subgroups of [22X2.M_22[122X but [13Xnot[113X a maximal
              subgroup of [21X[10X3x2.M22[110X[121X.[133X
  
        [8X[22Xname[122X[10XN[110X[8X[22Xp[122X[108X
              [33X[0;12Y[22XG[122X is isomorphic with the normalizers of the Sylow [22Xp[122X-subgroups of
              [22XH[122X.  An  example  is  [10X"M24N2"[110X  for  the  (self-normalizing) Sylow
              [22X2[122X-subgroup in the Mathieu group [22XM_24[122X.[133X
  
        [8X[22Xname[122X[10XN[110X[8X[22Xcnam[122X[108X
              [33X[0;12Y[22XG[122X  is  isomorphic  with  the normalizers of the cyclic subgroups
              generated  by the elements in the class with the name [22Xcnam[122X of [22XH[122X.
              An example is [10X"O7(3)N3A"[110X for the normalizer of an element in the
              class [10X3A[110X of the simple group [22XO_7(3)[122X.[133X
  
        [8X[22Xname[122X[10XC[110X[8X[22Xcnam[122X[108X
              [33X[0;12Y[22XG[122X  is  isomorphic  with  the  groups  in the centralizers of the
              elements  in  the  class  with the name [22Xcnam[122X of [22XH[122X. An example is
              [10X"M24C2A"[110X  for  the  centralizer of an element in the class [10X2A[110X in
              the Mathieu group [22XM_24[122X.[133X
  
        [33X[0;6YIn  these  cases,  [10XCTblLib.Test.RelativeNames[110X checks whether a library
        table  with  the admissible name [22Xname[122X exists and a class fusion to [22Xtbl[122X
        is stored on this table.[133X
  
        [33X[0;6YIn  the  case  of  Sylow  [22Xp[122X-normalizers,  it is also checked whether [22XG[122X
        contains  a  normal  Sylow  [22Xp[122X-subgroup  of the same order as the Sylow
        [22Xp[122X-subgroups  in  [22XH[122X. If the normal Sylow [22Xp[122X-subgroup of [22XG[122X is cyclic then
        it  is also checked whether [22XG[122X is the full Sylow [22Xp[122X-normalizer in [22XH[122X. (In
        general  this  information cannot be read off from the character table
        of [22XH[122X).[133X
  
        [33X[0;6YIn  the  case of normalizers (centralizers) of cyclic subgroups, it is
        also  checked  whether [22XH[122X really normalizes (centralizes) a subgroup of
        the given order, and whether the class fusion from [22Xtbl[122X to the table of
        [22XH[122X is compatible with the relative name.[133X
  
        [33X[0;6YIf  the  optional  argument  [22Xtblname[122X  is  given then only this name is
        tested.  If  there  is only one argument then all admissible names for
        [22Xtbl[122X are tested.[133X
  
  [8X[10XCTblLib.Test.FindRelativeNames( [110X[8X[22Xtbl[122X[10X )[110X[8X[108X
        [33X[0;6Yruns  over  the  class  fusions  stored  on  [22Xtbl[122X.  If  [22Xtbl[122X is the full
        centralizer/normalizer  of a cyclic subgroup in the table to which the
        class   fusion   points   then  the  function  proposes  to  make  the
        corresponding relative name an admissible name for [22Xtbl[122X.[133X
  
  [8X[10XCTblLib.Test.PowerMaps( [110X[8X[22Xtbl[122X[10X )[110X[8X[108X
        [33X[0;6Ychecks  whether  all  [22Xp[122X-th  power  maps  are  stored on [22Xtbl[122X, for prime
        divisors  [22Xp[122X  of  the  order  of [22XG[122X, and whether they are correct. (This
        includes the information about uniqueness of the power maps.)[133X
  
  [8X[10XCTblLib.Test.TableAutomorphisms( [110X[8X[22Xtbl[122X[10X )[110X[8X[108X
        [33X[0;6Ychecks  whether the table automorphisms are stored on [22Xtbl[122X, and whether
        they are correct. Also all available Brauer tables of [22Xtbl[122X are checked.[133X
  
  [8X[10XCTblLib.Test.CompatibleFactorFusions( [110X[8X[22Xtbl[122X[10X )[110X[8X[108X
        [33X[0;6Ychecks whether triangles and quadrangles of factor fusions from [22Xtbl[122X to
        other   library   tables  commute  (where  the  entries  in  the  list
        [10XCTblLib.IgnoreFactorFusionsCompatibility[110X are excluded from the tests),
        and   whether   the   factor  fusions  commute  with  the  actions  of
        corresponding outer automorphisms.[133X
  
  [8X[10XCTblLib.Test.FactorsModPCore( [110X[8X[22Xtbl[122X[10X )[110X[8X[108X
        [33X[0;6Ychecks,  for  all those prime divisors [22Xp[122X of the order of [22XG[122X such that [22XG[122X
        is not [22Xp[122X-solvable, whether the factor fusion to the character table of
        [22XG/O_p(G)[122X is stored on [22Xtbl[122X.[133X
  
        [33X[0;6YNote  that if [22XG[122X is not [22Xp[122X-solvable and [22XO_p(G)[122X is nontrivial then we can
        compute  the  [22Xp[122X-modular  Brauer table of [22XG[122X if that of the factor group
        [22XG/O_p(G)[122X is available. The availability of this table is indicated via
        the availability of the factor fusion from [22Xtbl[122X.[133X
  
  [8X[10XCTblLib.Test.Fusions( [110X[8X[22Xtbl[122X[10X )[110X[8X[108X
        [33X[0;6Ychecks  the  class  fusions  that  are  stored  on  the  table [22Xtbl[122X: No
        duplicates  shall  occur,  each  subgroup  fusion  or factor fusion is
        tested using [10XCTblLib.Test.SubgroupFusion[110X or [10XCTblLib.Test.FactorFusion[110X,
        respectively,  and  a  fusion  to the table of marks for [22Xtbl[122X is tested
        using [10XCTblLib.Test.FusionToTom[110X.[133X
  
  [8X[10XCTblLib.Test.Maxes( [110X[8X[22Xtbl[122X[10X )[110X[8X[108X
        [33X[0;6Ychecks  for those character tables [22Xtbl[122X that have the [2XMaxes[102X ([14X3.7-1[114X) set
        whether   the  character  tables  with  the  given  names  are  really
        available,  that  they  are ordered w.r.t. non-increasing group order,
        and that the fusions into [22Xtbl[122X are stored.[133X
  
  [8X[10XCTblLib.Test.ClassParameters( [110X[8X[22Xtbl[122X[10X )[110X[8X[108X
        [33X[0;6Ychecks  the  compatibility  of  class  parameters  of  alternating and
        symmetric  groups  (partitions describing cycle structures), using the
        underlying group stored in the corresponding table of marks.[133X
  
  [8X[10XCTblLib.Test.Constructions( [110X[8X[22Xtbl[122X[10X )[110X[8X[108X
        [33X[0;6Ychecks the [2XConstructionInfoCharacterTable[102X ([14X3.7-4[114X) status for the table
        [22Xtbl[122X: If this attribute value is set then tests depending on this value
        are  executed; if this attribute is not set then it is checked whether
        a description of [22Xtbl[122X via a construction would be appropriate.[133X
  
  [8X[10XCTblLib.Test.ExtensionInfo( [110X[8X[22Xtbl[122X[10X )[110X[8X[108X
        [33X[0;6Ychecks  whether  the  attribute [2XExtensionInfoCharacterTable[102X ([14X3.7-3[114X) is
        known  for  all  nonabelian  simple  character  tables  that  are  not
        duplicates.[133X
  
  [8X[10XCTblLib.Test.GroupForGroupInfo( [110X[8X[22Xtbl[122X[10X )[110X[8X[108X
        [33X[0;6Ychecks    that    the    entries    in    the    list    returned   by
        [2XGroupInfoForCharacterTable[102X ([14X3.3-1[114X) fit to the character table [22Xtbl[122X.[133X
  
  [33X[0;0YThe following tests concern only [13Xmodular[113X character tables. In all cases, let
  [22Xmodtbl[122X be a Brauer character table of a group [22XG[122X, say.[133X
  
  [8X[10XCTblLib.Test.BlocksInfo( [110X[8X[22Xmodtbl[122X[10X )[110X[8X[108X
        [33X[0;6Ychecks  whether the decomposition matrices of all blocks of the Brauer
        table   [22Xmodtbl[122X  are  integral,  as  well  as  the  inverses  of  their
        restrictions to basic sets.[133X
  
  [8X[10XCTblLib.Test.TensorDecomposition( [110X[8X[22Xmodtbl[122X[10X )[110X[8X[108X
        [33X[0;6Ychecks whether the tensor products of irreducible Brauer characters of
        the Brauer table [22Xmodtbl[122X decompose into Brauer characters.[133X
  
  [8X[10XCTblLib.Test.Indicators( [110X[8X[22Xmodtbl[122X[10X )[110X[8X[108X
        [33X[0;6Ychecks  the  [22X2[122X-nd indicators of the Brauer table [22Xmodtbl[122X: The indicator
        of  a  Brauer character is zero iff it has at least one nonreal value.
        In   odd  characteristic,  the  indicator  of  an  irreducible  Brauer
        character  is  equal  to  the  indicator  of  any ordinary irreducible
        character that contains it as a constituent, with odd multiplicity. In
        characteristic  two,  we  test  that  all  nontrivial real irreducible
        Brauer  characters  have  even  degree,  and  that  irreducible Brauer
        characters with indicator [22X-1[122X lie in the principal block.[133X
  
  [8X[10XCTblLib.Test.FactorBlocks( [110X[8X[22Xmodtbl[122X[10X )[110X[8X[108X
        [33X[0;6YIf  the  Brauer  table [22Xmodtbl[122X is encoded using references to tables of
        factor  groups  then we must make sure that the irreducible characters
        of  the  underlying  ordinary  table  and  the factors in question are
        sorted   compatibly.   (Note  that  we  simply  take  over  the  block
        information about the factors, without applying an explicit mapping.)[133X
  
  
  [1X4.9 [33X[0;0YMaintenance of the [5XGAP[105X[101X[1X Character Table Library[133X[101X
  
  [33X[0;0YIt  is  of  course desirable that the information in the [5XGAP[105X Character Table
  Library  is  consistent  with related data. For example, the ordering of the
  classes  of  maximal  subgroups  stored  in  the  [2XMaxes[102X  ([14X3.7-1[114X) list of the
  character  table  of a group [22XG[122X, say, should correspond to the ordering shown
  for  [22XG[122X  in  the  [5XAtlas[105X of Finite Groups [CCN+85], to the ordering of maximal
  subgroups  used  for  [22XG[122X  in  the  [5XAtlasRep[105X,  and  to the ordering of maximal
  subgroups  in  the  table  of  marks  of  [22XG[122X.  The fact that the related data
  collections  are  developed independently makes it difficult to achieve this
  kind  of  consistency. Sometimes it is unavoidable to [21Xadjust[121X data of the [5XGAP[105X
  Character Table Library to external data.[133X
  
  [33X[0;0YAn important issue is the consistency of class fusions. Usually such fusions
  are  determined  only  up  to  table automorphisms, and one candidate can be
  chosen.  However,  other conditions such as known Brauer tables may restrict
  the  choice.  The  point  is  that there are class fusions which predate the
  availability  of  Brauer tables in the Character Table Library (in fact many
  of  them  have been inherited from the table library of the [5XCAS[105X system), but
  they  are not compatible with the Brauer tables. For example, there are four
  possible  class fusion from [22XM_23[122X into [22XCo_3[122X, which lie in one orbit under the
  relevant groups of table automorphisms; two of these maps are not compatible
  with  the  [22X3[122X-modular  Brauer  tables of [22XM_23[122X and [22XCo_3[122X, and unfortunately the
  class  fusion  that  was stored on the [5XCAS[105X tables –and that was available in
  version  1.0  of  the  [5XGAP[105X  Character  Table  Library–  was  one  of the [13Xnot[113X
  compatible maps. One could argue that the class fusion has older rights, and
  that the Brauer tables should be adjusted to them, but the Brauer tables are
  published  in  the [5XAtlas[105X of Brauer Characters [JLPW95], which is an accepted
  standard.[133X
  
