/usr/share/gap/pkg/CtblLib/doc/chap5.txt

  
  5 Functions for Character Table Constructions
  
  The  functions  described  in  this  chapter  deal  with the construction of
  character  tables  from other character tables. So they fit to the functions
  in Section[14 X'Reference: Constructing Character Tables from Others'. But since
  they  are  used  in  situations that are typical for the GAP Character Table
  Library, they are described here.
  
  An  important  ingredient  of  the  constructions  is the description of the
  action of a group automorphism on the classes by a permutation. In practice,
  these  permutations are usually chosen from the group of table automorphisms
  of  the  character  table  in question, see[2 XAutomorphismsOfTable (Reference:
  AutomorphismsOfTable).
  
  Section[14 X5.1  deals  with  groups  of  the structure M.G.A, where the upwards
  extension  G.A acts suitably on the central extension M.G. Section[14 X5.2 deals
  with  groups  that  have  a factor group of type S_3. Section[14 X5.3 deals with
  upward  extensions  of a group by a Klein four group. Section[14 X5.4 deals with
  downward  extensions of a group by a Klein four group. Section[14 X5.6 describes
  the  construction  of  certain Brauer tables. Section[14 X5.7 deals with special
  cases  of  the  construction  of character tables of central extensions from
  known  character tables of suitable factor groups. Section[14 X5.8 documents the
  functions used to encode certain tables in the GAP Character Table Library.
  
  Examples can be found in [Breb] and [Bref].
  
  
  5.1 Character Tables of Groups of Structure M.G.A
  
  For  the functions in this section, let H be a group with normal subgroups N
  and  M  such that H/N is cyclic, M ≤ N holds, and such that each irreducible
  character  of N that does not contain M in its kernel induces irreducibly to
  H. (This is satisfied for example if N has prime index in H and M is a group
  of  prime order that is central in N but not in H.) Let G = N/M and A = H/N,
  so H has the structure M.G.A. For some examples, see [Bre11].
  
  5.1-1 PossibleCharacterTablesOfTypeMGA
  
  PossibleCharacterTablesOfTypeMGA( tblMG, tblG, tblGA, orbs, identifier )  function
  
  Let H, N, and M be as described at the beginning of the section.
  
  Let  tblMG, tblG, tblGA be the ordinary character tables of the groups M.G =
  N,  G,  and  G.A  = H/M, respectively, and orbs be the list of orbits on the
  class  positions  of  tblMG  that  is  induced  by  the  action of H on M.G.
  Furthermore, let the class fusions from tblMG to tblG and from tblG to tblGA
  be  stored  on  tblMG  and  tblG,  respectively (see[2 XStoreFusion (Reference:
  StoreFusion)).
  
  PossibleCharacterTablesOfTypeMGA  returns  a  list of records describing all
  possible ordinary character tables for groups H that are compatible with the
  arguments.  Note that in general there may be several possible groups H, and
  it  may  also  be  that  character tables are constructed for which no group
  exists.
  
  Each of the records in the result has the following components.
  
  table
        a possible ordinary character table for H, and
  
  MGfusMGA
        the fusion map from tblMG into the table stored in table.
  
  The  possible  tables  differ  w. r. t. some power maps, and perhaps element
  orders and table automorphisms; in particular, the MGfusMGA component is the
  same in all records.
  
  The returned tables have the Identifier (Reference: Identifier for character
  tables) value identifier. The classes of these tables are sorted as follows.
  First  come the classes contained in M.G, sorted compatibly with the classes
  in  tblMG,  then  the classes in H ∖ M.G follow, in the same ordering as the
  classes of G.A ∖ G.
  
  5.1-2 BrauerTableOfTypeMGA
  
  BrauerTableOfTypeMGA( modtblMG, modtblGA, ordtblMGA )  function
  
  Let  H,  N,  and  M  be  as  described  at the beginning of the section, let
  modtblMG  and modtblGA be the p-modular character tables of the groups N and
  H/M, respectively, and let ordtblMGA be the p-modular Brauer table of H, for
  some  prime  integer p. Furthermore, let the class fusions from the ordinary
  character  table of modtblMG to ordtblMGA and from ordtblMGA to the ordinary
  character table of modtblGA be stored.
  
  BrauerTableOfTypeMGA returns the p-modular character table of H.
  
  5.1-3 PossibleActionsForTypeMGA
  
  PossibleActionsForTypeMGA( tblMG, tblG, tblGA )  function
  
  Let  the  arguments  be  as  described  for PossibleCharacterTablesOfTypeMGA
  (5.1-1).  PossibleActionsForTypeMGA returns the set of orbit structures [122ΩX on
  the  class  positions of tblMG that can be induced by the action of H on the
  classes  of  M.G  in  the  sense  that  [122ΩX  is  the  set of orbits of a table
  automorphism      of     tblMG     (see[2 XAutomorphismsOfTable     (Reference:
  AutomorphismsOfTable)) that is compatible with the stored class fusions from
  tblMG  to  tblG  and  from tblG to tblGA. Note that the number of such orbit
  structures   can  be  smaller  than  the  number  of  the  underlying  table
  automorphisms.
  
  Information   about   the   progress  is  reported  if  the  info  level  of
  InfoCharacterTable   (Reference:   InfoCharacterTable)   is   at   least   1
  (see[2 XSetInfoLevel (Reference: InfoLevel)).
  
  
  5.2 Character Tables of Groups of Structure G.S_3
  
  
  5.2-1 CharacterTableOfTypeGS3
  
  CharacterTableOfTypeGS3( tbl, tbl2, tbl3, aut, identifier )  function
  CharacterTableOfTypeGS3( modtbl, modtbl2, modtbl3, ordtbls3, identifier )  function
  
  Let H be a group with a normal subgroup G such that H/G ≅ S_3, the symmetric
  group  on  three  points,  and  let G.2 and G.3 be preimages of subgroups of
  order  2  and 3, respectively, under the natural projection onto this factor
  group.
  
  In  the  first form, let tbl, tbl2, tbl3 be the ordinary character tables of
  the  groups  G,  G.2,  and  G.3, respectively, and aut be the permutation of
  classes  of  tbl3 induced by the action of H on G.3. Furthermore assume that
  the   class   fusions   from  tbl  to  tbl2  and  tbl3  are  stored  on  tbl
  (see[2 XStoreFusion  (Reference:  StoreFusion)).  In  particular, the two class
  fusions  must  be  compatible  in  the  sense that the induced action on the
  classes of tbl describes an action of S_3.
  
  In  the second form, let modtbl, modtbl2, modtbl3 be the p-modular character
  tables  of  the  groups  G,  G.2, and G.3, respectively, and ordtbls3 be the
  ordinary character table of H.
  
  CharacterTableOfTypeGS3 returns a record with the following components.
  
  table
        the ordinary or p-modular character table of H, respectively,
  
  tbl2fustbls3
        the fusion map from tbl2 into the table of H, and
  
  tbl3fustbls3
        the fusion map from tbl3 into the table of H.
  
  The  returned  table  of  H  has  the  Identifier (Reference: Identifier for
  character tables) value identifier. The classes of the table of H are sorted
  as  follows. First come the classes contained in G.3, sorted compatibly with
  the  classes  in  tbl3,  then  the  classes  in  H ∖ G.3 follow, in the same
  ordering as the classes of G.2 ∖ G.
  
  In  fact  the  code  is  applicable  in  the more general case that H/G is a
  Frobenius  group  F  =  K C with abelian kernel K and cyclic complement C of
  prime  order,  see [Bref].  Besides  F  =  S_3,  e. g.,  the case F = A_4 is
  interesting.
  
  5.2-2 PossibleActionsForTypeGS3
  
  PossibleActionsForTypeGS3( tbl, tbl2, tbl3 )  function
  
  Let  the  arguments  be  as  described  for CharacterTableOfTypeGS3 (5.2-1).
  PossibleActionsForTypeGS3  returns  the  set  of  those  table automorphisms
  (see[2 XAutomorphismsOfTable  (Reference:  AutomorphismsOfTable))  of tbl3 that
  can be induced by the action of H on the classes of tbl3.
  
  Information   about   the   progress  is  reported  if  the  info  level  of
  InfoCharacterTable   (Reference:   InfoCharacterTable)   is   at   least   1
  (see[2 XSetInfoLevel (Reference: InfoLevel)).
  
  
  5.3 Character Tables of Groups of Structure G.2^2
  
  The  following  functions are thought for constructing the possible ordinary
  character  tables of a group of structure G.2^2 from the known tables of the
  three normal subgroups of type G.2.
  
  
  5.3-1 PossibleCharacterTablesOfTypeGV4
  
  PossibleCharacterTablesOfTypeGV4( tblG, tblsG2, acts, identifier[, tblGfustblsG2] )  function
  PossibleCharacterTablesOfTypeGV4( modtblG, modtblsG2, ordtblGV4[, ordtblsG2fusordtblG4] )  function
  
  Let  H  be  a  group  with a normal subgroup G such that H/G is a Klein four
  group,  and  let G.2_1, G.2_2, and G.2_3 be the three subgroups of index two
  in H that contain G.
  
  In  the  first  version,  let tblG be the ordinary character table of G, let
  tblsG2  be a list containing the three character tables of the groups G.2_i,
  and  let  acts be a list of three permutations describing the action of H on
  the  conjugacy  classes  of the corresponding tables in tblsG2. If the class
  fusions  from  tblG  into  the  tables in tblsG2 are not stored on tblG (for
  example,  because  the  three  tables are equal) then the three maps must be
  entered in the list tblGfustblsG2.
  
  In  the  second  version, let modtblG be the p-modular character table of G,
  modtblsG  be  the  list  of p-modular Brauer tables of the groups G.2_i, and
  ordtblGV4  be  the  ordinary  character  table of H. In this case, the class
  fusions  from the ordinary character tables of the groups G.2_i to ordtblGV4
  can be entered in the list ordtblsG2fusordtblG4.
  
  PossibleCharacterTablesOfTypeGV4  returns  a  list of records describing all
  possible  (ordinary  or  p-modular)  character  tables for groups H that are
  compatible  with  the  arguments.  Note that in general there may be several
  possible  groups H, and it may also be that character tables are constructed
  for  which  no  group  exists.  Each  of  the  records in the result has the
  following components.
  
  table
        a possible (ordinary or p-modular) character table for H, and
  
  G2fusGV4
        the  list  of  fusion  maps  from  the tables in tblsG2 into the table
        component.
  
  The possible tables differ w.r.t. the irreducible characters and perhaps the
  table  automorphisms;  in  particular, the G2fusGV4 component is the same in
  all records.
  
  The returned tables have the Identifier (Reference: Identifier for character
  tables) value identifier. The classes of these tables are sorted as follows.
  First come the classes contained in G, sorted compatibly with the classes in
  tblG,  then  the  outer  classes in the tables in tblsG2 follow, in the same
  ordering as in these tables.
  
  5.3-2 PossibleActionsForTypeGV4
  
  PossibleActionsForTypeGV4( tblG, tblsG2 )  function
  
  Let  the  arguments  be  as  described  for PossibleCharacterTablesOfTypeGV4
  (5.3-1).  PossibleActionsForTypeGV4 returns the list of those triples [ π_1,
  π_2,  π_3  ]  of  permutations  for  which a group H may exist that contains
  G.2_1,  G.2_2,  G.2_3  as  index  2 subgroups which intersect in the index 4
  subgroup G.
  
  Information   about   the   progress   is   reported   if   the   level   of
  InfoCharacterTable   (Reference:   InfoCharacterTable)   is   at   least   1
  (see[2 XSetInfoLevel (Reference: InfoLevel)).
  
  
  5.4 Character Tables of Groups of Structure 2^2.G
  
  The  following  functions are thought for constructing the possible ordinary
  or  Brauer  character  tables  of  a group of structure 2^2.G from the known
  tables  of  the three factor groups modulo the normal order two subgroups in
  the central Klein four group.
  
  Note that in the ordinary case, only a list of possibilities can be computed
  whereas  in  the modular case, where the ordinary character table is assumed
  to be known, the desired table is uniquely determined.
  
  
  5.4-1 PossibleCharacterTablesOfTypeV4G
  
  PossibleCharacterTablesOfTypeV4G( tblG, tbls2G, id[, fusions] )  function
  PossibleCharacterTablesOfTypeV4G( tblG, tbl2G, aut, id )  function
  
  Let  H  be  a group with a central subgroup N of type 2^2, and let Z_1, Z_2,
  Z_3 be the order 2 subgroups of N.
  
  In  the  first  form,  let  tblG be the ordinary character table of H/N, and
  tbls2G  be  a list of length three, the entries being the ordinary character
  tables  of  the  groups H/Z_i. In the second form, let tbl2G be the ordinary
  character  table  of H/Z_1 and aut be a permutation; here it is assumed that
  the  groups  Z_i  are  permuted under an automorphism [122σX of order 3 of H, and
  that [122σX induces the permutation aut on the classes of tblG.
  
  The class fusions onto tblG are assumed to be stored on the tables in tbls2G
  or  tbl2G,  respectively,  except  if  they  are  explicitly entered via the
  optional argument fusions.
  
  PossibleCharacterTablesOfTypeV4G  returns the list of all possible character
  tables  for  H  in  this  situation. The returned tables have the Identifier
  (Reference: Identifier for character tables) value id.
  
  
  5.4-2 BrauerTableOfTypeV4G
  
  BrauerTableOfTypeV4G( ordtblV4G, modtbls2G )  function
  BrauerTableOfTypeV4G( ordtblV4G, modtbl2G, aut )  function
  
  Let H be a group with a central subgroup N of type 2^2, and let ordtblV4G be
  the  ordinary  character  table  of  H.  Let  Z_1,  Z_2,  Z_3 be the order 2
  subgroups  of  N.  In  the  first  form,  let  modtbls2G  be the list of the
  p-modular  Brauer  tables  of the factor groups H/Z_1, H/Z_2, and H/Z_3, for
  some  prime  integer  p.  In  the second form, let modtbl2G be the p-modular
  Brauer  table of H/Z_1 and aut be a permutation; here it is assumed that the
  groups  Z_i are permuted under an automorphism [122σX of order 3 of H, and that [122σX
  induces  the  permutation aut on the classes of the ordinary character table
  of H that is stored in ordtblV4G.
  
  The  class  fusions  from  ordtblV4G to the ordinary character tables of the
  tables in modtbls2G or modtbl2G are assumed to be stored.
  
  BrauerTableOfTypeV4G returns the p-modular character table of H.
  
  
  5.5 Character Tables of Subdirect Products of Index Two
  
  The  following function is thought for constructing the (ordinary or Brauer)
  character  tables of certain subdirect products from the known tables of the
  factor groups and normal subgroups involved.
  
  5.5-1 CharacterTableOfIndexTwoSubdirectProduct
  
  CharacterTableOfIndexTwoSubdirectProduct( tblH1, tblG1, tblH2, tblG2, identifier )  function
  Returns:  a record containing the character table of the subdirect product G
            that is described by the first four arguments.
  
  Let  tblH1,  tblG1, tblH2, tblG2 be the character tables of groups H_1, G_1,
  H_2, G_2, such that H_1 and H_2 have index two in G_1 and G_2, respectively,
  and such that the class fusions corresponding to these embeddings are stored
  on tblH1 and tblH1, respectively.
  
  In  this  situation,  the  direct  product  of G_1 and G_2 contains a unique
  subgroup  G of index two that contains the direct product of H_1 and H_2 but
  does not contain any of the groups G_1, G_2.
  
  The  function CharacterTableOfIndexTwoSubdirectProduct returns a record with
  the following components.
  
  table
        the character table of G,
  
  H1fusG
        the class fusion from tblH1 into the table of G, and
  
  H2fusG
        the class fusion from tblH2 into the table of G.
  
  If  the  first  four  arguments are ordinary character tables then the fifth
  argument  identifier  must  be  a  string;  this  is  used as the Identifier
  (Reference: Identifier for character tables) value of the result table.
  
  If  the  first  four  arguments  are  Brauer  character  tables for the same
  characteristic  then the fifth argument must be the ordinary character table
  of the desired subdirect product.
  
  5.5-2 ConstructIndexTwoSubdirectProduct
  
  ConstructIndexTwoSubdirectProduct( tbl, tblH1, tblG1, tblH2, tblG2, permclasses, permchars )  function
  
  ConstructIndexTwoSubdirectProduct  constructs  the irreducible characters of
  the  ordinary  character  table tbl of the subdirect product of index two in
  the  direct product of tblG1 and tblG2, which contains the direct product of
  tblH1 and tblH2 but does not contain any of the direct factors tblG1, tblG2.
  W. r. t. the    default    ordering    obtained    from    that   given   by
  CharacterTableDirectProduct  (Reference:  CharacterTableDirectProduct),  the
  columns  and  the  rows  of the matrix of irreducibles are permuted with the
  permutations permclasses and permchars, respectively.
  
  5.5-3 ConstructIndexTwoSubdirectProductInfo
  
  ConstructIndexTwoSubdirectProductInfo( tbl[, tblH1, tblG1, tblH2, tblG2] )  function
  Returns:  a   list   of   constriction   descriptions,   or  a  construction
            description, or fail.
  
  Called  with  one  argument tbl, an ordinary character table of the group G,
  say,  ConstructIndexTwoSubdirectProductInfo  analyzes  the  possibilities to
  construct  tbl from character tables of subgroups H_1, H_2 and factor groups
  G_1, G_2, using CharacterTableOfIndexTwoSubdirectProduct (5.5-1). The return
  value is a list of records with the following components.
  
  kernels
        the list of class positions of H_1, H_2 in tbl,
  
  kernelsizes
        the list of orders of H_1, H_2,
  
  factors
        the  list  of  Identifier (Reference: Identifier for character tables)
        values  of the GAP library tables of the factors G_2, G_1 of G by H_1,
        H_2; if no such table is available then the entry is fail, and
  
  subgroups
        the  list  of  Identifier (Reference: Identifier for character tables)
        values of the GAP library tables of the subgroups H_2, H_1 of G; if no
        such tables are available then the entries are fail.
  
  If  the  returned  list  is  empty then either tbl does not have the desired
  structure  as  a  subdirect  product,  or tbl is in fact a nontrivial direct
  product.
  
  Called  with  five  arguments, the ordinary character tables of G, H_1, G_1,
  H_2,  G_2,  ConstructIndexTwoSubdirectProductInfo returns a list that can be
  used  as  the ConstructionInfoCharacterTable (3.7-4) value for the character
  table    of    G    from    the    other   four   character   tables   using
  CharacterTableOfIndexTwoSubdirectProduct  (5.5-1);  if  this is not possible
  then fail is returned.
  
  
  5.6 Brauer Tables of Extensions by p-regular Automorphisms
  
  As  for  the  construction of Brauer character tables from known tables, the
  functions  PossibleCharacterTablesOfTypeMGA (5.1-1), CharacterTableOfTypeGS3
  (5.2-1), and PossibleCharacterTablesOfTypeGV4 (5.3-1) work for both ordinary
  and  Brauer  tables. The following function is designed specially for Brauer
  tables.
  
  5.6-1 IBrOfExtensionBySingularAutomorphism
  
  IBrOfExtensionBySingularAutomorphism( modtbl, act )  function
  
  Let modtbl be a p-modular Brauer table of the group G, say, and suppose that
  the group H, say, is an upward extension of G by an automorphism of order p.
  
  The second argument act describes the action of this automorphism. It can be
  either  a permutation of the columns of modtbl, or a list of the H-orbits on
  the  columns  of  modtbl, or the ordinary character table of H such that the
  class fusion from the ordinary table of modtbl into this table is stored. In
  all  these  cases,  IBrOfExtensionBySingularAutomorphism  returns the values
  lists of the irreducible p-modular Brauer characters of H.
  
  Note  that  the  table  head  of the p-modular Brauer table of H, in general
  without  the  Irr  (Reference:  Irr)  attribute, can be obtained by applying
  CharacterTableRegular  (Reference:  CharacterTableRegular)  to  the ordinary
  character  table  of H, but IBrOfExtensionBySingularAutomorphism can be used
  also  if  the  ordinary  character  table  of  H  is not known, and just the
  p-modular  character  table of G and the action of H on the classes of G are
  given.
  
  
  5.7 Character Tables of Coprime Central Extensions
  
  5.7-1 CharacterTableOfCommonCentralExtension
  
  CharacterTableOfCommonCentralExtension( tblG, tblmG, tblnG, id )  function
  
  Let  tblG  be  the ordinary character table of a group G, say, and let tblmG
  and tblnG be the ordinary character tables of central extensions m.G and n.G
  of  G by cyclic groups of prime orders m and n, respectively, with m not= n.
  We assume that the factor fusions from tblmG and tblnG to tblG are stored on
  the tables. CharacterTableOfCommonCentralExtension returns a record with the
  following components.
  
  tblmnG
        the  character table t, say, of the corresponding central extension of
        G by a cyclic group of order m n that factors through m.G and n.G; the
        Identifier  (Reference: Identifier for character tables) value of this
        table is id,
  
  IsComplete
        true  if  the  Irr  (Reference:  Irr)  value is stored in t, and false
        otherwise,
  
  irreducibles
        the list of irreducibles of t that are known; it contains the inflated
        characters  of  the factor groups m.G and n.G, plus those irreducibles
        that were found in tensor products of characters of these groups.
  
  Note  that  the  conjugacy  classes  and  the  power  maps of t are uniquely
  determined  by the input data. Concerning the irreducible characters, we try
  to  extract  them from the tensor products of characters of the given factor
  groups  by  reducing  with known irreducibles and applying the LLL algorithm
  (see[2 XReducedClassFunctions    (Reference:   ReducedClassFunctions)   and[2 XLLL
  (Reference: LLL)).
  
  
  5.8 Construction Functions used in the Character Table Library
  
  The  following  functions  are  used in the GAP Character Table Library, for
  encoding  table  constructions  via  the  mechanism  that  is  based  on the
  attribute ConstructionInfoCharacterTable (3.7-4). All construction functions
  take  as  their  first  argument  a  record  that  describes the table to be
  constructed,  and  the  function adds only those components that are not yet
  contained in this record.
  
  5.8-1 ConstructMGA
  
  ConstructMGA( tbl, subname, factname, plan, perm )  function
  
  ConstructMGA constructs the irreducible characters of the ordinary character
  table tbl of a group m.G.a where the automorphism a (a group of prime order)
  of  m.G  acts  nontrivially on the central subgroup m of m.G. subname is the
  name  of  the  subgroup  m.G  which  is  a  (not necessarily cyclic) central
  extension  of  the (not necessarily simple) group G, factname is the name of
  the  factor  group G.a. Then the faithful characters of tbl are induced from
  m.G.
  
  plan  is  a list, each entry being a list containing positions of characters
  of m.G that form an orbit under the action of a (the induction of characters
  is encoded this way).
  
  perm  is the permutation that must be applied to the list of characters that
  is  obtained on appending the faithful characters to the inflated characters
  of the factor group. A nonidentity permutation occurs for example for groups
  of  structure  12.G.2  that are encoded via the subgroup 12.G and the factor
  group  6.G.2,  where the faithful characters of 4.G.2 shall precede those of
  6.G.2, as in the Atlas.
  
  Examples  where ConstructMGA is used to encode library tables are the tables
  of 3.F_{3+}.2 (subgroup 3.F_{3+}, factor group F_{3+}.2) and 12_1.U_4(3).2_2
  (subgroup 12_1.U_4(3), factor group 6_1.U_4(3).2_2).
  
  5.8-2 ConstructMGAInfo
  
  ConstructMGAInfo( tblmGa, tblmG, tblGa )  function
  
  Let  tblmGa  be  the  ordinary character table of a group of structure m.G.a
  where  the  factor  group  of  prime order a acts nontrivially on the normal
  subgroup  of order m that is central in m.G, tblmG be the character table of
  m.G, and tblGa be the character table of the factor group G.a.
  
  ConstructMGAInfo  returns  the  list  that  is  to  be stored in the library
  version  of  tblmGa:  the  first  entry  is  the  string "ConstructMGA", the
  remaining  four  entries  are  the  last  four  arguments  for  the  call to
  ConstructMGA (5.8-1).
  
  5.8-3 ConstructGS3
  
  ConstructGS3( tbls3, tbl2, tbl3, ind2, ind3, ext, perm )  function
  ConstructGS3Info( tbl2, tbl3, tbls3 )  function
  
  ConstructGS3  constructs  the  irreducibles  of  an ordinary character table
  tbls3  of  type  G.S_3  from  the  tables  with  names  tbl2 and tbl3, which
  correspond  to  the  groups  G.2  and  G.3,  respectively. ind2 is a list of
  numbers  referring  to  irreducibles  of tbl2. ind3 is a list of pairs, each
  referring to irreducibles of tbl3. ext is a list of pairs, each referring to
  one  irreducible  character  of  tbl2 and one of tbl3. perm is a permutation
  that must be applied to the irreducibles after the construction.
  
  ConstructGS3Info returns a record with the components ind2, ind3, ext, perm,
  and list, as are needed for ConstructGS3.
  
  5.8-4 ConstructV4G
  
  ConstructV4G( tbl, facttbl, aut )  function
  
  Let  tbl  be  the  character  table  of a group of type 2^2.G where an outer
  automorphism  of  order 3 permutes the three involutions in the central 2^2.
  Let  aut  be the permutation of classes of tbl induced by that automorphism,
  and facttbl be the name of the character table of the factor group 2.G. Then
  ConstructV4G   constructs  the  irreducible  characters  of  tbl  from  that
  information.
  
  5.8-5 ConstructProj
  
  ConstructProj( tbl, irrinfo )  function
  ConstructProjInfo( tbl, kernel )  function
  
  ConstructProj  constructs  the irreducible characters of the record encoding
  the  ordinary  character  table  tbl from projective characters of tables of
  factor  groups, which are stored in the ProjectivesInfo (3.7-2) value of the
  smallest  factor;  the  information  about  the  name of this factor and the
  projectives to take is stored in irrinfo.
  
  ConstructProjInfo takes an ordinary character table tbl and a list kernel of
  class  positions  of  a  cyclic  kernel  of order dividing 12, and returns a
  record with the components
  
  tbl
        a  character table that is permutation isomorphic with tbl, and sorted
        such  that classes that differ only by multiplication with elements in
        the classes of kernel are consecutive,
  
  projectives
        a  record being the entry for the projectives list of the table of the
        factor  of  tbl by kernel, describing this part of the irreducibles of
        tbl, and
  
  info
        the  value of irrinfo that is needed for constructing the irreducibles
        of  the  tbl  component  of  the  result  (not the irreducibles of the
        argument tbl!) via ConstructProj.
  
  5.8-6 ConstructDirectProduct
  
  ConstructDirectProduct( tbl, factors[, permclasses, permchars] )  function
  
  The  direct  product  of  the library character tables described by the list
  factors  of  table  names  is  constructed using CharacterTableDirectProduct
  (Reference:  CharacterTableDirectProduct),  and  all its components that are
  not yet stored on tbl are added to tbl.
  
  The  ComputedClassFusions  (Reference: ComputedClassFusions) value of tbl is
  enlarged by the factor fusions from the direct product to the factors.
  
  If  the optional arguments permclasses, permchars are given then the classes
  and characters of the result are sorted accordingly.
  
  factors must have length at least two; use ConstructPermuted (5.8-11) in the
  case of only one factor.
  
  5.8-7 ConstructCentralProduct
  
  ConstructCentralProduct( tbl, factors, Dclasses[, permclasses, permchars] )  function
  
  The library table tbl is completed with help of the table obtained by taking
  the  direct  product  of the tables with names in the list factors, and then
  factoring  out  the  normal  subgroup  that is given by the list Dclasses of
  class positions.
  
  If  the optional arguments permclasses, permchars are given then the classes
  and characters of the result are sorted accordingly.
  
  5.8-8 ConstructSubdirect
  
  ConstructSubdirect( tbl, factors, choice )  function
  
  The library table tbl is completed with help of the table obtained by taking
  the  direct  product  of the tables with names in the list factors, and then
  taking the table consisting of the classes in the list choice.
  
  Note that in general, the restriction to the classes of a normal subgroup is
  not  sufficient  for  describing  the  irreducible characters of this normal
  subgroup.
  
  5.8-9 ConstructWreathSymmetric
  
  ConstructWreathSymmetric( tbl, subname, n[, permclasses, permchars] )  function
  
  The  wreath  product  of  the  library character table with identifier value
  subname   with  the  symmetric  group  on  n  points  is  constructed  using
  CharacterTableWreathSymmetric   (Reference:  CharacterTableWreathSymmetric),
  and all its components that are not yet stored on tbl are added to tbl.
  
  If  the optional arguments permclasses, permchars are given then the classes
  and characters of the result are sorted accordingly.
  
  5.8-10 ConstructIsoclinic
  
  ConstructIsoclinic( tbl, factors[, nsg[, centre]][, permclasses, permchars] )  function
  
  constructs  first  the direct product of library tables as given by the list
  factors  of  admissible  character  table  names,  and  then  constructs the
  isoclinic table of the result.
  
  If   the   argument   nsg   is   present   and  a  record  or  a  list  then
  CharacterTableIsoclinic  (Reference:  CharacterTableIsoclinic)  gets called,
  and nsg (as well as centre if present) is passed to this function.
  
  In  both  cases,  if the optional arguments permclasses, permchars are given
  then the classes and characters of the result are sorted accordingly.
  
  5.8-11 ConstructPermuted
  
  ConstructPermuted( tbl, libnam[, permclasses, permchars] )  function
  
  The  library  table  tbl  is  computed  from the library table with the name
  libnam,  by  permuting  the  classes  and the characters by the permutations
  permclasses and permchars, respectively.
  
  So  tbl  and  the  library  table  with  the  name  libnam  are  permutation
  equivalent.  With  the more general function ConstructAdjusted (5.8-12), one
  can derive character tables that are not necessarily permutation equivalent,
  by additionally replacing some defining data.
  
  The  two  permutations  are  optional. If they are missing then the lists of
  irreducible  characters  and  the  power  maps  of  the two character tables
  coincide.  However, different class fusions may be stored on the two tables.
  This  is used for example in situations where a group has several classes of
  isomorphic  maximal  subgroups  whose class fusions are different; different
  character  tables  (with different identifiers) are stored for the different
  classes,  each  with  appropriate class fusions, and all these tables except
  the  one for the first class of subgroups can be derived from this table via
  ConstructPermuted.
  
  5.8-12 ConstructAdjusted
  
  ConstructAdjusted( tbl, libnam, pairs[, permclasses, permchars] )  function
  
  The  defining  attribute  values  of  the library table tbl are given by the
  attribute values described by the list pairs and –for those attributes which
  do  not  appear  in pairs– by the attribute values of the library table with
  the  name  libnam,  whose  classes  and characters have been permuted by the
  optional permutations permclasses and permchars, respectively.
  
  This  construction  can  be  used  to  derive a character table from another
  library  table  (the  one  with  the  name  libnam)  that is not permutation
  equivalent  to  this  table.  For  example, it may happen that the character
  tables  of  a  split and a nonsplit extension differ only by some power maps
  and  element  orders.  In  this  case,  one can encode one of the tables via
  ConstructAdjusted, by prescribing just the power maps in the list pairs.
  
  If  no  replacement  of  components  is  needed  then  one should better use
  ConstructPermuted  (5.8-11),  because  the  system can then exploit the fact
  that the two tables are permutation equivalent.
  
  5.8-13 ConstructFactor
  
  ConstructFactor( tbl, libnam, kernel )  function
  
  The  library table tbl is completed with help of the library table with name
  libnam, by factoring out the classes in the list kernel.
  

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