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<Chapter Label="chap:ctblcons">
<Heading>Using Table Automorphisms for Constructing Character Tables in &GAP;</Heading>

Date: June 27th, 2004



This chapter has three aims.
First it shows how character table automorphisms can be utilized to
construct certain character tables from others
using the &GAP; system <Cite Key="GAP"/>;
the &GAP; functions used for that are part of the &GAP; Character Table
Library <Cite Key="CTblLib"/>.
Second it documents several constructions of character tables which are
contained in the &GAP; Character Table Library.
Third it serves as a testfile for the involved &GAP; functions.


<Section Label="sect:overview_ctblcons">
<Heading>Overview</Heading>

Several types of constructions of character tables of finite groups
from known tables of smaller groups are described in
Section <Ref Sect="sect:constr"/>.
Selecting suitable character table automorphisms is an important
ingredient of these constructions.



Section <Ref Sect="sect:theory"/> collects the few
representation theoretical facts on which these constructions are based.



The remaining sections show examples of the constructions in &GAP;.
These examples use the &GAP; Character Table Library,
therefore we load this package first.



<Example><![CDATA[
gap> LoadPackage( "ctbllib", "1.1.4", false );
true
]]></Example>

</Section>


<Section Label="sect:theory">
<Heading>Theoretical Background</Heading>


<Subsection Label="subsect:Character Table Automorphisms">
<Heading>Character Table Automorphisms</Heading>

Let <M>G</M> be a finite group,
<M>&Irr;(G)</M> be the matrix of ordinary irreducible characters of <M>G</M>,
<M>Cl(G)</M> be the set of conjugacy classes of elements in <M>G</M>,
<M>g^G</M> the <M>G</M>-conjugacy class of <M>g \in G</M>,
and let

<Display Mode="M">
 pow_p \colon Cl(G) \rightarrow Cl(G),
 g^G \mapsto (g^p)^G
</Display>

be the <M>p</M>-th power map, for each prime integer <M>p</M>.



A <E>table automorphism</E> of <M>G</M> is a permutation
<M>\sigma \colon Cl(G) \rightarrow Cl(G)</M> with the properties that
<M>\chi \circ \sigma \in &Irr;(G)</M> holds for all <M>\chi \in &Irr;(G)</M>
and that <M>\sigma</M> commutes with <M>pow_p</M>,
for all prime integers <M>p</M> that divide the order of <M>G</M>.
Note that for prime integers <M>p</M> that are coprime to the order of <M>G</M>,
<M>pow_p</M> commutes with each <M>\sigma</M> that permutes <M>&Irr;(G)</M>,
since <M>pow_p</M> acts as a field automorphism on the character values.



In &GAP;, a character table covers the irreducible characters
–a matrix <M>M</M> of character values–
as well as the power maps of the underlying group
–each power map <M>pow_p</M> being represented as a list
<M>pow_p^{\prime}</M> of positive integers denoting the positions of the image
classes.
The group of table automorphisms of a character table is represented
as a permutation group on the column positions of the table;
it can be computed with the &GAP; function
<Ref Func="AutomorphismsOfTable" BookName="ref"/>.



In the following, we will mainly use that each <E>group automorphism</E>
<M>\sigma</M>
of <M>G</M> induces a table automorphism that maps the class of each element
in <M>G</M> to the class of its image under <M>\sigma</M>.

</Subsection>


<Subsection Label="sect:Permutation Equivalence of Character Tables">
<Heading>Permutation Equivalence of Character Tables</Heading>

Two character tables with matrices <M>M_1</M>, <M>M_2</M> of irreducibles
and <M>p</M>-th power maps <M>pow_{{1,p}}</M>, <M>pow_{{2,p}}</M> are
<E>permutation equivalent</E>
if permutations <M>\psi</M> and <M>\pi</M> of row and column positions
of the <M>M_i</M> exist such that
<M>[ M_1 ]_{{i,j}} = [ M_2 ]_{{i \psi, j \pi}}</M>
holds for all indices <M>i</M>, <M>j</M>,
and such that
<M>\pi \cdot pow_{{2,p}}^{\prime} = pow_{{1,p}}^{\prime} \cdot \pi</M>
holds for all primes <M>p</M> that divide the (common) group order.
The first condition is equivalent to the existence of a permutation <M>\pi</M>
such that permuting the columns of <M>M_1</M> with <M>\pi</M> maps
the set of rows of <M>M_1</M> to the set of rows of <M>M_2</M>.



<M>\pi</M> is of course determined only up to table automorphisms of the two
character tables, that is,
two transforming permutations <M>\pi_1</M>, <M>\pi_2</M>
satisfy that <M>\pi_1 \cdot \pi_2^{-1}</M> is a table automorphism of the first
table, and <M>\pi_1^{-1} \cdot \pi_2</M> is a table automorphism of the second.



Clearly two isomorphic groups have permutation equivalent character tables.











The &GAP; library function
<Ref Func="TransformingPermutationsCharacterTables" BookName="ref"/>
returns a record that contains transforming permutations of rows and columns
if the two argument tables are permutation equivalent,
and <K>fail</K> otherwise.



In the example sections, the following function for computing representatives
from a list of character tables w.r.t. permutation equivalence
will be used.
More precisely, the input is either a list of character tables
or a list of records which have a component <C>table</C> whose value
is a character table, and the output is a sublist of the input.



<Ignore Remark="gapfilecomments">
#############################################################################
##
#W ctblcons.g examples of character table constructions Thomas Breuer
##
#Y Copyright (C) 2006, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
##
</Ignore>
<Example><![CDATA[
gap> RepresentativesCharacterTables:= function( list )
> local reps, entry, r;
>
> reps:= [];
> for entry in list do
> if ForAll( reps, r -> ( IsCharacterTable( r ) and
> TransformingPermutationsCharacterTables( entry, r ) = fail )
> or ( IsRecord( r ) and TransformingPermutationsCharacterTables(
> entry.table, r.table ) = fail ) ) then
> Add( reps, entry );
> fi;
> od;
> return reps;
> end;;
]]></Example>

</Subsection>


<Subsection Label="subsect:class_fusions">
<Heading>Class Fusions</Heading>

For two groups <M>H</M>, <M>G</M> such that <M>H</M> is isomorphic with
a subgroup of <M>G</M>,
any embedding <M>\iota \colon H \rightarrow G</M> induces a class function

<Display Mode="M">
 fus_{\iota} \colon Cl(H) \rightarrow Cl(G),
 h^G \mapsto (\iota(h))^G
</Display>

the <E>class fusion</E> of <M>H</M> in <M>G</M> via <M>\iota</M>.
Analogously, for a normal subgroup <M>N</M> of <M>G</M>,
any epimorphism <M>\pi \colon G \rightarrow G/N</M> induces a class function

<Display Mode="M">
 fus_{\pi} \colon Cl(G) \rightarrow Cl(G/N),
 g^G \mapsto (\pi(g))^G
</Display>

the <E>class fusion</E> of <M>G</M> onto <M>G/N</M> via <M>\pi</M>.



When one works only with character tables and not with groups,
these class fusions are the objects that describe subgroup and factor group
relations between character tables.
Technically, class fusions are necessary for restricting, inducing, and
inflating characters from one character table to another.
If one is faced with the problem to compute the class fusion between the
character tables of two groups <M>H</M> and <M>G</M> for which it is known
that <M>H</M> can be embedded into <M>G</M>
then one can use character-theoretic necessary conditions,
concerning that the restriction of all irreducible characters of <M>G</M> to
<M>H</M> (via the class fusion) must decompose into the irreducible
characters of <M>H</M>,
and that the class fusion must commute with the power maps of <M>H</M> and
<M>G</M>.



With this character-theoretic approach, one can clearly determine
possible class fusions only up to character table automorphisms.
Note that one can interpret each character table automorphism of <M>G</M>
as a class fusion from the table of <M>G</M> to itself.



If <M>N</M> is a normal subgroup in <M>G</M> then the class fusion of
<M>N</M> in <M>G</M> determines the orbits of the conjugation action of
<M>G</M> on the classes of <M>N</M>.
Often the knowledge of these orbits suffices to identify the subgroup of
table automorphisms of <M>N</M> that corresponds to this action of <M>G</M>;
for example, this is always the case if <M>N</M> has index <M>2</M> in
<M>G</M>.



&GAP; library functions for dealing with class fusions, power maps,
and character table automorphisms are described in the chapter
<Q>Maps Concerning Character Tables</Q> in the &GAP; Reference Manual.

</Subsection>


<Subsection Label="subsect:isoclinism">
<Heading>Constructing Character Tables of Certain Isoclinic Groups</Heading>

As is stated in <Cite Key="CCN85" Where="p. xxiii"/>,
two groups <M>G</M>, <M>H</M> are called <E>isoclinic</E> if they can be
embedded into a group <M>K</M> such that <M>K</M> is generated by <M>Z(K)</M>
and <M>G</M>, and also by <M>Z(K)</M> and <M>H</M>.
In the following, two special cases of isoclinism will be used,
where the character tables of the isoclinic groups are closely related.



<List>
(1)
<Item>
 <M>G \cong 2 \times U</M> for a group <M>U</M> that has a central
 subgroup <M>N</M> of order <M>2</M>,
 and <M>H</M> is the central product of <M>U</M> and a cyclic group
 of order four.
 Here we can set <M>K = 2 \times H</M>.
</Item>
(2)
<Item>
 <M>G \cong 2 \times U</M> for a group <M>U</M> that has a normal
 subgroup <M>N</M> of index <M>2</M>,
 and <M>H</M> is the subdirect product of <M>U</M> and a cyclic
 group of order four,
 Here we can set <M>K = 4 \times U</M>.
</Item>
</List>



<Alt Only='HTML'><![CDATA[
<center>
<img src="ctblcons01.png" alt="two constructions of K"/>
</center>
]]></Alt>

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 ╱│╲ ╱│╲ ╱│╲
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 ╱ ⟨z⟩─╱────────╱──. ╱ ╱ ╲ │ ╱│╲
 .──│──╱─────── G │ ╱ ╱ ╲│╱ │ ╲
 ╱│╲ │ ╱ ╲ │ ╱ ╱ S . U
 ╱ │ ╲│╱ ╲│╱ ╱ ╱ ╲ │ ╱
 ╱ │ N ──────────── U = S ⟨z⟩ ╱ ╲│╱
 ╱ . ╱ ╲ ╱ N
. │ ╱ ╲ ╱ ╱
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Starting from the group <M>K</M> containing both <M>G</M> and <M>H</M>,
we first note that each irreducible representation of <M>G</M> or <M>H</M>
extends to <M>K</M>.
More specifically,
if <M>\rho_G</M> is an irreducible representation of <M>G</M> then
we can define an extension <M>\rho</M> of <M>K</M> by defining it suitably
on <M>Z(K)</M>
and then form <M>\rho_H</M>, the restriction of <M>\rho</M> to <M>H</M>.



In our two cases, we set <M>S = G \cap H</M>,
so <M>K = S \cup G \setminus S \cup H \setminus S \cup z S</M> holds for some
element <M>z \in Z(K) \setminus ( G \cup H )</M> of order four,
and <M>G = S \cup g S</M> for some <M>g \in G \setminus S</M>,
and <M>H = S \cup h S</M> where <M>h = z \cdot g \in H \setminus S</M>.
For defining <M>\rho_H</M>,
it suffices to consider <M>\rho(h) = \rho(z) \rho(g)</M>,
where <M>\rho(z) = \epsilon_{\rho}(z) \cdot I</M> is a scalar matrix.



As for the character table heads of <M>G</M> and <M>H</M>,
we have <M>s^G = s^H</M> and <M>z (g \cdot s)^G = (h \cdot s)^H</M> for each
<M>s \in S</M>,
so this defines a bijection of the conjugacy classes of <M>G</M> and <M>H</M>.
For a prime integer <M>p</M>,
<M>(h \cdot s)^p = (z \cdot g \cdot s)^p = z^p \cdot (g \cdot s)^p</M> holds
for all <M>s \in S</M>,
so the <M>p</M>-th power maps of <M>G</M> and <M>H</M> are related as follows:
Inside <M>S</M> they coincide for any <M>p</M>.
If <M>p \equiv 1 \bmod 4</M> they coincide also outside <M>S</M>,
if <M>p \equiv -1 \bmod 4</M> the images differ by exchanging the classes
of <M>(h \cdot s)^p</M> and <M>z^2 \cdot (h \cdot s)^p</M> (if these elements
lie in different classes),
and for <M>p = 2</M> the images (which lie inside <M>S</M>) differ by
exchanging the classes of <M>(h \cdot s)^2</M> and
<M>z^2 \cdot (g \cdot s)^2</M>
(if these elements lie in different classes).



Let <M>\rho</M> be an irreducible representation of <M>K</M>.
Then <M>\rho_G</M> and <M>\rho_H</M> are related as follows:
<M>\rho_G(s) = \rho_H(s)</M>
and <M>\rho(z) \cdot \rho_G(g \cdot s) = \rho_H(h \cdot s)</M>
for all <M>s \in S</M>.
If <M>\chi_G</M> and <M>\chi_H</M> are the characters afforded by
<M>\rho_G</M> and <M>\rho_H</M>, respectively,
then <M>\chi_G(s) = \chi_H(s)</M> and
<M>\epsilon_{\rho}(z) \cdot \chi_G(g \cdot s) = \chi_H(h \cdot s)</M> hold
for all <M>s \in S</M>.
In the case <M>\chi_G(z^2) = \chi(1)</M> we have
<M>\epsilon_{\rho}(z) = \pm 1</M>,
and both cases actually occur if one considers all irreducible
representations of <M>K</M>.
In the case <M>\chi_G(z^2) = - \chi(1)</M> we have
<M>\epsilon_{\rho}(z) = \pm i</M>,
and again both cases occur.
So we obtain the irreducible characters of <M>H</M> from those of <M>G</M> by
multiplying the values outside <M>S</M> in all those characters by <M>i</M>
that do not have <M>z^2</M> in their kernels.



In &GAP;, the function
<Ref Func="CharacterTableIsoclinic" BookName="ref"/>
can be used for computing the character table of <M>H</M> from that of
<M>G</M>, and vice versa.
(Note that in the above two cases, also the groups <M>U</M> and <M>H</M> are
isoclinic by definition, but
<Ref Func="CharacterTableIsoclinic" BookName="ref"/>
does not transfer the character table of <M>U</M> to that of <M>H</M>.)



One could construct the character tables mentioned above by forming the
character tables of certain factor groups or normal subgroups of direct
products.
However,
the construction via <Ref Func="CharacterTableIsoclinic" BookName="ref"/>
has the advantage that the result stores from which sources it arose,
and this information can be used to derive also the Brauer character tables,
provided that the Brauer character tables of the source tables are known.

</Subsection>


<Subsection Label="subsect:isoclinicP">
<Heading>Character Tables of Isoclinic Groups of the Structure <M>p.G.p</M>
(October 2016)</Heading>

Since the release of &GAP; 4.11,
<Ref Func="CharacterTableIsoclinic" BookName="ref"/>
admits the construction of the character tables of the
isoclinic variants of groups of the structure <M>p.G.p</M>,
also for odd primes <M>p</M>.



This feature will be used in the construction of the character table of
<M>9.U_3(8).3_3</M>,
in order to construct the table of the subgroup <M>3.(3 \times U_3(8))</M>
and of the factor group <M>(3 \times U_3(8)).3_3</M>,
see Section <Ref Subsect="subsect:9.U_3(8).3_3"/>.
These constructions are a straightforward generalization of those described
in detail in Section <Ref Subsect="subsect:isoclinism"/>.



There are several examples of &ATLAS; groups of the structure <M>3.G.3</M>.
The character table of one such group is shown in the &ATLAS;,
the tables of their isoclinic variants can now be obtained from
<Ref Func="CharacterTableIsoclinic" BookName="ref"/>.



For example, the group GL<M>(3,4)</M> has the structure <M>3.L_3(4).3</M>.
There are three pairwise nonisomorphic isoclinic variants
of groups of this structure.

<Example><![CDATA[
gap> t:= CharacterTable( "3.L3(4).3" );
CharacterTable( "3.L3(4).3" )
gap> iso1:= CharacterTableIsoclinic( t );
CharacterTable( "Isoclinic(3.L3(4).3,1)" )
gap> iso2:= CharacterTableIsoclinic( t, rec( k:= 2 ) );
CharacterTable( "Isoclinic(3.L3(4).3,2)" )
gap> TransformingPermutationsCharacterTables( t, iso1 );
fail
gap> TransformingPermutationsCharacterTables( t, iso2 );
fail
gap> TransformingPermutationsCharacterTables( iso1, iso2 );
fail
]]></Example>



The character table of GL<M>(3,4)</M> is in fact the one which is shown
in the &ATLAS;.



<Example><![CDATA[
gap> IsRecord( TransformingPermutationsCharacterTables( t,
> CharacterTable( GL( 3, 4 ) ) ) );
true
]]></Example>

</Subsection>


<Subsection Label="subsect:isoclinicATLAS">
<Heading>Isoclinic Double Covers of Almost Simple Groups</Heading>

The function
<Ref Func="CharacterTableIsoclinic" BookName="ref"/>
can also be used to switch between the character tables of double covers
of groups of the type <M>G.2</M>, where <M>G</M> is a perfect group,
see <Cite Key="CCN85" Where="Section 6.7"/>.
Typical examples are the double covers of symmetric groups.



Note that these double covers may be isomorphic.
This happens for <M>2.S_6</M>.
More generally, this happens for all semilinear groups
<M>\Sigma</M>L<M>(2,p^2)</M>, for odd primes <M>p</M>.
The smallest examples are <M>\Sigma</M>L<M>(2,9) = 2.A_6.2_1</M> and
<M>\Sigma</M>L<M>(2,25) = 2.L_2(25).2_2</M>.
This implies that the character table and its isoclinic variant are
permutation isomorphic.



<Example><![CDATA[
gap> t:= CharacterTable( "2.A6.2_1" );
CharacterTable( "2.A6.2_1" )
gap> TransformingPermutationsCharacterTables( t,
> CharacterTableIsoclinic( t ) );
rec( columns := (4,6)(5,7)(11,12)(14,16)(15,17),
 group := Group([ (16,17), (14,15) ]),
 rows := (3,5)(4,6)(10,11)(12,15,13,14) )
gap> t:= CharacterTable( "2.L2(25).2_2" );
CharacterTable( "2.L2(25).2_2" )
gap> TransformingPermutationsCharacterTables( t,
> CharacterTableIsoclinic( t ) );
rec( columns := (7,9)(8,10)(20,21)(23,24)(25,27)(26,28),
 group := <permutation group with 4 generators>,
 rows := (3,5)(4,6)(14,15)(16,17)(19,22,20,21) )
]]></Example>



For groups of the type <M>4.G.2</M>, two different situations can occur.
Either the distinguished central cyclic subgroup of order four in <M>4.G</M>
is inverted by the elements in <M>4.G.2 \setminus 4.G</M>,
or this subgroup is central in <M>4.G.2</M>.
In the first case, calling
<Ref Func="CharacterTableIsoclinic" BookName="ref"/>
with the character table of <M>4.G.2</M> yields a character table with the
same set of irreducibles,
only the <M>2</M>-power map will in general differ from that of the input
table.
In the second case, the one argument version of
<Ref Func="CharacterTableIsoclinic" BookName="ref"/>
returns a permutation isomorphic table.
By supplying additional arguments, there is a chance to construct tables of
different groups.



We demonstrate this phenomenon with the various groups of the structure
<M>4.L_3(4).2</M>.



<Example><![CDATA[
gap> tbls:= [];;
gap> for m in [ "4_1", "4_2" ] do
> for a in [ "2_1", "2_2", "2_3" ] do
> Add( tbls, CharacterTable( Concatenation( m, ".L3(4).", a ) ) );
> od;
> od;
gap> tbls;
[ CharacterTable( "4_1.L3(4).2_1" ), CharacterTable( "4_1.L3(4).2_2" )
 , CharacterTable( "4_1.L3(4).2_3" ),
 CharacterTable( "4_2.L3(4).2_1" ), CharacterTable( "4_2.L3(4).2_2" )
 , CharacterTable( "4_2.L3(4).2_3" ) ]
gap> case1:= Filtered( tbls, t -> Size( ClassPositionsOfCentre( t ) ) = 2 );
[ CharacterTable( "4_1.L3(4).2_1" ), CharacterTable( "4_1.L3(4).2_2" )
 , CharacterTable( "4_2.L3(4).2_1" ),
 CharacterTable( "4_2.L3(4).2_3" ) ]
gap> case2:= Filtered( tbls, t -> Size( ClassPositionsOfCentre( t ) ) = 4 );
[ CharacterTable( "4_1.L3(4).2_3" ),
 CharacterTable( "4_2.L3(4).2_2" ) ]
]]></Example>



The centres of the groups <M>4_1.L_3(4).2_1</M>, <M>4_1.L_3(4).2_2</M>,
<M>4_2.L_3(4).2_1</M>, and <M>4_2.L_3(4).2_3</M> have order two,
that is, these groups belong to the first case.
Each of these groups is not permutation equivalent to its isoclinic
variant but has the same irreducible characters.



<Example><![CDATA[
gap> isos1:= List( case1, CharacterTableIsoclinic );;
gap> List( [ 1 .. 4 ], i -> Irr( case1[i] ) = Irr( isos1[i] ) );
[ true, true, true, true ]
gap> List( [ 1 .. 4 ],
> i -> TransformingPermutationsCharacterTables( case1[i], isos1[i] ) );
[ fail, fail, fail, fail ]
]]></Example>



The groups <M>4_1.L_3(4).2_3</M> and <M>4_2.L_3(4).2_2</M> belong to the
second case because their centres have order four.



<Example><![CDATA[
gap> isos2:= List( case2, CharacterTableIsoclinic );;
gap> List( [ 1, 2 ],
> i -> TransformingPermutationsCharacterTables( case2[i], isos2[i] ) );
[ rec( columns := (26,27,28,29)(30,31,32,33)(38,39,40,41)(42,43,44,45)
 , group := <permutation group with 5 generators>,
 rows := (16,17)(18,19)(20,21)(22,23)(28,29)(32,33)(36,37)(40,
 41) ),
 rec( columns := (28,29,30,31)(32,33)(34,35,36,37)(38,39,40,41)(42,
 43,44,45)(46,47,48,49), group := <permutation group with
 3 generators>, rows := (15,16)(17,18)(20,21)(22,23)(24,25)(26,
 27)(28,29)(34,35)(38,39)(42,43)(46,47) ) ]
gap> isos3:= List( case2, t -> CharacterTableIsoclinic( t,
> ClassPositionsOfCentre( t ) ) );;
gap> List( [ 1, 2 ],
> i -> TransformingPermutationsCharacterTables( case2[i], isos3[i] ) );
[ fail, fail ]
]]></Example>

</Subsection>


<Subsection Label="subsect:theorClifford">
<Heading>Characters of Normal Subgroups</Heading>

Let <M>G</M> be a group and <M>N</M> be a normal subgroup of <M>G</M>.
We will need the following well-known facts about the relation
between the irreducible characters of <M>G</M> and <M>N</M>.



For an irreducible (Brauer) character <M>\chi</M> of <M>N</M> and
<M>g \in G</M>,
we define <M>\chi^g</M> by <M>\chi^g(n) = \chi(n^g)</M> for all
<M>n \in N</M>,
and set <M>I_G(\chi) = \{ g \in G; \chi^g = \chi \}</M>
(see <Cite Key="Feit82" Where="p. 86"/>).



If <M>I_G(\chi) = N</M> then the induced character <M>\chi^G</M> is an
irreducible (Brauer) character of <M>G</M>
(see <Cite Key="Feit82" Where="Lemma III 2.11"/>
or <Cite Key="Nav98" Where="Theorem 8.9"/>
or <Cite Key="LP10" Where="Corollary 4.3.8"/>).



If <M>G/N</M> is cyclic and if <M>I_G(\chi) = G</M> then <M>\chi = \psi_N</M>
for an irreducible (Brauer) character <M>\psi</M> of <M>G</M>,
and each irreducible (Brauer) character <M>\theta</M> with the property
<M>\chi = \theta_N</M> is of the form <M>\theta = \psi \cdot \epsilon</M>,
where <M>\epsilon</M> is an irreducible (Brauer) character of <M>G/N</M>
(see <Cite Key="Feit82" Where="Theorem III 2.14"/>
or <Cite Key="Nav98" Where="Theorem 8.12"/>
or <Cite Key="LP10" Where="Theorem 3.6.13"/>).



Clifford's theorem
(<Cite Key="Feit82" Where="Theorem III 2.12"/> or
<Cite Key="Nav98" Where="Corollary 8.7"/> or
<Cite Key="LP10" Where="Theorem 3.6.2"/>) states
that the restriction of an irreducible (Brauer) character of <M>G</M> to
<M>N</M> has the form <M>e \sum_{i=1}^t \varphi_i</M> for a positive integer
<M>e</M> and irreducible (Brauer) characters <M>\varphi_i</M> of <M>N</M>,
where <M>t</M> is the index of <M>I_G(\varphi_1)</M> in <M>G</M>.



Now assume that <M>G</M> is a normal subgroup in a larger group <M>H</M>,
that <M>G/N</M> is an abelian chief factor of <M>H</M> and that <M>\psi</M>
is an ordinary irreducible character of <M>G</M> such that
<M>I_H(\psi) = H</M>.
Then either <M>t = 1</M> and <M>e^2</M> is one of <M>1</M>, <M>|G/N|</M>,
or <M>t = |G/N|</M> and <M>e = 1</M>
(see <Cite Key="Isa76" Where="Theorem 6.18"/>).

</Subsection>
</Section>


<Section Label="sect:constr">
<Heading>The Constructions</Heading>


<Subsection Label="subsect:theorMGA">
<Heading>Character Tables of Groups of the Structure <M>M.G.A</M></Heading>

(This kind of table construction is described in <Cite Key="Bre11"/>.)



Let <M>N</M> denote a downward extension of the finite group <M>G</M>
by a finite group <M>M</M>,
let <M>H</M> denote an automorphic (upward) extension of <M>N</M> by
a finite cyclic group <M>A</M> such that <M>M</M> is normal in <M>H</M>,
and set <M>F = H / M</M>.
We consider the situation that each irreducible character of <M>N</M>
that does not contain <M>M</M> in its kernel induces irreducibly to <M>H</M>.
Equivalently, the action of <M>A = \langle a \rangle</M> on the characters of
<M>N</M>,
via <M>\chi \mapsto \chi^a</M>, has only orbits of length exactly <M>|A|</M>
on the set <M>\{ \chi \in &Irr;(N); M \nsubseteq \ker(\chi) \}</M>.



<Alt Only='HTML'><![CDATA[
<center>
<img src="ctblcons02.png" alt="groups of the structure M.G.A"/>
</center>
]]></Alt>

<Alt Only='Text'>
<Verb>
 H
 │
 │ A G ───▶ F
 │
 N ▲ ▲
 │ │ │
 │ G │ │
 │
 M N ───▶ H
 │
 1
</Verb>
</Alt>

<Alt Only='LaTeX'>

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\end{picture}
\end{center}
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</Alt>



This occurs for example if <M>M</M> is central in <M>N</M> and <M>A</M> acts
fixed-point freely on <M>M</M>,
we have <M>|M| \equiv 1 \bmod |A|</M> in this case.
If <M>M</M> has prime order then it is sufficient that <M>A</M> does not
centralize <M>M</M>.



The ordinary (or <M>p</M>-modular) irreducible characters of <M>H</M> are
then given by the ordinary (or <M>p</M>-modular) irreducible characters of
<M>F</M> and <M>N</M>,
the class fusions from the table of <M>N</M> onto the table of <M>G</M>
and from the table of <M>G</M> into that of <M>F</M>,
and the permutation <M>\pi</M> that is induced by the action of <M>A</M> on
the conjugacy classes of <M>N</M>.



In general, the action of <M>A</M> on the classes of <M>M</M> is not the
right thing to look at, one really must consider the action on the relevant
characters of <M>M.G</M>.
For example, take <M>H</M> the quaternion group or the dihedral group of
order eight, <M>N</M> a cyclic subgroup of index two,
and <M>M</M> the centre of <M>H</M>;
here <M>A</M> acts trivially on <M>M</M>, but the relevant fact is that the
action of <M>A</M> swaps those two irreducible characters of <M>N</M> that
take the value <M>-1</M> on the involution in <M>M</M>
–these are the faithful irreducible characters of <M>N</M>.



If the orders of <M>M</M> and <M>A</M> are coprime then also the power maps
of <M>H</M> can be computed from the above data.
For each prime <M>p</M> that divides the orders of both <M>M</M> and
<M>A</M>,
the <M>p</M>-th power map is in general not uniquely determined by these
input data.
In this case, we can compute the (finitely many) candidates for the
character table of <M>H</M> that are described by these data.
One possible reason for ambiguities is the existence of several isoclinic
but nonisomorphic groups that can arise from the input tables
(cf. Section <Ref Subsect="subsect:isoclinism"/>,
see Section <Ref Subsect="subsect:HN2"/> for an example).



With the &GAP; function
<Ref Func="PossibleActionsForTypeMGA" BookName="ctbllib"/>,
one can compute the possible orbit structures induced by <M>G.A</M> on the
classes of <M>M.G</M>, and
<Ref Func="PossibleCharacterTablesOfTypeMGA" BookName="ctbllib"/>
computes the possible ordinary character tables for a given orbit structure.
For constructing the <M>p</M>-modular Brauer table of a group <M>H</M> of
the structure <M>M.G.A</M>,
the &GAP; function
<Ref Func="BrauerTableOfTypeMGA" BookName="ctbllib"/>
takes the ordinary
character table of <M>H</M> and the <M>p</M>-modular tables of the subgroup
<M>M.G</M> and the factor group <M>G.A</M> as its input.
The <M>p</M>-modular table of <M>G</M> is not explicitly needed in the
construction,
it is implicitly given by the class fusions from <M>M.G</M> into <M>M.G.A</M>
and from <M>M.G.A</M> onto <M>G.A</M>;
these class fusions must of course be available.



The &GAP; Character Table Library contains many tables of groups of the
structure <M>M.G.A</M> as described above, which are encoded by references to
the tables of the groups <M>M.G</M> and <M>G.A</M>, plus the fusion and
action information.
This reduces the space needed for storing these character tables.



For examples, see Section <Ref Sect="sect:explMGA"/>.

</Subsection>


<Subsection Label="sect:Character Tables of Groups of the Structure G.S_3">
<Heading>Character Tables of Groups of the Structure <M>G.S_3</M></Heading>

Let <M>G</M> be a finite group, and <M>H</M> be an upward extension of
<M>G</M> such that the factor group <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 <M>c</M>.
(Typical cases for <M>F</M> are the symmetric group <M>S_3</M> on three
points and the alternating group <M>A_4</M> on four points.)
Let <M>N</M> and <M>U</M> denote the preimages of <M>K</M> and <M>C</M> under
the natural epimorphism from <M>H</M> onto <M>F</M>.





<Alt Only='HTML'><![CDATA[
<center>
<img src="ctblcons03.png" alt="Groups of the structure G.3.2"/>
</center>
]]></Alt>

<Alt Only='Text'>
<Verb>
 H
 ╱ ╲
 ╱ ╲ N ───▶ H
 N ╲
 ╲ U ▲ ▲
 ╲ ╱ │ │
 ╲ ╱ │ │
 G
 │ G ───▶ U
 │
 1
</Verb>
</Alt>

<Alt Only='LaTeX'>

<![CDATA[
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</Alt>



For certain isomorphism types of <M>F</M>,
the ordinary (or <M>p</M>-modular) character table of <M>H</M> can be
computed from the ordinary (or <M>p</M>-modular) character tables of
<M>G</M>, <M>U</M>, and <M>N</M>,
the class fusions from the table of <M>G</M> into those of <M>U</M> and
<M>N</M>,
and the permutation <M>\pi</M> induced by <M>H</M> on the conjugacy classes
of <M>N</M>.
This holds for example for <M>F = S_3</M> and in the ordinary case also
for <M>F = A_4</M>.



Each class of <M>H</M> is either a union of <M>\pi</M>-orbits or an
<M>H</M>-class of <M>U \setminus G</M>;
the latter classes are in bijection with the <M>U</M>-classes of
<M>U \setminus G</M>, they are just <M>|K|</M> times larger
since the <M>|K|</M> conjugates of <M>U</M> in <M>H</M> are fused.
The power maps of <M>H</M> are uniquely determined from the power maps of
<M>U</M> and <M>N</M>,
because each element in <M>F</M> lies in <M>K</M> or in an <M>F</M>-conjugate
of <M>C</M>.



Concerning the computation of the ordinary irreducible characters of
<M>H</M>,
we could induce the irreducible characters of <M>U</M> and <M>N</M> to
<M>H</M>,
and then take the union of the irreducible characters among those
and the irreducible differences of those.
(For the case <M>F = S_3</M>, this approach has been described in the
Appendix of <Cite Key="HL94"/>.)



The &GAP; function
<Ref Func="CharacterTableOfTypeGS3" BookName="ctbllib"/>
proceeds in a different way,
which is suitable also for the construction of <M>p</M>-modular character
tables of <M>H</M>.



By the facts listed in Section <Ref Subsect="subsect:theorClifford"/>,
for an irreducible (Brauer) character <M>\chi</M> of <M>N</M>,
we have <M>I_H(\chi)</M> equal to either <M>N</M> or <M>H</M>.
In the former case, <M>\chi</M> induces irreducibly to <M>H</M>.
In the latter case, there are extensions <M>\psi^{(i)}</M>,
<M>1 \leq i \leq |C|</M> (or <M>|C|_{p^\prime}</M>), to <M>H</M>,
and we have the following possibilities,
depending on the restriction <M>\chi_G</M>.



If <M>\chi_G = e \varphi</M>, for an irreducible character <M>\varphi</M> of
<M>G</M>, then <M>I_U(\varphi) = U</M> holds,
hence the <M>\psi^{(i)}_U</M> are <M>|C|</M> (or <M>|C|_{p^\prime}</M>)
extensions of <M>\chi_G</M> to <M>U</M>.
Moreover, we have either <M>e = 1</M> or <M>e^2 = |K|</M>.
In the case <M>e = 1</M>, this determines the values of the <M>\psi^{(i)}</M>
on the classes of <M>U</M> outside <M>G</M>.
In the case <M>e ¬eq; 1</M>,
we have the problem to combine <M>e</M> extensions
of <M>\varphi</M> to a character of <M>U</M> that extends to <M>H</M>.



(One additional piece of information in the case of ordinary character tables
is that the norm of this linear combination equals <M>1 + (|K|-1)/|C|</M>,

which determines the <M>\psi^{(i)}_U</M> if <M>F = A_4 \cong 2^2:3</M> or
<M>F = 2^3:7</M> holds;
in the former case, the sum of each two out of the three different extensions
of <M>\varphi</M> extends to <M>U</M>;
in the latter case, the sum of all different extensions plus one of the
extensions extends.
Note that for <M>F = S_3</M>, the case <M>e ¬eq; 1</M> does not occur.)



The remaining case is that <M>\chi_G</M> is not a multiple of an irreducible
character of <M>G</M>.
Then <M>\chi_G = \varphi_1 + \varphi_2 + \ldots + \varphi_{|K|}</M>,
for pairwise different irreducible characters <M>\varphi_i</M>,
<M>1 \leq i \leq |K|</M>,
of <M>G</M> with the property <M>\varphi_i^N = \chi</M>.
The action of <M>U</M> on <M>G</M> fixes at least one of the
<M>\varphi_i</M>,
since <M>|K| \equiv 1 \bmod |C|</M>.
Without loss of generality, let <M>I_U(\varphi_1) = U</M>,
and let <M>\varphi_1^{(i)}</M>, <M>1 \leq i \leq |C|</M>,
be the extensions of <M>\varphi_1</M> to <M>U</M>.
(In fact exactly <M>\varphi_1</M> is fixed by <M>U</M>
since otherwise <M>k \in K</M>
would exist with <M>\varphi_1^k ¬eq; \varphi_1</M> and such that also
<M>\varphi_1^k</M> would be invariant in <M>U</M>;
but then <M>\varphi_1</M> would be invariant under both <M>C</M> and
<M>C^k</M>, which generate <M>F</M>.
So each of the <M>|K|</M> constituents is invariant in exactly one of the
<M>|K|</M> subgroups of type <M>U</M> above <M>G</M>.)



Then
<M>((\varphi_1^{(i)})^H)_N = \varphi_1^N = \chi</M>,
hence the values of <M>\psi^{(i)}</M> on the classes of <M>U \setminus G</M>
are given by those of <M>(\varphi_1^{(i)})^H</M>.
(These are exactly the values of <M>\varphi_1^{(i)}</M>.
So in both cases, we take the values of <M>\chi</M> on <M>N</M>,
and on the classes of <M>U \setminus G</M> the values of the extensions
of the unique extendible constituent of <M>\chi_G</M>.)



For examples, see Section <Ref Sect="sect:GS3"/>.

</Subsection>


<Subsection Label="subsect:theorGV4">
<Heading>Character Tables of Groups of the Structure <M>G.2^2</M></Heading>

Let <M>G</M> be a finite group, and <M>H</M> be an upward extension of
<M>G</M> such that the factor group <M>H / G</M> is a Klein four group.
We assume that the ordinary character tables of <M>G</M> and of the three
index two subgroups <M>U_1</M>, <M>U_2</M>, and <M>U_3</M>
(of the structures <M>G.2_1</M>, <M>G.2_2</M>, and <M>G.2_3</M>,
respectively) of <M>H</M> above <M>G</M> are known,
as well as the class fusions of <M>G</M> into these groups.
The idea behind the method that is described in this section is that
in this situation,
there are only few possibilities for the ordinary character table of
<M>H</M>.



<Alt Only='HTML'><![CDATA[
<center>
<img src="ctblcons04.png" alt="groups of the structure G.V4"/>
</center>
]]></Alt>

<Alt Only='Text'>
<Verb>
 H
 ╱│╲
 ╱ │ ╲ G ───▶ U_1 ───▶ H
 ╱ │ ╲
 U_1 U_2 U_3 G ───▶ U_2 ───▶ H
 ╲ │ ╱
 ╲ │ ╱ G ───▶ U_3 ───▶ H
 ╲│╱
 G
 │
 │
 1
</Verb>
</Alt>

<Alt Only='LaTeX'>

<![CDATA[
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</Alt>



Namely, the action of <M>H</M> on the classes of <M>G.2_i</M> is given by
a table automorphism <M>\pi_i</M> of <M>G.2_i</M>,
and <M>H</M> realizes compatible choices of such
automorphisms <M>\pi_1</M>, <M>\pi_2</M>, <M>\pi_3</M> in the sense that
the orbits of all three <M>\pi_i</M> on the classes of <M>G</M> inside the
groups <M>G.2_i</M> coincide.
Furthermore, if <M>G.2_i</M> has <M>n_i</M> conjugacy classes then an action
<M>\pi_i</M> that is a product of <M>f_i</M> disjoint transpositions leads
to a character table candidate for <M>G.2^2</M> that has <M>2 n_i - 3 f_i</M>
classes,
so also the <M>f_i</M> must be compatible.



Taking the <Q>inner</Q> classes, i.e., the orbit sums of the classes inside
<M>G</M> under the <M>\pi_i</M>, plus the union of the <M>\pi_i</M>-orbits of
the classes of <M>G.2_i \setminus G</M> gives a possibility for the classes
of <M>H</M>.
Furthermore, the power maps of the groups <M>G.2_i</M> determine the
power maps of the candidate table constructed this way.



Concerning the computation of the irreducible characters of <M>H</M>,
we consider also the case of <M>p</M>-modular characters tables,
where we assume that the ordinary character table of <M>H</M> is already
known and the only task is to compute the irreducible <M>p</M>-modular
Brauer characters.



Let <M>\chi</M> be an irreducible (<M>p</M>-modular Brauer) character of
<M>G</M>.
By the facts that are listed in
Section <Ref Subsect="subsect:theorClifford"/>,
there are three possibilities.

<List>
1.
<Item>
 <M>I_H(\chi) = G</M>; then <M>\chi^H</M> is irreducible.
</Item>
2.
<Item>
 <M>I_H(\chi) = G.2_i</M> for <M>i</M> one of <M>1</M>, <M>2</M>, <M>3</M>;
 then <M>I_{G.2_i}(\chi) = G.2_i</M> for this <M>i</M>,
 so <M>\chi</M> extends to <M>G.2_i</M>;
 none of these extensions extends to <M>H</M>
 (because otherwise <M>\chi</M> would be invariant in <M>H</M>),
 so they induce irreducible characters of <M>H</M>.
</Item>
3.
<Item>
 <M>I_H(\chi) = H</M>;
 then <M>\chi</M> extends to each of the three groups <M>G.2_i</M>,
 and either all these extensions induce the same character of <M>H</M>
 (which vanishes on <M>H \setminus G</M>) or they are invariant in <M>H</M>
 and thus extend to <M>H</M>.
</Item>
</List>



In the latter part of case 3. (except if <M>p = 2</M>),
the problem is to combine the values of six irreducible characters
of the groups <M>G.2_i</M> to four characters of <M>H</M>.
This yields essentially two choices, and we try to exclude one possibility
by forming scalar products with the <M>2</M>-nd symmetrizations of the known
irreducibles.
If several possibilities remain then we get several possible tables.



So we end up with a list of possible character tables of <M>H</M>.



The first step is to specify a list of possible triples
<M>(\pi_1, \pi_2, \pi_3)</M>, using the table automorphisms of the groups
<M>G.2_i</M>;
this can be done using the &GAP; function
<Ref Func="PossibleActionsForTypeGV4" BookName="ctbllib"/>.
Then the &GAP; function
<Ref Func="PossibleCharacterTablesOfTypeGV4" BookName="ctbllib"/>
can be used
for computing the character table candidates for each given triple of
permutations; it may of course happen that some triples of automorphisms
are excluded in this second step.



For examples, see Section <Ref Sect="sect:xplGV4"/>.

</Subsection>


<Subsection Label="subsect:theorV4G">
<Heading>Character Tables of Groups of the Structure <M>2^2.G</M>
(August 2005)</Heading>

Let <M>G</M> be a finite group, and <M>H</M> be a central extension of
<M>G</M> by a Klein four group <M>Z = \langle z_1, z_2 \rangle</M>;
set <M>z_3 = z_1 z_2</M> and <M>Z_i = \langle z_i \rangle</M>,
for <M>1 \leq i \leq 3</M>.
We assume that the ordinary character tables of the three factor groups
<M>2_i.G = H / Z_i</M> of <M>H</M> are known,
as well as the class fusions from these groups to <M>G</M>.
The idea behind the method described in this section is that
in this situation,
there are only few possibilities for the ordinary character table of
<M>H</M>.



<Alt Only='HTML'><![CDATA[
<center>
<img src="ctblcons05.png" alt="groups of the structure V4.G"/>
</center>
]]></Alt>

<Alt Only='Text'>
<Verb>
 H
 │
 G │
 │
 Z
 ╱│╲
 ╱ │ ╲ H ───▶ H / Z_1 ───▶ G
 ╱ │ ╲
 Z_1 Z_2 Z_3 H ───▶ H / Z_2 ───▶ G
 ╲ │ ╱
 ╲ │ ╱ H ───▶ H / Z_3 ───▶ G
 ╲│╱
 1
</Verb>
</Alt>

<Alt Only='LaTeX'>

<![CDATA[
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\put(0,0){\begin{picture}(40,30)(0,0)
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\put(20,14){\circle*{1}} \put(17,19){\makebox(0,0){$G$}}
\put(13, 7){\circle*{1}} \put(10, 7){\makebox(0,0){$Z_1$}}
\put(20, 7){\circle*{1}} \put(17, 7){\makebox(0,0){$Z_2$}}
\put(27, 7){\circle*{1}} \put(30, 7){\makebox(0,0){$Z_3$}}
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\put(40,2){\begin{picture}(35,30)
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\put(18,25){\makebox(0,0){$H/Z_1$}}
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</Alt>



Namely,
the irreducible (<M>p</M>-modular) characters of <M>H</M> are exactly the
inflations of the irreducible (<M>p</M>-modular) characters of the three
factor groups <M>H / Z_i</M>.
(Note that for any noncyclic central subgroup <M>C</M> of <M>H</M>
and any <M>\chi \in &Irr;(H)</M>, we have <M>|\ker(\chi) \cap C| > 1</M>.
To see this, let <M>N = \ker(\chi)</M>.
Then clearly <M>|N| > 1</M>,
and <M>\chi</M> can be regarded as a faithful irreducible character of
<M>H/N</M>.
If <M>N \cap C</M> would be trivial then <M>N C / N \cong C</M> would be
a noncyclic central subgroup of <M>H/N</M>.
This cannot happen by <Cite Key="Isa76" Where="Thm. 2.32 (a)"/>,
so the statement can be regarded as an obvious refinement of this theorem.)
So all we have to construct is the character table head of <M>H</M>
–classes and power maps– and the factor fusions from <M>H</M>
to these groups.



For fixed <M>h \in H</M>, we consider the question in which <M>H</M>-classes
the elements <M>h</M>, <M>h z_1</M>, <M>h z_2</M>, and <M>h z_3</M> lie.
There are three possibilities.



<Enum>
<Item>
 The four elements are all conjugate in <M>H</M>.
 Then in each of the three groups <M>H/Z_i</M>,
 the two preimages of <M>h Z \in H/Z</M> are conjugate.
</Item>
<Item>
 We are not in case 1. but two of the four elements are conjugate in
 <M>H</M>, i. e.,
 <M>g^{-1} h g = h z_i</M> for some <M>g \in H</M> and some <M>i</M>;
 then <M>g^{-1} h z_j g = h z_i z_j</M> for each <M>j</M>,
 so the four elements lie in exactly two <M>H</M>-classes.
 This implies that for <M>i ¬eq; j</M>, the elements <M>h</M> and
 <M>h z_j</M> are not <M>H</M>-conjugate,
 so <M>h Z_i</M> is not conjugate to <M>h z_j Z_i</M> in <M>H/Z_i</M>
 and <M>h Z_j</M> is conjugate to <M>h z_i Z_j</M> in <M>H/Z_j</M>.
</Item>
<Item>
 The four elements are pairwise nonconjugate in <M>H</M>.
 Then in each of the three groups <M>H/Z_i</M>,
 the two preimages of <M>h Z \in H/Z</M> are nonconjugate.
</Item>
</Enum>



We observe that the question which case actually applies for <M>h \in H</M>
can be decided from the three factor fusions from <M>H/Z_i</M> to <M>G</M>.
So we attempt to construct the table head of <M>H</M> and the three factor
fusions from <M>H</M> to the groups <M>H/Z_i</M>, as follows.
Each class <M>g^G</M> of <M>G</M> yields either one or two or four preimage
classes in <M>H</M>.



In case 1., we get one preimage class in <M>H</M>,
and have no choice for the factor fusions.



In case 2., we get two preimage classes,
there is exactly one group <M>H/Z_i</M> in which <M>g^G</M> has two preimage
classes
–which are in bijection with the two preimage classes of
<M>H</M>–
and for the other two groups <M>H/Z_j</M>, the factor fusions from <M>H</M>
map the two classes of <M>H</M> to the unique preimage class of <M>g^G</M>.
(In the following picture, this is shown for <M>i = 1</M>.)



<Alt Only='HTML'><![CDATA[
<center>
<img src="ctblcons06.png" alt="Two preimages for a class"/>
</center>
]]></Alt>

<Alt Only='Text'>
<Verb>
 H/Z hZ H/Z hZ H/Z hZ
 ╱ ╲ │ │
 ╱ ╲ │ │
 H/Z_1 hZ_1 (hz_2)Z_1 H/Z_2 hZ_2 H/Z_3 hZ_3
 │ │ ╱ ╲ ╱ ╲
 │ │ ╱ ╲ ╱ ╲
 H h hz_2 h hz_2 h hz_2
</Verb>
</Alt>

<Alt Only='LaTeX'>

<![CDATA[
\begin{center}
\setlength{\unitlength}{3pt}
\begin{picture}(110,30)
\put( 3, 5){\makebox(0,0){$H$}}
\put( 3,15){\makebox(0,0){$H/Z_1$}}
\put( 3,25){\makebox(0,0){$H/Z$}}
\put(10, 5){\circle*{1}} \put(12, 5){\makebox(0,0){$h$}}
\put(10,15){\circle*{1}} \put(14,15){\makebox(0,0){$h Z_1$}}
\put(20, 5){\circle*{1}} \put(24, 5){\makebox(0,0){$h z_2$}}
\put(20,15){\circle*{1}} \put(25,15){\makebox(0,0){$h z_2 Z_1$}}
\put(15,25){\circle*{1}} \put(18,25){\makebox(0,0){$h Z$}}
\put(10, 5){\line(0,1){10}}
\put(20, 5){\line(0,1){10}}
\put(10,15){\line(1,2){5}}
\put(20,15){\line(-1,2){5}}
\put(43, 5){\makebox(0,0){$H$}}
\put(43,15){\makebox(0,0){$H/Z_2$}}
\put(43,25){\makebox(0,0){$H/Z$}}
\put(50, 5){\circle*{1}} \put(52, 5){\makebox(0,0){$h$}}
\put(55,15){\circle*{1}} \put(59,15){\makebox(0,0){$h Z_2$}}
\put(60, 5){\circle*{1}} \put(64, 5){\makebox(0,0){$h z_2$}}
\put(55,25){\circle*{1}} \put(58,25){\makebox(0,0){$h Z$}}
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\put(60,5){\line(-1,2){5}}
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\put(83, 5){\makebox(0,0){$H$}}
\put(83,15){\makebox(0,0){$H/Z_3$}}
\put(83,25){\makebox(0,0){$H/Z$}}
\put(90, 5){\circle*{1}} \put(92, 5){\makebox(0,0){$h$}}
\put(95,15){\circle*{1}} \put(99,15){\makebox(0,0){$h Z_3$}}
\put(100, 5){\circle*{1}} \put(104, 5){\makebox(0,0){$h z_2$}}
\put(95,25){\circle*{1}} \put(98,25){\makebox(0,0){$h Z$}}
\put(90,5){\line(1,2){5}}
\put(100,5){\line(-1,2){5}}
\put(95,15){\line(0,1){10}}
\end{picture}
\end{center}
]]>

</Alt>



In case 3., the three factor fusions are in general not uniquely determined:
We get four classes, which are defined as two pairs of preimages of the two
preimages of <M>g^G</M> in <M>H/Z_1</M> and in <M>H/Z_2</M>
–so we choose the relevant images in the two factor fusions to
<M>H/Z_1</M> and <M>H/Z_2</M>, respectively.
Note that the class of <M>h</M> in <M>H</M> is the unique class
that maps to the class of <M>h Z_1</M> in <M>H/Z_1</M> and to the class of
<M>h Z_2</M> in <M>H/Z_2</M>, and so on,
and we define four classes of <M>H</M> via the four possible combinations of
image classes in <M>H/Z_1</M> and <M>H/Z_2</M> (see the picture below).



<Alt Only='HTML'><![CDATA[
<center>
<img src="ctblcons07.png" alt="Four preimages for a class (part 1)"/>
</center>
]]></Alt>

<Alt Only='Text'>
<Verb>
 H/Z hZ H/Z hZ
 ╱ ╲ ╱ ╲
 ╱ ╲ ╱ ╲
 H/Z_1 hZ_1 (hz_2)Z_1 H/Z_2 hZ_2 (hz_1)Z_2
 ╱│ │╲ ╱ ╲╱ ╲
 ╱ │ │ ╲ ╱ ╱╲ ╲
 H h hz_1 hz_2 hz_3 h hz_1 hz_2 hz_3
</Verb>
</Alt>

<Alt Only='LaTeX'>

<![CDATA[
\begin{center}
\setlength{\unitlength}{3pt}
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\put( 3, 5){\makebox(0,0){$H$}}
\put( 3,15){\makebox(0,0){$H/Z_1$}}
\put( 3,25){\makebox(0,0){$H/Z$}}
\put(10, 5){\circle*{1}} \put(12, 5){\makebox(0,0){$h$}}
\put(20, 5){\circle*{1}} \put(23, 5){\makebox(0,0){$h z_1$}}
\put(30, 5){\circle*{1}} \put(33, 5){\makebox(0,0){$h z_2$}}
\put(40, 5){\circle*{1}} \put(43, 5){\makebox(0,0){$h z_3$}}
\put(15,15){\circle*{1}} \put(19,15){\makebox(0,0){$h Z_1$}}
\put(35,15){\circle*{1}} \put(40,15){\makebox(0,0){$h z_2 Z_1$}}
\put(25,25){\circle*{1}} \put(29,25){\makebox(0,0){$h Z$}}
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\put(20, 5){\line(-1,2){5}}
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\put(40, 5){\line(-1,2){5}}
\put(15,15){\line(1,1){10}}
\put(35,15){\line(-1,1){10}}
\put(63, 5){\makebox(0,0){$H$}}
\put(63,15){\makebox(0,0){$H/Z_2$}}
\put(63,25){\makebox(0,0){$H/Z$}}
\put(70, 5){\circle*{1}} \put(72, 5){\makebox(0,0){$h$}}
\put(80, 5){\circle*{1}} \put(84, 5){\makebox(0,0){$h z_1$}}
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\put(100, 5){\circle*{1}} \put(103, 5){\makebox(0,0){$h z_3$}}
\put(75,15){\circle*{1}} \put(80,15){\makebox(0,0){$h Z_2$}}
\put(95,15){\circle*{1}} \put(100,15){\makebox(0,0){$h z_1 Z_2$}}
\put(85,25){\circle*{1}} \put(89,25){\makebox(0,0){$h Z$}}
\put(70, 5){\line(1,2){5}}
\put(80, 5){\line(3,2){15}}
\put(90, 5){\line(-3,2){15}}
\put(100, 5){\line(-1,2){5}}
\put(75,15){\line(1,1){10}}
\put(95,15){\line(-1,1){10}}
\end{picture}
\end{center}
]]>

</Alt>



Due to the fact that in general we do not know which of the two
preimage classes of <M>g^G</M> in <M>H/Z_3</M> is the class of <M>h Z_3</M>,
there are in general the following <E>two</E> possibilities
for the fusion from <M>H</M> to <M>H/Z_3</M>.



<Alt Only='HTML'><![CDATA[
<center>
<img src="ctblcons08.png" alt="Four preimages for a class (part 2)"/>
</center>
]]></Alt>

<Alt Only='Text'>
<Verb>
 H/Z hZ H/Z hZ
 ╱ ╲ ╱ ╲
 ╱ ╲ ╱ ╲
 H/Z_3 hZ_3 (hz_1)Z_3 H/Z_3 hZ_3 (hz_1)Z_3
 ╱ └──╱│────┐ ┌───│╲──┘ ╲
 ╱ ╱ │ │ │ │ ╲ ╲
 H h hz_1 hz_2 hz_3 h hz_1 hz_2 hz_3
</Verb>
</Alt>

<Alt Only='LaTeX'>

<![CDATA[
\begin{center}
\setlength{\unitlength}{3pt}
\begin{picture}(110,30)
\put( 3, 5){\makebox(0,0){$H$}}
\put( 3,15){\makebox(0,0){$H/Z_3$}}
\put( 3,25){\makebox(0,0){$H/Z$}}
\put(10, 5){\circle*{1}} \put(12, 5){\makebox(0,0){$h$}}
\put(20, 5){\circle*{1}} \put(24, 5){\makebox(0,0){$h z_1$}}
\put(30, 5){\circle*{1}} \put(33, 5){\makebox(0,0){$h z_2$}}
\put(40, 5){\circle*{1}} \put(43, 5){\makebox(0,0){$h z_3$}}
\put(15,15){\circle*{1}} \put(20,15){\makebox(0,0){$h Z_3$}}
\put(35,15){\circle*{1}} \put(40,15){\makebox(0,0){$h z_1 Z_3$}}
\put(25,25){\circle*{1}} \put(29,25){\makebox(0,0){$h Z$}}
\put(10, 5){\line(1,2){5}}
\put(20, 5){\line(3,2){15}}
\put(30, 5){\line(1,2){5}}
\put(40, 5){\line(-5,2){25}}
\put(15,15){\line(1,1){10}}
\put(35,15){\line(-1,1){10}}
\put(63, 5){\makebox(0,0){$H$}}
\put(63,15){\makebox(0,0){$H/Z_3$}}
\put(63,25){\makebox(0,0){$H/Z$}}
\put(70, 5){\circle*{1}} \put(74, 5){\makebox(0,0){$h$}}
\put(80, 5){\circle*{1}} \put(83, 5){\makebox(0,0){$h z_1$}}
\put(90, 5){\circle*{1}} \put(93, 5){\makebox(0,0){$h z_2$}}
\put(100, 5){\circle*{1}} \put(103, 5){\makebox(0,0){$h z_3$}}
\put(75,15){\circle*{1}} \put(81,15){\makebox(0,0){$h z_1 Z_3$}}
\put(95,15){\circle*{1}} \put(99,15){\makebox(0,0){$h Z_3$}}
\put(85,25){\circle*{1}} \put(89,25){\makebox(0,0){$h Z$}}
\put(70, 5){\line(5,2){25}}
\put(80, 5){\line(-1,2){5}}
\put(90, 5){\line(-3,2){15}}
\put(100, 5){\line(-1,2){5}}
\put(75,15){\line(1,1){10}}
\put(95,15){\line(-1,1){10}}
\end{picture}
\end{center}
]]>

</Alt>



This means that we can inflate the irreducible characters of <M>H/Z_1</M> and
of <M>H/Z_2</M> to <M>H</M> but that for the inflations of those irreducible
characters of <M>H/Z_3</M> to <M>H</M> that are not characters of <M>G</M>,
the values on classes where case 3. applies are determined only up to
sign.



The &GAP; function
<Ref Func="PossibleCharacterTablesOfTypeV4G" BookName="ctbllib"/>
computes the
candidates for the table of <M>H</M> from the tables of the groups
<M>H/Z_i</M>
by setting up the character table head of <M>H</M> using the class fusions
from <M>H/Z_1</M> and <M>H/Z_2</M> to <M>G</M>,
and then forming the possible class fusions from <M>H</M> to <M>H/Z_3</M>.



If case 3. applies for a class <M>g^G</M> with <M>g</M> of <E>odd</E>
element order
then exactly one preimage class in <M>H</M> has odd element order,
and we can identify this class in the groups <M>H/Z_i</M>,
which resolves the ambiguity in this situation.
More generally, if <M>g = k^2</M> holds for some <M>k \in G</M> then all
preimages of <M>k^G</M> in <M>H</M> square to the same class of <M>H</M>,
so again this class can be identified.
In fact
<Ref Func="PossibleCharacterTablesOfTypeV4G" BookName="ctbllib"/>
checks whether the
<M>p</M>-th power maps of the candidate table for <M>H</M> and the
<M>p</M>-th power map of <M>H/Z_3</M>
together with the fusion candidate form a commutative diagram.



An additional criterion used by
<Ref Func="PossibleCharacterTablesOfTypeV4G" BookName="ctbllib"/>
is given by the property that the product of two characters inflated from
<M>H/Z_1</M> and <M>H/Z_2</M>, respectively, that are not characters of
<M>G</M> is a character of <M>H</M> that contains <M>Z_3</M> in its kernel,
so it is checked whether the scalar products of these characters
with all characters that are inflated from <M>H/Z_3</M> via the candidate
fusion are nonnegative integers.



Once the fusions from <M>H</M> to the groups <M>H/Z_i</M> are known,
the computation of the irreducible <M>p</M>-modular characters of <M>H</M>
from those of the groups <M>H/Z_i</M> is straightforward.



The only open question is why this construction is described in this note.
That is, how is it related to table automorphisms?



The answer is that in several interesting cases,
the three subgroups <M>Z_1</M>, <M>Z_2</M>, <M>Z_3</M> are conjugate
under an order three automorphism <M>\sigma</M>, say, of <M>H</M>.
In this situation, the three factor groups <M>2_i.G = H/Z_i</M> are
isomorphic,
and we can describe the input tables and fusions by the character table of
<M>2_1.G</M>, the factor fusion from this group to <M>G</M>,
and the automorphism <M>\sigma' of G
that is induced by <M>\sigma</M>.
Assume that <M>\sigma(Z_1) = Z_2</M> holds, and choose <M>h \in H</M>.
Then <M>\sigma(h Z_1) = \sigma(h) Z_2</M> is mapped to
<M>\sigma(h) Z = \sigma'(h Z) under the factor fusion from 2_2.G
to <M>G</M>.
Let us start with the character table of <M>2_1.G</M>,
and fix the class fusion to the character table of <M>G</M>.
We may choose the identity map as isomorphism from the table of <M>2_1.G</M>
to the tables of <M>2_2.G</M> and <M>2_3.G</M>,
which implies that the class of <M>h Z_1</M> is identified with the class
of <M>h Z_2</M> and in turn the class fusion from the table of <M>2_2.G</M>
to that of <M>G</M> can be chosen as the class fusion from the table of
<M>2_1.G</M> followed by the permutation of classes of <M>G</M> induced by
<M>\sigma';
analogously, the fusion from the table of <M>2_3.G</M> is obtained
by applying this permutation twice to the class fusion from the table of
<M>2_1.G</M>.



For examples, see Section <Ref Sect="sect:xplV4G"/>.

</Subsection>


<Subsection Label="subsect:theorpsing">
<Heading><M>p</M>-Modular Tables of Extensions by <M>p</M>-singular Automorphisms</Heading>

Let <M>G</M> be a finite group, and <M>H</M> be an upward extension of
<M>G</M> by an automorphism of prime order <M>p</M>, say.
<M>H</M> induces a table automorphism of the <M>p</M>-modular character table
of <M>G</M>;
let <M>\pi</M> denote the corresponding permutation of classes of <M>G</M>.
The columns of the <M>p</M>-modular character table of <M>H</M> are given by
the orbits of <M>\pi</M>,
and the irreducible Brauer characters of <M>H</M> are exactly the orbit sums
of <M>\pi</M> on the irreducible Brauer characters of <M>G</M>.



Note that for computing the <M>p</M>-modular character table of <M>H</M> from
that of <M>G</M>,
it is sufficient to know the orbits of <M>\pi</M> and not <M>\pi</M> itself.
Also the ordinary character table of <M>H</M> is not needed,
but since &GAP; stores Brauer character tables relative to their ordinary
tables, we are interested mainly in cases where the ordinary character
tables of <M>G</M> and <M>H</M> and the <M>p</M>-modular character table of
<M>G</M> are known.
Assuming that the class fusion between the ordinary tables of <M>G</M> and
<M>H</M> is stored on the table of <M>G</M>,
the orbits of the action of <M>H</M> on the <M>p</M>-regular classes of
<M>G</M> can be read off from it.



The &GAP; function
<Ref Func="IBrOfExtensionBySingularAutomorphism" BookName="ctbllib"/>
can be used to compute the <M>p</M>-modular irreducibles of <M>H</M>.



For examples, see Section <Ref Sect="sect:xplpsing"/>.

</Subsection>


<Subsection Label="subsect:theorsubdir">
<Heading>Character Tables of Subdirect Products of Index Two (July 2007)</Heading>

Let <M>C_2</M> denote the cyclic group of order two,
let <M>G_1</M>, <M>G_2</M> be two finite groups,
and for <M>i \in \{ 1, 2 \}</M>,
let <M>\varphi_i\colon G_i \rightarrow C_2</M>
be an epimorphism with kernel <M>H_i</M>.
Let <M>G</M> be the subdirect product (pullback) of <M>G_1</M> and <M>G_2</M>
w.r.t. the epimorphisms <M>\varphi_i</M>, i.e.,

<Display Mode="M">
 G = \{ (g_1, g_2) \in G_1 \times G_2; \varphi_1(g_1) = \varphi_2(g_2) \} .
</Display>

The group <M>G</M> has index two in the direct product <M>G_1 \times G_2</M>,
and <M>G</M> contains <M>H_1 \times H_2</M> as a subgroup of index two.



In the following,
we describe how the ordinary (or <M>p</M>-modular) character table of
<M>G</M> can be computed from the ordinary (or <M>p</M>-modular)
character tables of the groups <M>G_i</M> and <M>H_i</M>,
and the class fusions from <M>H_i</M> to <M>G_i</M>.



(For the case that one of the groups <M>G_i</M> is a cyclic group of order
four,
an alternative way to construct the character table of <M>G</M> is described
in Section <Ref Subsect="subsect:isoclinism"/>.
For the case that one of the groups <M>G_i</M> acts fixed point freely on the
nontrivial irreducible characters of <M>H_i</M>, an alternative construction
is described in Section <Ref Subsect="subsect:theorMGA"/>.)



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