#############################################################################
##
#W construc.gd GAP 4 package CTblLib Thomas Breuer
##
## 1. Character Tables of Groups of Structure $M.G.A$
## 2. Character Tables of Groups of Structure $G.S_3$
## 3. Character Tables of Groups of Structure $G.2^2$
## 4. Character Tables of Groups of Structure $2^2.G$
## 5. Character Tables of Subdirect Products of Index Two
## 6. Brauer Tables of Extensions by $p$-regular Automorphisms
## 7. Construction Functions used in the Character Table Library
## 8. Character Tables of Coprime Central Extensions
## 9. Miscellaneous
##
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##
## <#GAPDoc Label="construc:intro">
## 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 <Ref Sect="Constructing Character Tables from Others"
## BookName="ref"/>.
## But since they are used in situations that are typical for the &GAP;
## Character Table Library, they are described here.
## <P/>
## 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 <Ref Func="AutomorphismsOfTable" BookName="ref"/>.
## <P/>
## Section <Ref Sect="sec:construc:MGA"/> deals with
## groups of the structure <M>M.G.A</M>,
## where the upwards extension <M>G.A</M> acts suitably
## on the central extension <M>M.G</M>.
## Section <Ref Sect="sec:construc:GS3"/> deals with
## groups that have a factor group of type <M>S_3</M>.
## Section <Ref Sect="sec:construc:GV4"/> deals with
## upward extensions of a group by a Klein four group.
## Section <Ref Sect="sec:construc:V4G"/> deals with
## downward extensions of a group by a Klein four group.
## Section <Ref Sect="sec:construc:preg"/> describes
## the construction of certain Brauer tables.
## Section <Ref Sect="sec:construc:cenex"/> deals with
## special cases of the construction of character tables of central
## extensions from known character tables of suitable factor groups.
## Section <Ref Sect="sec:construc:functions"/> documents
## the functions used to encode certain tables in the &GAP;
## Character Table Library.
## <P/>
## Examples can be found in <Cite Key="CCE"/> and <Cite Key="Auto"/>.
## <#/GAPDoc>
##
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##
## 1. Character Tables of Groups of Structure <M>M.G.A</M>
##
#############################################################################
##
#F PossibleCharacterTablesOfTypeMGA( <tblMG>, <tblG>, <tblGA>, <orbs>,
#F <identifier> )
##
## <#GAPDoc Label="PossibleCharacterTablesOfTypeMGA">
## <ManSection>
## <Func Name="PossibleCharacterTablesOfTypeMGA"
## Arg="tblMG, tblG, tblGA, orbs, identifier"/>
##
## <Description>
## Let <M>H</M>, <M>N</M>, and <M>M</M> be as described at the beginning of
## the section.
## <P/>
## Let <A>tblMG</A>, <A>tblG</A>, <A>tblGA</A> be the ordinary character
## tables of the groups <M>M.G = N</M>, <M>G</M>, and <M>G.A = H/M</M>,
## respectively,
## and <A>orbs</A> be the list of orbits on the class positions of
## <A>tblMG</A> that is induced by the action of <M>H</M> on <M>M.G</M>.
## Furthermore, let the class fusions from <A>tblMG</A> to <A>tblG</A>
## and from <A>tblG</A> to <A>tblGA</A> be stored on <A>tblMG</A>
## and <A>tblG</A>, respectively
## (see <Ref Func="StoreFusion" BookName="ref"/>).
## <P/>
## <Ref Func="PossibleCharacterTablesOfTypeMGA"/> returns a list of records
## describing all possible ordinary character tables
## for groups <M>H</M> that are compatible with the arguments.
## Note that in general there may be several possible groups <M>H</M>,
## and it may also be that <Q>character tables</Q> are constructed
## for which no group exists.
## <P/>
## Each of the records in the result has the following components.
## <List>
## <Mark><C>table</C></Mark>
## <Item>
## a possible ordinary character table for <M>H</M>, and
## </Item>
## <Mark><C>MGfusMGA</C></Mark>
## <Item>
## the fusion map from <A>tblMG</A> into the table stored in <C>table</C>.
## </Item>
## </List>
## The possible tables differ w. r. t. some power maps,
## and perhaps element orders and table automorphisms;
## in particular, the <C>MGfusMGA</C> component is the same in all records.
## <P/>
## The returned tables have the
## <Ref Attr="Identifier" Label="for character tables" BookName="ref"/>
## value <A>identifier</A>.
## The classes of these tables are sorted as follows.
## First come the classes contained in <M>M.G</M>,
## sorted compatibly with the classes in <A>tblMG</A>,
## then the classes in <M>H \setminus M.G</M> follow,
## in the same ordering as the classes of <M>G.A \setminus G</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "PossibleCharacterTablesOfTypeMGA" );
#############################################################################
##
#F BrauerTableOfTypeMGA( <modtblMG>, <modtblGA>, <ordtblMGA> )
##
## <#GAPDoc Label="BrauerTableOfTypeMGA">
## <ManSection>
## <Func Name="BrauerTableOfTypeMGA" Arg="modtblMG, modtblGA, ordtblMGA"/>
##
## <Description>
## Let <M>H</M>, <M>N</M>, and <M>M</M> be as described at the beginning of
## the section,
## let <A>modtblMG</A> and <A>modtblGA</A> be the <M>p</M>-modular character
## tables of the groups <M>N</M> and <M>H/M</M>, respectively, and let
## <A>ordtblMGA</A> be the <M>p</M>-modular Brauer table of <M>H</M>,
## for some prime integer <M>p</M>.
## Furthermore, let the class fusions from the ordinary character table of
## <A>modtblMG</A> to <A>ordtblMGA</A> and from <A>ordtblMGA</A> to the
## ordinary character table of <A>modtblGA</A> be stored.
## <P/>
## <Ref Func="BrauerTableOfTypeMGA"/> returns the <M>p</M>-modular character
## table of <M>H</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "BrauerTableOfTypeMGA" );
#############################################################################
##
#F PossibleActionsForTypeMGA( <tblMG>, <tblG>, <tblGA> )
##
## <#GAPDoc Label="PossibleActionsForTypeMGA">
## <ManSection>
## <Func Name="PossibleActionsForTypeMGA" Arg="tblMG, tblG, tblGA"/>
##
## <Description>
## Let the arguments be as described for
## <Ref Func="PossibleCharacterTablesOfTypeMGA"/>.
## <Ref Func="PossibleActionsForTypeMGA"/> returns the set of
## orbit structures <M>\Omega</M> on the class positions of <A>tblMG</A>
## that can be induced by the action of <M>H</M> on the classes of
## <M>M.G</M> in the sense that <M>\Omega</M> is the set of orbits
## of a table automorphism of <A>tblMG</A>
## (see <Ref Func="AutomorphismsOfTable" BookName="ref"/>)
## that is compatible with the stored class fusions from <A>tblMG</A>
## to <A>tblG</A> and from <A>tblG</A> to <A>tblGA</A>.
## Note that the number of such orbit structures can be smaller
## than the number of the underlying table automorphisms.
## <P/>
## Information about the progress is reported if the info level of
## <Ref InfoClass="InfoCharacterTable" BookName="ref"/> is at least <M>1</M>
## (see <Ref Func="SetInfoLevel" BookName="ref"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "PossibleActionsForTypeMGA" );
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##
## 2. Character Tables of Groups of Structure <M>G.S_3</M>
##
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##
#F CharacterTableOfTypeGS3( <tbl>, <tbl2>, <tbl3>, <aut>, <identifier> )
#F CharacterTableOfTypeGS3( <modtbl>, <modtbl2>, <modtbl3>, <ordtbls3>,
#F <identifier> )
##
## <#GAPDoc Label="CharacterTableOfTypeGS3">
## <ManSection>
## <Heading>CharacterTableOfTypeGS3</Heading>
## <Func Name="CharacterTableOfTypeGS3"
## Arg="tbl, tbl2, tbl3, aut, identifier"/>
## <Func Name="CharacterTableOfTypeGS3"
## Arg="modtbl, modtbl2, modtbl3, ordtbls3, identifier"
## Label="for Brauer tables"/>
##
## <Description>
## Let <M>H</M> be a group with a normal subgroup <M>G</M>
## such that <M>H/G \cong S_3</M>, the symmetric group on three points,
## and let <M>G.2</M> and <M>G.3</M> be preimages of subgroups of order
## <M>2</M> and <M>3</M>, respectively,
## under the natural projection onto this factor group.
## <P/>
## In the first form, let <A>tbl</A>, <A>tbl2</A>, <A>tbl3</A> be
## the ordinary character tables of the groups <M>G</M>, <M>G.2</M>,
## and <M>G.3</M>, respectively,
## and <A>aut</A> be the permutation of classes of <A>tbl3</A>
## induced by the action of <M>H</M> on <M>G.3</M>.
## Furthermore assume that the class fusions from <A>tbl</A> to <A>tbl2</A>
## and <A>tbl3</A> are stored on <A>tbl</A>
## (see <Ref Func="StoreFusion" BookName="ref"/>).
## In particular, the two class fusions must be compatible in the sense that
## the induced action on the classes of <A>tbl</A> describes an action of
## <M>S_3</M>.
## <P/>
## In the second form, let <A>modtbl</A>, <A>modtbl2</A>, <A>modtbl3</A> be
## the <M>p</M>-modular character tables of the groups <M>G</M>, <M>G.2</M>,
## and <M>G.3</M>, respectively,
## and <A>ordtbls3</A> be the ordinary character table of <M>H</M>.
## <P/>
## <Ref Func="CharacterTableOfTypeGS3"/> returns a record with the following
## components.
## <List>
## <Mark><C>table</C></Mark>
## <Item>
## the ordinary or <M>p</M>-modular character table of <M>H</M>,
## respectively,
## </Item>
## <Mark><C>tbl2fustbls3</C></Mark>
## <Item>
## the fusion map from <A>tbl2</A> into the table of <M>H</M>, and
## </Item>
## <Mark><C>tbl3fustbls3</C></Mark>
## <Item>
## the fusion map from <A>tbl3</A> into the table of <M>H</M>.
## </Item>
## </List>
## <P/>
## The returned table of <M>H</M> has the
## <Ref Attr="Identifier" Label="for character tables" BookName="ref"/>
## value <A>identifier</A>.
## The classes of the table of <M>H</M> are sorted as follows.
## First come the classes contained in <M>G.3</M>,
## sorted compatibly with the classes in <A>tbl3</A>,
## then the classes in <M>H \setminus G.3</M> follow,
## in the same ordering as the classes of <M>G.2 \setminus G</M>.
## <P/>
## In fact the code is applicable in the more general case that <M>H/G</M>
## is a Frobenius group <M>F = K C</M> with abelian kernel <M>K</M>
## and cyclic complement <M>C</M> of prime order,
## see <Cite Key="Auto"/>.
## Besides <M>F = S_3</M>,
## e. g., the case <M>F = A_4</M> is interesting.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "CharacterTableOfTypeGS3" );
#############################################################################
##
#F PossibleActionsForTypeGS3( <tbl>, <tbl2>, <tbl3> )
##
## <#GAPDoc Label="PossibleActionsForTypeGS3">
## <ManSection>
## <Func Name="PossibleActionsForTypeGS3" Arg="tbl, tbl2, tbl3"/>
##
## <Description>
## Let the arguments be as described for
## <Ref Func="CharacterTableOfTypeGS3"/>.
## <Ref Func="PossibleActionsForTypeGS3"/> returns the set of those
## table automorphisms
## (see <Ref Func="AutomorphismsOfTable" BookName="ref"/>)
## of <A>tbl3</A> that can be induced by the action of <M>H</M>
## on the classes of <A>tbl3</A>.
## <P/>
## Information about the progress is reported if the info level of
## <Ref InfoClass="InfoCharacterTable" BookName="ref"/> is at least <M>1</M>
## (see <Ref Func="SetInfoLevel" BookName="ref"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "PossibleActionsForTypeGS3" );
#############################################################################
##
## 3. Character Tables of Groups of Structure <M>G.2^2</M>
##
## <#GAPDoc Label="construc:GV4">
## The following functions are thought for constructing the possible
## ordinary character tables of a group of structure <M>G.2^2</M>
## from the known tables of the three normal subgroups of type <M>G.2</M>.
## <#/GAPDoc>
##
#############################################################################
##
#F PossibleCharacterTablesOfTypeGV4( <tblG>, <tblsG2>, <acts>, <identifier>
#F [, <tblGfustblsG2>] )
#F PossibleCharacterTablesOfTypeGV4( <modtblG>, <modtblsG2>, <ordtblGV4>
#F [, <ordtblsG2fusordtblG4>] )
##
## <#GAPDoc Label="PossibleCharacterTablesOfTypeGV4">
## <ManSection>
## <Heading>PossibleCharacterTablesOfTypeGV4</Heading>
## <Func Name="PossibleCharacterTablesOfTypeGV4"
## Arg="tblG, tblsG2, acts, identifier[, tblGfustblsG2]"/>
## <Func Name="PossibleCharacterTablesOfTypeGV4"
## Arg="modtblG, modtblsG2, ordtblGV4[, ordtblsG2fusordtblG4]"
## Label="for Brauer tables"/>
##
## <Description>
## Let <M>H</M> be a group with a normal subgroup <M>G</M>
## such that <M>H/G</M> is a Klein four group,
## and let <M>G.2_1</M>, <M>G.2_2</M>, and <M>G.2_3</M> be
## the three subgroups of index two in <M>H</M> that contain <M>G</M>.
## <P/>
## In the first version, let <A>tblG</A> be the ordinary character table
## of <M>G</M>,
## let <A>tblsG2</A> be a list containing the three character tables of the
## groups <M>G.2_i</M>, and
## let <A>acts</A> be a list of three permutations describing
## the action of <M>H</M> on the conjugacy classes
## of the corresponding tables in <A>tblsG2</A>.
## If the class fusions from <A>tblG</A> into the tables in <A>tblsG2</A>
## are not stored on <A>tblG</A>
## (for example, because the three tables are equal)
## then the three maps must be entered in the list <A>tblGfustblsG2</A>.
## <P/>
## In the second version, let <A>modtblG</A> be the <M>p</M>-modular
## character table of <M>G</M>, <A>modtblsG</A> be the list of
## <M>p</M>-modular Brauer tables of the groups <M>G.2_i</M>,
## and <A>ordtblGV4</A> be the ordinary character table of <M>H</M>.
## In this case, the class fusions from the ordinary character tables of
## the groups <M>G.2_i</M> to <A>ordtblGV4</A> can be entered in the list
## <A>ordtblsG2fusordtblG4</A>.
## <P/>
## <Ref Func="PossibleCharacterTablesOfTypeGV4"/> returns a list of records
## describing all possible (ordinary or <M>p</M>-modular) character tables
## for groups <M>H</M> that are compatible with the arguments.
## Note that in general there may be several possible groups <M>H</M>,
## and it may also be that <Q>character tables</Q> are constructed
## for which no group exists.
## Each of the records in the result has the following components.
## <P/>
## <List>
## <Mark><C>table</C></Mark>
## <Item>
## a possible (ordinary or <M>p</M>-modular) character table for <M>H</M>,
## and
## </Item>
## <Mark><C>G2fusGV4</C></Mark>
## <Item>
## the list of fusion maps from the tables in <A>tblsG2</A> into the
## <C>table</C> component.
## </Item>
## </List>
## <P/>
## The possible tables differ w.r.t. the irreducible characters and perhaps
## the table automorphisms;
## in particular, the <C>G2fusGV4</C> component is the same in all records.
## <P/>
## The returned tables have the
## <Ref Attr="Identifier" Label="for character tables" BookName="ref"/>
## value <A>identifier</A>.
## The classes of these tables are sorted as follows.
## First come the classes contained in <M>G</M>, sorted compatibly with the
## classes in <A>tblG</A>,
## then the outer classes in the tables in <A>tblsG2</A> follow,
## in the same ordering as in these tables.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "PossibleCharacterTablesOfTypeGV4" );
#############################################################################
##
#F PossibleActionsForTypeGV4( <tblG>, <tblsG2> )
##
## <#GAPDoc Label="PossibleActionsForTypeGV4">
## <ManSection>
## <Func Name="PossibleActionsForTypeGV4" Arg="tblG, tblsG2"/>
##
## <Description>
## Let the arguments be as described for
## <Ref Func="PossibleCharacterTablesOfTypeGV4"/>.
## <Ref Func="PossibleActionsForTypeGV4"/> returns the list of those triples
## <M>[ \pi_1, \pi_2, \pi_3 ]</M> of permutations
## for which a group <M>H</M> may exist
## that contains <M>G.2_1</M>, <M>G.2_2</M>, <M>G.2_3</M>
## as index <M>2</M> subgroups
## which intersect in the index <M>4</M> subgroup <M>G</M>.
## <P/>
## Information about the progress is reported if the level of
## <Ref InfoClass="InfoCharacterTable" BookName="ref"/> is at least <M>1</M>
## (see <Ref Func="SetInfoLevel" BookName="ref"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "PossibleActionsForTypeGV4" );
#############################################################################
##
## 4. Character Tables of Groups of Structure <M>2^2.G</M>
##
## <#GAPDoc Label="construc:V4G">
## The following functions are thought for constructing the possible
## ordinary or Brauer character tables of a group of structure <M>2^2.G</M>
## from the known tables of the three factor groups
## modulo the normal order two subgroups in the central Klein four group.
## <P/>
## 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.
## <#/GAPDoc>
##
#############################################################################
##
#F PossibleCharacterTablesOfTypeV4G( <tblG>, <tbls2G>, <id>[, <fusions>] )
#F PossibleCharacterTablesOfTypeV4G( <tblG>, <tbl2G>, <aut>, <id> )
##
## <#GAPDoc Label="PossibleCharacterTablesOfTypeV4G">
## <ManSection>
## <Heading>PossibleCharacterTablesOfTypeV4G</Heading>
## <Func Name="PossibleCharacterTablesOfTypeV4G"
## Arg="tblG, tbls2G, id[, fusions]"/>
## <Func Name="PossibleCharacterTablesOfTypeV4G"
## Arg="tblG, tbl2G, aut, id"
## Label="for conj. ordinary tables, and an autom."/>
##
## <Description>
## Let <M>H</M> be a group with a central subgroup <M>N</M>
## of type <M>2^2</M>,
## and let <M>Z_1</M>, <M>Z_2</M>, <M>Z_3</M> be
## the order <M>2</M> subgroups of <M>N</M>.
## <P/>
## In the first form, let <A>tblG</A> be the ordinary character table
## of <M>H/N</M>,
## and <A>tbls2G</A> be a list of length three,
## the entries being the ordinary character tables
## of the groups <M>H/Z_i</M>.
## In the second form, let <A>tbl2G</A> be the ordinary character table of
## <M>H/Z_1</M> and <A>aut</A> be a permutation;
## here it is assumed that the groups <M>Z_i</M> are permuted under an
## automorphism <M>\sigma</M> of order <M>3</M> of <M>H</M>,
## and that <M>\sigma</M> induces the permutation <A>aut</A>
## on the classes of <A>tblG</A>.
## <P/>
## The class fusions onto <A>tblG</A> are assumed to be stored
## on the tables in <A>tbls2G</A> or <A>tbl2G</A>, respectively,
## except if they are explicitly entered via the optional argument
## <A>fusions</A>.
## <P/>
## <Ref Func="PossibleCharacterTablesOfTypeV4G"/> returns the list of all
## possible character tables for <M>H</M> in this situation.
## The returned tables have the
## <Ref Attr="Identifier" Label="for character tables" BookName="ref"/>
## value <A>id</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
#T which criteria are used?
##
DeclareGlobalFunction( "PossibleCharacterTablesOfTypeV4G" );
#############################################################################
##
#F BrauerTableOfTypeV4G( <ordtblV4G>, <modtbls2G> )
#F BrauerTableOfTypeV4G( <ordtblV4G>, <modtbl2G>, <aut> )
##
## <#GAPDoc Label="BrauerTableOfTypeV4G">
## <ManSection>
## <Heading>BrauerTableOfTypeV4G</Heading>
## <Func Name="BrauerTableOfTypeV4G" Arg="ordtblV4G, modtbls2G"
## Label="for three factors"/>
## <Func Name="BrauerTableOfTypeV4G" Arg="ordtblV4G, modtbl2G, aut"
## Label="for one factor and an autom."/>
##
## <Description>
## Let <M>H</M> be a group with a central subgroup <M>N</M>
## of type <M>2^2</M>,
## and let <A>ordtblV4G</A> be the ordinary character table of <M>H</M>.
## Let <M>Z_1</M>, <M>Z_2</M>, <M>Z_3</M> be
## the order <M>2</M> subgroups of <M>N</M>.
## In the first form,
## let <A>modtbls2G</A> be the list of the <M>p</M>-modular Brauer tables
## of the factor groups <M>H/Z_1</M>, <M>H/Z_2</M>, and <M>H/Z_3</M>,
## for some prime integer <M>p</M>.
## In the second form, let <A>modtbl2G</A> be the <M>p</M>-modular Brauer
## table of <M>H/Z_1</M> and <A>aut</A> be a permutation;
## here it is assumed that the groups <M>Z_i</M> are permuted under an
## automorphism <M>\sigma</M> of order <M>3</M> of <M>H</M>,
## and that <M>\sigma</M> induces the permutation <A>aut</A>
## on the classes of the ordinary character table of <M>H</M> that is stored
## in <A>ordtblV4G</A>.
## <P/>
## The class fusions from <A>ordtblV4G</A> to the ordinary character tables
## of the tables in <A>modtbls2G</A> or <A>modtbl2G</A> are assumed to be
## stored.
## <P/>
## <Ref Func="BrauerTableOfTypeV4G" Label="for three factors"/> returns
## the <M>p</M>-modular character table of <M>H</M>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "BrauerTableOfTypeV4G" );
#############################################################################
##
## 5. Character Tables of Subdirect Products of Index Two
##
## <#GAPDoc Label="construc:subdirindex2">
## 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.
## <#/GAPDoc>
##
#############################################################################
##
#F CharacterTableOfIndexTwoSubdirectProduct( <tblH1>, <tblG1>,
#F <tblH2>, <tblG2>, <identifier> )
##
## <#GAPDoc Label="CharacterTableOfIndexTwoSubdirectProduct">
## <ManSection>
## <Func Name="CharacterTableOfIndexTwoSubdirectProduct"
## Arg='tblH1, tblG1, tblH2, tblG2, identifier'/>
##
## <Returns>
## a record containing the character table of the subdirect product <M>G</M>
## that is described by the first four arguments.
## </Returns>
##
## <Description>
## Let <A>tblH1</A>, <A>tblG1</A>, <A>tblH2</A>, <A>tblG2</A> be the
## character tables of groups <M>H_1</M>, <M>G_1</M>, <M>H_2</M>,
## <M>G_2</M>, such that <M>H_1</M> and <M>H_2</M> have index two
## in <M>G_1</M> and <M>G_2</M>, respectively, and such that the class
## fusions corresponding to these embeddings are stored on <A>tblH1</A> and
## <A>tblH1</A>, respectively.
## <P/>
## In this situation, the direct product of <M>G_1</M> and <M>G_2</M>
## contains a unique subgroup <M>G</M> of index two
## that contains the direct product of <M>H_1</M> and <M>H_2</M>
## but does not contain any of the groups <M>G_1</M>, <M>G_2</M>.
## <P/>
## The function <Ref Func="CharacterTableOfIndexTwoSubdirectProduct"/>
## returns a record with the following components.
## <List>
## <Mark><C>table</C></Mark>
## <Item>
## the character table of <M>G</M>,
## </Item>
## <Mark><C>H1fusG</C></Mark>
## <Item>
## the class fusion from <A>tblH1</A> into the table of <M>G</M>, and
## </Item>
## <Mark><C>H2fusG</C></Mark>
## <Item>
## the class fusion from <A>tblH2</A> into the table of <M>G</M>.
## </Item>
## </List>
## <P/>
## If the first four arguments are <E>ordinary</E> character tables
## then the fifth argument <A>identifier</A> must be a string;
## this is used as the
## <Ref Attr="Identifier" Label="for character tables" BookName="ref"/>
## value of the result table.
## <P/>
## If the first four arguments are <E>Brauer</E> character tables for the
## same characteristic then the fifth argument must be the ordinary
## character table of the desired subdirect product.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "CharacterTableOfIndexTwoSubdirectProduct" );
#############################################################################
##
#F ConstructIndexTwoSubdirectProduct( <tbl>, <tblH1>, <tblG1>, <tblH2>,
#F <tblG2>, <permclasses>, <permchars> )
##
## <#GAPDoc Label="ConstructIndexTwoSubdirectProduct">
## <ManSection>
## <Func Name="ConstructIndexTwoSubdirectProduct"
## Arg='tbl, tblH1, tblG1, tblH2, tblG2, permclasses, permchars'/>
##
## <Description>
## <Ref Func="ConstructIndexTwoSubdirectProduct"/> constructs the
## irreducible characters of the ordinary character table <A>tbl</A> of the
## subdirect product of index two in the direct product of <A>tblG1</A> and
## <A>tblG2</A>, which contains the direct product of <A>tblH1</A> and
## <A>tblH2</A> but does not contain any of the direct factors <A>tblG1</A>,
## <A>tblG2</A>.
## W. r. t. the default ordering obtained from that given by
## <Ref Oper="CharacterTableDirectProduct" BookName="ref"/>,
## the columns and the rows of the matrix of irreducibles are permuted with
## the permutations <A>permclasses</A> and <A>permchars</A>, respectively.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ConstructIndexTwoSubdirectProduct" );
#############################################################################
##
#F ConstructIndexTwoSubdirectProductInfo( <tbl>[, <tblH1>, <tblG1>,
#F <tblH2>, <tblG2>] )
##
## <#GAPDoc Label="ConstructIndexTwoSubdirectProductInfo">
## <ManSection>
## <Func Name="ConstructIndexTwoSubdirectProductInfo"
## Arg='tbl[, tblH1, tblG1, tblH2, tblG2 ]'/>
##
## <Returns>
## a list of constriction descriptions, or a construction description,
## or <K>fail</K>.
## </Returns>
## <Description>
## Called with one argument <A>tbl</A>, an ordinary character table of the
## group <M>G</M>, say,
## <Ref Func="ConstructIndexTwoSubdirectProductInfo"/> analyzes the
## possibilities to construct <A>tbl</A> from character tables of subgroups
## <M>H_1</M>, <M>H_2</M> and factor groups <M>G_1</M>, <M>G_2</M>,
## using <Ref Func="CharacterTableOfIndexTwoSubdirectProduct"/>.
## The return value is a list of records with the following components.
## <List>
## <Mark><C>kernels</C></Mark>
## <Item>
## the list of class positions of <M>H_1</M>, <M>H_2</M> in <A>tbl</A>,
## </Item>
## <Mark><C>kernelsizes</C></Mark>
## <Item>
## the list of orders of <M>H_1</M>, <M>H_2</M>,
## </Item>
## <Mark><C>factors</C></Mark>
## <Item>
## the list of
## <Ref Attr="Identifier" Label="for character tables" BookName="ref"/>
## values of the &GAP; library tables of the factors <M>G_2</M>,
## <M>G_1</M> of <M>G</M> by <M>H_1</M>, <M>H_2</M>;
## if no such table is available then the entry is <K>fail</K>, and
## </Item>
## <Mark><C>subgroups</C></Mark>
## <Item>
## the list of
## <Ref Attr="Identifier" Label="for character tables" BookName="ref"/>
## values of the &GAP; library tables of the subgroups <M>H_2</M>,
## <M>H_1</M> of <M>G</M>;
## if no such tables are available then the entries are <K>fail</K>.
## </Item>
## </List>
## <P/>
## If the returned list is empty then either <A>tbl</A> does not have the
## desired structure as a subdirect product,
## <E>or</E> <A>tbl</A> is in fact a nontrivial direct product.
## <P/>
## Called with five arguments, the ordinary character tables of <M>G</M>,
## <M>H_1</M>, <M>G_1</M>, <M>H_2</M>, <M>G_2</M>,
## <Ref Func="ConstructIndexTwoSubdirectProductInfo"/> returns a list that
## can be used as the <Ref Attr="ConstructionInfoCharacterTable"/> value
## for the character table of <M>G</M> from the other four character tables
## using <Ref Func="CharacterTableOfIndexTwoSubdirectProduct"/>;
## if this is not possible then <K>fail</K> is returned.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ConstructIndexTwoSubdirectProductInfo" );
#############################################################################
##
## 6. Brauer Tables of Extensions by <M>p</M>-regular Automorphisms
##
## <#GAPDoc Label="construc:preg">
## As for the construction of Brauer character tables from known tables,
## the functions <Ref Func="PossibleCharacterTablesOfTypeMGA"/>,
## <Ref Func="CharacterTableOfTypeGS3"/>,
## and <Ref Func="PossibleCharacterTablesOfTypeGV4"/>
## work for both ordinary and Brauer tables.
## The following function is designed specially for Brauer tables.
## <#/GAPDoc>
##
#############################################################################
##
#F IBrOfExtensionBySingularAutomorphism( <modtbl>, <perm> )
#F IBrOfExtensionBySingularAutomorphism( <modtbl>, <orbits> )
#F IBrOfExtensionBySingularAutomorphism( <modtbl>, <ordexttbl> )
##
## <#GAPDoc Label="IBrOfExtensionBySingularAutomorphism">
## <ManSection>
## <Func Name="IBrOfExtensionBySingularAutomorphism" Arg="modtbl, act"/>
##
## <Description>
## Let <A>modtbl</A> be a <M>p</M>-modular Brauer table
## of the group <M>G</M>, say,
## and suppose that the group <M>H</M>, say,
## is an upward extension of <M>G</M> by an automorphism of order <M>p</M>.
## <P/>
## The second argument <A>act</A> describes the action of this automorphism.
## It can be either a permutation of the columns of <A>modtbl</A>,
## or a list of the <M>H</M>-orbits on the columns of <A>modtbl</A>,
## or the ordinary character table of <M>H</M>
## such that the class fusion from the ordinary table of <A>modtbl</A> into
## this table is stored.
## In all these cases, <Ref Func="IBrOfExtensionBySingularAutomorphism"/>
## returns the values lists of the irreducible <M>p</M>-modular
## Brauer characters of <M>H</M>.
## <P/>
## Note that the table head of the <M>p</M>-modular Brauer table of
## <M>H</M>, in general without the <Ref Attr="Irr" BookName="ref"/>
## attribute, can be obtained
## by applying <Ref Func="CharacterTableRegular" BookName="ref"/> to the
## ordinary character table of <M>H</M>,
## but <Ref Func="IBrOfExtensionBySingularAutomorphism"/> can be used
## also if the ordinary character table of <M>H</M> is not known,
## and just the <M>p</M>-modular character table of <M>G</M> and the action
## of <M>H</M> on the classes of <M>G</M> are given.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "IBrOfExtensionBySingularAutomorphism" );
#############################################################################
##
## 7. Construction Functions used in the Character Table Library
##
## <#GAPDoc Label="construc:functions">
## The following functions are used in the &GAP; Character Table Library,
## for encoding table constructions via the mechanism that is based on the
## attribute <Ref Attr="ConstructionInfoCharacterTable"/>.
## 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.
## <#/GAPDoc>
##
#############################################################################
##
#F ConstructMGA( <tbl>, <subname>, <factname>, <plan>, <perm> )
##
## <#GAPDoc Label="ConstructMGA">
## <ManSection>
## <Func Name="ConstructMGA" Arg="tbl, subname, factname, plan, perm"/>
##
## <Description>
## <Ref Func="ConstructMGA"/> constructs the irreducible characters of the
## ordinary character table <A>tbl</A> of a group <M>m.G.a</M>
## where the automorphism <M>a</M> (a group of prime order) of <M>m.G</M>
## acts nontrivially on the central subgroup <M>m</M> of <M>m.G</M>.
## <A>subname</A> is the name of the subgroup <M>m.G</M> which is a
## (not necessarily cyclic) central extension of the
## (not necessarily simple) group <M>G</M>,
## <A>factname</A> is the name of the factor group <M>G.a</M>.
## Then the faithful characters of <A>tbl</A> are induced from <M>m.G</M>.
## <P/>
## <A>plan</A> is a list, each entry being a list containing positions of
## characters of <M>m.G</M> that form an orbit under the action of <M>a</M>
## (the induction of characters is encoded this way).
## <P/>
## <A>perm</A> 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
## <M>12.G.2</M> that are encoded via the subgroup <M>12.G</M>
## and the factor group <M>6.G.2</M>,
## where the faithful characters of <M>4.G.2</M> shall precede those
## of <M>6.G.2</M>,
## as in the &ATLAS;.
## <P/>
## Examples where <Ref Func="ConstructMGA"/> is used
## to encode library tables are the tables of <M>3.F_{{3+}}.2</M>
## (subgroup <M>3.F_{{3+}}</M>, factor group <M>F_{{3+}}.2</M>)
## and <M>12_1.U_4(3).2_2</M>
## (subgroup <M>12_1.U_4(3)</M>, factor group <M>6_1.U_4(3).2_2</M>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ConstructMGA" );
DeclareSynonym( "ConstructMixed", ConstructMGA );
#############################################################################
##
#F ConstructMGAInfo( <tblmGa>, <tblmG>, <tblGa> )
##
## <#GAPDoc Label="ConstructMGAInfo">
## <ManSection>
## <Func Name="ConstructMGAInfo" Arg="tblmGa, tblmG, tblGa"/>
##
## <Description>
## Let <A>tblmGa</A> be the ordinary character table of a group of structure
## <M>m.G.a</M>
## where the factor group of prime order <M>a</M> acts nontrivially on
## the normal subgroup of order <M>m</M> that is central in <M>m.G</M>,
## <A>tblmG</A> be the character table of <M>m.G</M>,
## and <A>tblGa</A> be the character table of the factor group <M>G.a</M>.
## <P/>
## <Ref Func="ConstructMGAInfo"/> returns the list that is to be stored
## in the library version of <A>tblmGa</A>:
## the first entry is the string <C>"ConstructMGA"</C>,
## the remaining four entries are the last four arguments for the call to
## <Ref Func="ConstructMGA"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ConstructMGAInfo" );
#############################################################################
##
#F ConstructGS3( <tbls3>, <tbl2>, <tbl3>, <ind2>, <ind3>, <ext>, <perm> )
#F ConstructGS3Info( <tbl2>, <tbl3>, <tbls3> )
##
## <#GAPDoc Label="ConstructGS3">
## <ManSection>
## <Func Name="ConstructGS3"
## Arg="tbls3, tbl2, tbl3, ind2, ind3, ext, perm"/>
## <Func Name="ConstructGS3Info" Arg="tbl2, tbl3, tbls3"/>
##
## <Description>
## <Ref Func="ConstructGS3"/> constructs the irreducibles
## of an ordinary character table <A>tbls3</A> of type <M>G.S_3</M>
## from the tables with names <A>tbl2</A> and <A>tbl3</A>,
## which correspond to the groups <M>G.2</M> and <M>G.3</M>, respectively.
## <A>ind2</A> is a list of numbers referring to irreducibles of
## <A>tbl2</A>.
## <A>ind3</A> is a list of pairs, each referring to irreducibles of
## <A>tbl3</A>.
## <A>ext</A> is a list of pairs, each referring to one irreducible
## character of <A>tbl2</A> and one of <A>tbl3</A>.
## <A>perm</A> is a permutation that must be applied to the irreducibles
## after the construction.
## <P/>
## <Ref Func="ConstructGS3Info"/> returns a record with the components
## <C>ind2</C>, <C>ind3</C>, <C>ext</C>, <C>perm</C>, and <C>list</C>,
## as are needed for <Ref Func="ConstructGS3"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ConstructGS3" );
DeclareGlobalFunction( "ConstructGS3Info" );
#############################################################################
##
#F ConstructV4G( <tbl>, <facttbl>, <aut> )
#F ConstructV4GInfo( <tbl>, <facttbl>, <aut> )
##
## <#GAPDoc Label="ConstructV4G">
## <ManSection>
## <Func Name="ConstructV4G" Arg="tbl, facttbl, aut"/>
##
## <Description>
## Let <A>tbl</A> be the character table of a group of type <M>2^2.G</M>
## where an outer automorphism of order <M>3</M> permutes
## the three involutions in the central <M>2^2</M>.
## Let <A>aut</A> be the permutation of classes of <A>tbl</A>
## induced by that automorphism,
## and <A>facttbl</A> be the name of the character table
## of the factor group <M>2.G</M>.
## Then <Ref Func="ConstructV4G"/> constructs the irreducible characters
## of <A>tbl</A> from that information.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
## `ConstructV4GInfo' returns a list that is needed for `ConstructV4G'.
## The arguments are the character tables <tbl> and <facttbl> of $2^2.G$ and
## $2.G$, respectively, and the permutation <aut> of classes of <tbl> that
## is induced by the outer automorphism of order $3$.
##
DeclareGlobalFunction( "ConstructV4G" );
DeclareGlobalFunction( "ConstructV4GInfo" );
#############################################################################
##
#F ConstructProj( <tbl>, <irrinfo> )
#F ConstructProjInfo( <tbl>, <kernel> )
##
## <#GAPDoc Label="ConstructProj">
## <ManSection>
## <Func Name="ConstructProj" Arg="tbl, irrinfo"/>
## <Func Name="ConstructProjInfo" Arg="tbl, kernel"/>
##
## <Description>
## <Ref Func="ConstructProj"/> constructs the irreducible characters
## of the record encoding the ordinary character table <A>tbl</A>
## from projective characters of tables of factor groups,
## which are stored in the <Ref Func="ProjectivesInfo"/> value
## of the smallest factor;
## the information about the name of this factor and the projectives to
## take is stored in <A>irrinfo</A>.
## <P/>
## <Ref Func="ConstructProjInfo"/> takes an ordinary character table
## <A>tbl</A> and a list <A>kernel</A> of class positions of a cyclic kernel
## of order dividing <M>12</M>,
## and returns a record with the components
## <P/>
## <List>
## <Mark><C>tbl</C></Mark>
## <Item>
## a character table that is permutation isomorphic with <A>tbl</A>,
## and sorted such that classes that differ only by multiplication with
## elements in the classes of <A>kernel</A> are consecutive,
## </Item>
## <Mark><C>projectives</C></Mark>
## <Item>
## a record being the entry for the <C>projectives</C> list of the table
## of the factor of <A>tbl</A> by <A>kernel</A>,
## describing this part of the irreducibles of <A>tbl</A>, and
## </Item>
## <Mark><C>info</C></Mark>
## <Item>
## the value of <A>irrinfo</A> that is needed for constructing the
## irreducibles of the <C>tbl</C> component of the result (<E>not</E> the
## irreducibles of the argument <A>tbl</A>!)
## via <Ref Func="ConstructProj"/>.
## </Item>
## </List>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ConstructProj" );
DeclareGlobalFunction( "ConstructProjInfo" );
#############################################################################
##
#F ConstructDirectProduct( <tbl>, <factors>[, <permclasses>, <permchars>] )
##
## <#GAPDoc Label="ConstructDirectProduct">
## <ManSection>
## <Func Name="ConstructDirectProduct"
## Arg="tbl, factors[, permclasses, permchars]"/>
##
## <Description>
## The direct product of the library character tables described by the list
## <A>factors</A> of table names is constructed using
## <Ref Func="CharacterTableDirectProduct" BookName="ref"/>,
## and all its components that are not yet stored on <A>tbl</A> are
## added to <A>tbl</A>.
## <P/>
## The <Ref Attr="ComputedClassFusions" BookName="ref"/> value of <A>tbl</A>
## is enlarged by the factor fusions from the direct product to the factors.
## <P/>
## If the optional arguments <A>permclasses</A>, <A>permchars</A> are given
## then the classes and characters of the result are sorted accordingly.
## <P/>
## <A>factors</A> must have length at least two;
## use <Ref Func="ConstructPermuted"/> in the case of only one factor.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ConstructDirectProduct" );
#############################################################################
##
#F ConstructCentralProduct( <tbl>, <factors>, <Dclasses>
#F [, <permclasses>, <permchars>] )
##
## <#GAPDoc Label="ConstructCentralProduct">
## <ManSection>
## <Func Name="ConstructCentralProduct"
## Arg="tbl, factors, Dclasses[, permclasses, permchars]"/>
##
## <Description>
## The library table <A>tbl</A> is completed with help of the table
## obtained by taking the direct product of the tables with names in the
## list <A>factors</A>, and then factoring out the normal subgroup that is
## given by the list <A>Dclasses</A> of class positions.
## <P/>
## If the optional arguments <A>permclasses</A>, <A>permchars</A> are given
## then the classes and characters of the result are sorted accordingly.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ConstructCentralProduct" );
#############################################################################
##
#F ConstructSubdirect( <tbl>, <factors>, <choice> )
##
## <#GAPDoc Label="ConstructSubdirect">
## <ManSection>
## <Func Name="ConstructSubdirect" Arg="tbl, factors, choice"/>
##
## <Description>
## The library table <A>tbl</A> is completed with help of the table
## obtained by taking the direct product of the tables with names in the
## list <A>factors</A>, and then taking the table consisting of the classes
## in the list <A>choice</A>.
## <P/>
## 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.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ConstructSubdirect" );
#############################################################################
##
#F ConstructWreathSymmetric( <tbl>, <subname>, <n>
#F [, <permclasses>, <permchars>] )
##
## <#GAPDoc Label="ConstructWreathSymmetric">
## <ManSection>
## <Func Name="ConstructWreathSymmetric"
## Arg="tbl, subname, n[, permclasses, permchars]"/>
##
## <Description>
## The wreath product of the library character table with identifier value
## <A>subname</A> with the symmetric group on <A>n</A> points is constructed
## using <Ref Func="CharacterTableWreathSymmetric" BookName="ref"/>,
## and all its components that are not yet stored on <A>tbl</A> are
## added to <A>tbl</A>.
## <P/>
## If the optional arguments <A>permclasses</A>, <A>permchars</A> are given
## then the classes and characters of the result are sorted accordingly.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ConstructWreathSymmetric" );
#############################################################################
##
#F ConstructIsoclinic( <tbl>, <factors>[, <nsg>[, <centre>]]
#F [, <permclasses>, <permchars>] )
#F ConstructIsoclinic( <tbl>, <factors>[, <arec>]
#F [, <permclasses>, <permchars>] )
##
## <#GAPDoc Label="ConstructIsoclinic">
## <ManSection>
## <Func Name="ConstructIsoclinic"
## Arg="tbl, factors[, nsg[, centre]][, permclasses, permchars]"/>
##
## <Description>
## constructs first the direct product of library tables as given by the
## list <A>factors</A> of admissible character table names,
## and then constructs the isoclinic table of the result.
## <P/>
## If the argument <A>nsg</A> is present and a record or a list then
## <Ref Func="CharacterTableIsoclinic" BookName="ref"/> gets called,
## and <A>nsg</A> (as well as <A>centre</A> if present) is passed to this
## function.
## <P/>
## In both cases,
## if the optional arguments <A>permclasses</A>, <A>permchars</A> are given
## then the classes and characters of the result are sorted accordingly.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ConstructIsoclinic" );
#############################################################################
##
#F ConstructPermuted( <tbl>, <libnam>[, <permclasses>, <permchars>] )
##
## <#GAPDoc Label="ConstructPermuted">
## <ManSection>
## <Func Name="ConstructPermuted"
## Arg="tbl, libnam[, permclasses, permchars]"/>
##
## <Description>
## The library table <A>tbl</A> is computed from
## the library table with the name <A>libnam</A>,
## by permuting the classes and the characters by the permutations
## <A>permclasses</A> and <A>permchars</A>, respectively.
## <P/>
## So <A>tbl</A> and the library table with the name <A>libnam</A> are
## permutation equivalent.
## With the more general function <Ref Func="ConstructAdjusted"/>,
## one can derive character tables that are not necessarily permutation
## equivalent, by additionally replacing some defining data.
## <P/>
## 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 <Ref Func="ConstructPermuted"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ConstructPermuted" );
#############################################################################
##
#F ConstructAdjusted( <tbl>, <libnam>, <pairs>
#F [, <permclasses>, <permchars>] )
##
## <#GAPDoc Label="ConstructAdjusted">
## <ManSection>
## <Func Name="ConstructAdjusted"
## Arg="tbl, libnam, pairs[, permclasses, permchars]"/>
##
## <Description>
## The defining attribute values of the library table <A>tbl</A> are given
## by the attribute values described by the list <A>pairs</A> and
## –for those attributes which do not appear in <A>pairs</A>–
## by the attribute values of the library table with the name <A>libnam</A>,
## whose classes and characters have been permuted by the optional
## permutations <A>permclasses</A> and <A>permchars</A>, respectively.
## <P/>
## This construction can be used to derive a character table from another
## library table (the one with the name <A>libnam</A>) that is <E>not</E>
## 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
## <Ref Func="ConstructAdjusted"/>, by prescribing just the power maps in
## the list <A>pairs</A>.
## <P/>
## If no replacement of components is needed then one should better use
## <Ref Func="ConstructPermuted"/>,
## because the system can then exploit the fact that the two tables are
## permutation equivalent.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ConstructAdjusted" );
#############################################################################
##
#F ConstructFactor( <tbl>, <libnam>, <kernel> )
##
## <#GAPDoc Label="ConstructFactor">
## <ManSection>
## <Func Name="ConstructFactor" Arg="tbl, libnam, kernel"/>
##
## <Description>
## The library table <A>tbl</A> is completed with help of the library table
## with name <A>libnam</A>,
## by factoring out the classes in the list <A>kernel</A>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "ConstructFactor" );
#############################################################################
##
#F ConstructClifford( <tbl>, <cliffordtable> )
##
## constructs the irreducibles of the ordinary character table <tbl> from
## the Clifford matrices stored in `<tbl>.cliffordTable'.
##
DeclareGlobalFunction( "ConstructClifford" );
#############################################################################
##
## 8. Character Tables of Coprime Central Extensions
##
#############################################################################
##
#F CharacterTableOfCommonCentralExtension( <tblG>, <tblmG>, <tblnG>, <id> )
##
## <#GAPDoc Label="CharacterTableOfCommonCentralExtension">
## <ManSection>
## <Func Name="CharacterTableOfCommonCentralExtension"
## Arg="tblG, tblmG, tblnG, id"/>
##
## <Description>
## Let <A>tblG</A> be the ordinary character table of a group <M>G</M>, say,
## and let <A>tblmG</A> and <A>tblnG</A> be the ordinary character tables
## of central extensions <M>m.G</M> and <M>n.G</M> of <M>G</M>
## by cyclic groups of prime orders <M>m</M> and <M>n</M>, respectively,
## with <M>m \neq n</M>.
## We assume that the factor fusions from <A>tblmG</A> and <A>tblnG</A>
## to <A>tblG</A> are stored on the tables.
## <Ref Func="CharacterTableOfCommonCentralExtension"/> returns a record
## with the following components.
## <P/>
## <List>
## <Mark><C>tblmnG</C></Mark>
## <Item>
## the character table <M>t</M>, say, of the corresponding central
## extension of <M>G</M> by a cyclic group of order <M>m n</M>
## that factors through <M>m.G</M> and <M>n.G</M>;
## the
## <Ref Attr="Identifier" Label="for character tables" BookName="ref"/>
## value of this table is <A>id</A>,
## </Item>
## <Mark><C>IsComplete</C></Mark>
## <Item>
## <K>true</K> if the <Ref Attr="Irr" BookName="ref"/> value is stored
## in <M>t</M>, and <K>false</K> otherwise,
## </Item>
## <Mark><C>irreducibles</C></Mark>
## <Item>
## the list of irreducibles of <M>t</M> that are known;
## it contains the inflated characters of the factor groups <M>m.G</M> and
## <M>n.G</M>, plus those irreducibles that were found in tensor
## products of characters of these groups.
## </Item>
## </List>
## <P/>
## Note that the conjugacy classes and the power maps of <M>t</M>
## 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 <Ref Func="ReducedClassFunctions" BookName="ref"/>
## and <Ref Func="LLL" BookName="ref"/>).
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "CharacterTableOfCommonCentralExtension" );
#############################################################################
##
#F IrreduciblesForCharacterTableOfCommonCentralExtension(
#F <tblmnG>, <factirreducibles>, <zpos>, <needed> )
##
## This function implements a heuristic for finding the missing irreducible
## characters of a character table whose table head is constructed with
## `CharacterTableOfCommonCentralExtension'
## (see~"CharacterTableOfCommonCentralExtension").
## Currently reducing tensor products and applying the LLL algorithm are
## the only ingredients.
##
DeclareGlobalFunction(
"IrreduciblesForCharacterTableOfCommonCentralExtension" );
#############################################################################
##
## 9. Miscellaneous
##
#############################################################################
##
#F PossibleActionsForTypeGA( <tblG>, <tblGA> )
##
## Let <tblG> and <tblGA> be the ordinary character tables of a group
## <M>G</M> and of an extension <M>\tilde{G}</M> of <M>G</M>
## by an automorphism of order <M>A</M>, say.
##
## `PossibleActionsForTypeGA' returns the list of all those permutations
## that may describe the action of <M>\tilde{G}</M> on the classes
## of <tblG>, that is, all table automorphisms of <tblG> that have order
## dividing <M>A</M> and permute the classes of <tblG>
## compatibly with the fusion from <tblG> into <tblGA>.
##
DeclareGlobalFunction( "PossibleActionsForTypeGA" );
#T Replace the function by one that takes a perm. group and a fusion map!
#T The following two functions belong to the package for interactive
#T character table constructions;
#T but they are needed for `CharacterTableOfCommonCentralExtension'.
#############################################################################
##
#F ReducedX( <tbl>, <redresult>, <chars> )
##
## Let <tbl> be an ordinary character table, <redresult> be a result record
## returned by `Reduced' when called with first argument <tbl>, and <chars>
## be a list of characters of <tbl>.
## `ReducedX' first reduces <chars> with the `irreducibles' component of
## <redresult>; if new irreducibles are obtained this way then the
## characters in the `remainders' component of <redresult> are reduced with
## them; this process is iterated until no more irreducibles are found.
## The function returns a record with the following components.
##
## \beginitems
## `irreducibles' &
## all irreducible characters found during the process, including the
## `irreducibles' component of <redresult>,
##
## `remainders' &
## the reducible characters that are left from <chars> and the
## `remainders' component of <redresult>.
## \enditems
##
DeclareGlobalFunction( "ReducedX" );
#############################################################################
##
#F TensorAndReduce( <tbl>, <chars1>, <chars2>, <irreducibles>, <needed> )
##
## Let <tbl> be an ordinary character table, <chars1> and <chars2> be two
## lists of characters of <tbl>, <irreducibles> be a list of irreducible
## characters of <tbl>, and <needed> be a nonnegative integer.
## `TensorAndReduce' forms the tensor products of the characters in <chars1>
## with the characters in <chars2>, and reduces them with the characters in
## <irreducibles> and with all irreducible characters that are found this
## way.
## The function returns a record with the following components.
##
## \beginitems
## `irreducibles' &
## all new irreducible characters found during the process,
##
## `remainders' &
## the reducible characters that are left from the tensor products.
## \enditems
##
## When at least <needed> new irreducibles are found then the process is
## stopped immediately, without forming more tensor products.
##
## For example, <chars1> and <chars2> can be chosen as lists of irreducible
## characters with prescribed kernels such that the tensor products have a
## prescribed kernel, too.
## In this situation, <irreducibles> can be restricted to the list of those
## known irreducible characters that can be constituents of the tensor
## products, and <needed> can be chosen as the number of all missing
## irreducibles of that kind.
##
DeclareGlobalFunction( "TensorAndReduce" );
#############################################################################
##
#E
