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<a id="X7B77FD307F0DE563" name="X7B77FD307F0DE563"></a>
<div class="ChapSects"><a href="chap2_mj.html#X7B77FD307F0DE563">2 Using Table Automorphisms for Constructing Character Tables in GAP</a>
<div class="ContSect"> <a href="chap2_mj.html#X8389AD927B74BA4A">2.1 Overview</a>

</div>
<div class="ContSect"> <a href="chap2_mj.html#X7B6AEBDF7B857E2E">2.2 Theoretical Background</a>

<div class="ContSSBlock">
 <a href="chap2_mj.html#X78EBF9BA7A34A9C2">2.2-1 Character Table Automorphisms</a>

 <a href="chap2_mj.html#X832525DE7AB34F16">2.2-2 Permutation Equivalence of Character Tables</a>

 <a href="chap2_mj.html#X7906869F7F190E76">2.2-3 Class Fusions</a>

 <a href="chap2_mj.html#X80C37276851D5E39">2.2-4 Constructing Character Tables of Certain Isoclinic Groups</a>

 <a href="chap2_mj.html#X7AEFFEEC84511FD0">2.2-5 Character Tables of Isoclinic Groups of the Structure \(p.G.p\)
(October 2016)</a>

 <a href="chap2_mj.html#X78F41D2A78E70BEE">2.2-6 Isoclinic Double Covers of Almost Simple Groups</a>

 <a href="chap2_mj.html#X834B42A07E98FBC6">2.2-7 Characters of Normal Subgroups</a>

</div></div>
<div class="ContSect"> <a href="chap2_mj.html#X787F430E7FDB8765">2.3 The Constructions</a>

<div class="ContSSBlock">
 <a href="chap2_mj.html#X82E75B6880EC9E6C">2.3-1 Character Tables of Groups of the Structure \(M.G.A\)</a>

 <a href="chap2_mj.html#X7CCABDDE864E6300">2.3-2 Character Tables of Groups of the Structure \(G.S_3\)</a>

 <a href="chap2_mj.html#X7D3EF3BC83BE05CF">2.3-3 Character Tables of Groups of the Structure \(G.2^2\)</a>

 <a href="chap2_mj.html#X81464C4B8178C85A">2.3-4 Character Tables of Groups of the Structure \(2^2.G\)
(August 2005)</a>

 <a href="chap2_mj.html#X86CF6A607B0827EE">2.3-5 \(p\)-Modular Tables of Extensions by \(p\)-singular Automorphisms</a>

 <a href="chap2_mj.html#X788591D78451C024">2.3-6 Character Tables of Subdirect Products of Index Two (July 2007)</a>

</div></div>
<div class="ContSect"> <a href="chap2_mj.html#X817D2134829FA8FA">2.4 Examples for the Type \(M.G.A\)</a>

<div class="ContSSBlock">
 <a href="chap2_mj.html#X7F2DBAB48437052C">2.4-1 Character Tables of Dihedral Groups</a>

 <a href="chap2_mj.html#X7925DBFA7C5986B5">2.4-2 An \(M.G.A\) Type Example with \(M\) noncentral in \(M.G\) (May 2004)</a>

 <a href="chap2_mj.html#X7ED45AB379093A70">2.4-3 Atlas Tables of the Type \(M.G.A\)</a>

 <a href="chap2_mj.html#X7A236EDE7A7A28F9">2.4-4 More Atlas Tables of the Type \(M.G.A\)</a>

 <a href="chap2_mj.html#X794EC2FD7F69B4E6">2.4-5 The Character Tables of \(4_2.L_3(4).2_3\) and \(12_2.L_3(4).2_3\)</a>

 <a href="chap2_mj.html#X7E3E748E85AEDDB3">2.4-6 The Character Tables of \(12_1.U_4(3).2_2'\) and
\(12_2.U_4(3).2_3'\) (December 2015)

 <a href="chap2_mj.html#X8379003582D06130">2.4-7 Groups of the Structures \(3.U_3(8).3_1\) and \(3.U_3(8).6\)
(February 2017)</a>

 <a href="chap2_mj.html#X7B46C77B850D3B4D">2.4-8 The Character Table of \((2^2 \times F_4(2)):2 < B\)
(March 2003)</a>

 <a href="chap2_mj.html#X8254AA4A843F99BE">2.4-9 The Character Table of \(2.(S_3 \times Fi_{22}.2) < 2.B\) (March 2003)</a>

 <a href="chap2_mj.html#X7AF125168239D208">2.4-10 The Character Table of \((2 \times 2.Fi_{22}):2 < Fi_{24}\) (November 2008)</a>

 <a href="chap2_mj.html#X79C93F7D87D9CF1D">2.4-11 The Character Table of \(S_3 \times 2.U_4(3).2_2 \leq 2.Fi_{22}\) (September 2002)</a>

 <a href="chap2_mj.html#X83724BCE86FCD77B">2.4-12 The Character Table of \(4.HS.2 \leq HN.2\) (May 2002)</a>

 <a href="chap2_mj.html#X7E9A88DA7CBF6426">2.4-13 The Character Tables of \(4.A_6.2_3\), \(12.A_6.2_3\),
and \(4.L_2(25).2_3\)</a>

 <a href="chap2_mj.html#X7BD79BA37C3E729B">2.4-14 The Character Table of \(4.L_2(49).2_3\) (December 2020)</a>

 <a href="chap2_mj.html#X817A961487D2DFD1">2.4-15 The Character Table of \(4.L_2(81).2_3\) (December 2020)</a>

 <a href="chap2_mj.html#X7AF324AF7A54798F">2.4-16 The Character Table of \(9.U_3(8).3_3\) (March 2017)</a>

 <a href="chap2_mj.html#X7E0C603880157C4E">2.4-17 Pseudo Character Tables of the Type \(M.G.A\) (May 2004)</a>

 <a href="chap2_mj.html#X844185EF7A8F2A99">2.4-18 Some Extra-ordinary \(p\)-Modular Tables of the Type \(M.G.A\)
(September 2005)</a>

</div></div>
<div class="ContSect"> <a href="chap2_mj.html#X7F50C782840F06E4">2.5 Examples for the Type \(G.S_3\)</a>

<div class="ContSSBlock">
 <a href="chap2_mj.html#X7F0DC29F874AA09F">2.5-1 Small Examples</a>

 <a href="chap2_mj.html#X80F9BC057980A9E9">2.5-2 Atlas Tables of the Type \(G.S_3\)</a>

</div></div>
<div class="ContSect"> <a href="chap2_mj.html#X7EA489E07D7C7D86">2.6 Examples for the Type \(G.2^2\)</a>

<div class="ContSSBlock">
 <a href="chap2_mj.html#X8054FDE679053B1C">2.6-1 The Character Table of \(A_6.2^2\)</a>

 <a href="chap2_mj.html#X7FEC3AB081487AF2">2.6-2 Atlas Tables of the Type \(G.2^2\) – Easy Cases</a>

 <a href="chap2_mj.html#X869B65D3863EDEC3">2.6-3 The Character Table of \(S_4(9).2^2\) (September 2011)</a>

 <a href="chap2_mj.html#X7B38006380618543">2.6-4 The Character Tables of Groups of the Type \(2.L_3(4).2^2\)
(June 2010)</a>

 <a href="chap2_mj.html#X79818ABD7E972370">2.6-5 The Character Tables of Groups of the Type \(6.L_3(4).2^2\)
(October 2011)</a>

 <a href="chap2_mj.html#X878889308653435F">2.6-6 The Character Tables of Groups of the Type \(2.U_4(3).2^2\)
(February 2012)</a>

 <a href="chap2_mj.html#X7DC42AE57E9EED4D">2.6-7 The Character Tables of Groups of the Type \(4_1.L_3(4).2^2\)
(October 2011)</a>

 <a href="chap2_mj.html#X7E9AF180869B4786">2.6-8 The Character Tables of Groups of the Type \(4_2.L_3(4).2^2\)
(October 2011)</a>

 <a href="chap2_mj.html#X7EAF9CD07E536120">2.6-9 The Character Table of Aut\((L_2(81))\)</a>

 <a href="chap2_mj.html#X78AED04685EDCC19">2.6-10 The Character Table of \(O_8^+(3).2^2_{111}\)</a>

</div></div>
<div class="ContSect"> <a href="chap2_mj.html#X845BAA2A7FD768B0">2.7 Examples for the Type \(2^2.G\)</a>

<div class="ContSSBlock">
 <a href="chap2_mj.html#X87EEBDB987249117">2.7-1 The Character Table of \(2^2.Sz(8)\)</a>

 <a href="chap2_mj.html#X83652A0282A64D14">2.7-2 Atlas Tables of the Type \(2^2.G\) (September 2005)</a>

 <a href="chap2_mj.html#X7F63DDF77870F967">2.7-3 The Character Table of \(2^2.O_8^+(3)\) (March 2009)</a>

 <a href="chap2_mj.html#X86A1607787DE6BB9">2.7-4 The Character Table of the Schur Cover of \(L_3(4)\)
(September 2005)</a>

</div></div>
<div class="ContSect"> <a href="chap2_mj.html#X8711DBB083655A25">2.8 Examples of Extensions by \(p\)-singular Automorphisms</a>

<div class="ContSSBlock">
 <a href="chap2_mj.html#X81C08739850E4AAE">2.8-1 Some \(p\)-Modular Tables of Groups of the Type \(M.G.A\)</a>

 <a href="chap2_mj.html#X7FED618F83ACB7C2">2.8-2 Some \(p\)-Modular Tables of Groups of the Type \(G.S_3\)</a>

 <a href="chap2_mj.html#X7EEF6A7F8683177A">2.8-3 \(2\)-Modular Tables of Groups of the Type \(G.2^2\)</a>

 <a href="chap2_mj.html#X875F8DD77C0997FA">2.8-4 The \(3\)-Modular Table of \(U_3(8).3^2\)</a>

</div></div>
<div class="ContSect"> <a href="chap2_mj.html#X7A4D6044865E516B">2.9 Examples of Subdirect Products of Index Two</a>

<div class="ContSSBlock">
 <a href="chap2_mj.html#X850FF694801700CF">2.9-1 Certain Dihedral Groups as Subdirect Products of Index Two</a>

 <a href="chap2_mj.html#X80C5D6FA83D7E2CF">2.9-2 The Character Table of \((D_{10} \times HN).2 < M\) (June 2008)</a>

 <a href="chap2_mj.html#X85EECFD47EC252A2">2.9-3 A Counterexample (August 2015)</a>

</div></div>
</div>

<h3>2 Using Table Automorphisms for Constructing Character Tables in GAP</h3>

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 <a href="chapBib_mj.html#biBGAP">[GAP24]</a>; the GAP functions used for that are part of the GAP Character Table Library <a href="chapBib_mj.html#biBCTblLib">[Bre25]</a>. 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.

<a id="X8389AD927B74BA4A" name="X8389AD927B74BA4A"></a>

<h4>2.1 Overview</h4>

Several types of constructions of character tables of finite groups from known tables of smaller groups are described in Section <a href="chap2_mj.html#X787F430E7FDB8765">2.3</a>. Selecting suitable character table automorphisms is an important ingredient of these constructions.

Section <a href="chap2_mj.html#X7B6AEBDF7B857E2E">2.2</a> 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.

<div class="example"><pre>
gap> LoadPackage( "ctbllib", "1.1.4", false );
true
</pre></div>

<a id="X7B6AEBDF7B857E2E" name="X7B6AEBDF7B857E2E"></a>

<h4>2.2 Theoretical Background</h4>

<a id="X78EBF9BA7A34A9C2" name="X78EBF9BA7A34A9C2"></a>

<h5>2.2-1 Character Table Automorphisms</h5>

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

\[
 pow_p \colon Cl(G) \rightarrow Cl(G),
 g^G \mapsto (g^p)^G
\]

be the \(p\)-th power map, for each prime integer \(p\).

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

In GAP, a character table covers the irreducible characters –a matrix \(M\) of character values– as well as the power maps of the underlying group –each power map \(pow_p\) being represented as a list \(pow_p^{\prime}\) 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 <code class="func">AutomorphismsOfTable</code> (<a href="../../../doc/ref/chap71_mj.html#X7C2753DE8094F4BA">Reference: AutomorphismsOfTable</a>).

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

<a id="X832525DE7AB34F16" name="X832525DE7AB34F16"></a>

<h5>2.2-2 Permutation Equivalence of Character Tables</h5>

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

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

Clearly two isomorphic groups have permutation equivalent character tables.

The GAP library function <code class="func">TransformingPermutationsCharacterTables</code> (<a href="../../../doc/ref/chap71_mj.html#X849731AA7EC9FA73">Reference: TransformingPermutationsCharacterTables</a>) returns a record that contains transforming permutations of rows and columns if the two argument tables are permutation equivalent, and <code class="keyw">fail</code> 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 <code class="code">table</code> whose value is a character table, and the output is a sublist of the input.

<div class="example"><pre>
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;;
</pre></div>

<a id="X7906869F7F190E76" name="X7906869F7F190E76"></a>

<h5>2.2-3 Class Fusions</h5>

For two groups \(H\), \(G\) such that \(H\) is isomorphic with a subgroup of \(G\), any embedding \(\iota \colon H \rightarrow G\) induces a class function

\[
 fus_{\iota} \colon Cl(H) \rightarrow Cl(G),
 h^G \mapsto (\iota(h))^G
\]

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

\[
 fus_{\pi} \colon Cl(G) \rightarrow Cl(G/N),
 g^G \mapsto (\pi(g))^G
\]

the class fusion of \(G\) onto \(G/N\) via \(\pi\).

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 \(H\) and \(G\) for which it is known that \(H\) can be embedded into \(G\) then one can use character-theoretic necessary conditions, concerning that the restriction of all irreducible characters of \(G\) to \(H\) (via the class fusion) must decompose into the irreducible characters of \(H\), and that the class fusion must commute with the power maps of \(H\) and \(G\).

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 \(G\) as a class fusion from the table of \(G\) to itself.

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

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.

<a id="X80C37276851D5E39" name="X80C37276851D5E39"></a>

<h5>2.2-4 Constructing Character Tables of Certain Isoclinic Groups</h5>

As is stated in <a href="chapBib_mj.html#biBCCN85">[CCN+85, p. xxiii]</a>, two groups \(G\), \(H\) are called isoclinic if they can be embedded into a group \(K\) such that \(K\) is generated by \(Z(K)\) and \(G\), and also by \(Z(K)\) and \(H\). In the following, two special cases of isoclinism will be used, where the character tables of the isoclinic groups are closely related.

<dl>
<dt>(1)</dt>
<dd>\(G \cong 2 \times U\) for a group \(U\) that has a central subgroup \(N\) of order \(2\), and \(H\) is the central product of \(U\) and a cyclic group of order four. Here we can set \(K = 2 \times H\).

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

</dd>
</dl>
<center> <img src="ctblcons01.png" alt="two constructions of K"/> </center>

Starting from the group \(K\) containing both \(G\) and \(H\), we first note that each irreducible representation of \(G\) or \(H\) extends to \(K\). More specifically, if \(\rho_G\) is an irreducible representation of \(G\) then we can define an extension \(\rho\) of \(K\) by defining it suitably on \(Z(K)\) and then form \(\rho_H\), the restriction of \(\rho\) to \(H\).

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

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

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

In GAP, the function <code class="func">CharacterTableIsoclinic</code> (<a href="../../../doc/ref/chap71_mj.html#X85BE46F784B83938">Reference: CharacterTableIsoclinic</a>) can be used for computing the character table of \(H\) from that of \(G\), and vice versa. (Note that in the above two cases, also the groups \(U\) and \(H\) are isoclinic by definition, but <code class="func">CharacterTableIsoclinic</code> (<a href="../../../doc/ref/chap71_mj.html#X85BE46F784B83938">Reference: CharacterTableIsoclinic</a>) does not transfer the character table of \(U\) to that of \(H\).)

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 <code class="func">CharacterTableIsoclinic</code> (<a href="../../../doc/ref/chap71_mj.html#X85BE46F784B83938">Reference: CharacterTableIsoclinic</a>) 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.

<a id="X7AEFFEEC84511FD0" name="X7AEFFEEC84511FD0"></a>

<h5>2.2-5 Character Tables of Isoclinic Groups of the Structure \(p.G.p\)
(October 2016)</h5>

Since the release of GAP 4.11, <code class="func">CharacterTableIsoclinic</code> (<a href="../../../doc/ref/chap71_mj.html#X85BE46F784B83938">Reference: CharacterTableIsoclinic</a>) admits the construction of the character tables of the isoclinic variants of groups of the structure \(p.G.p\), also for odd primes \(p\).

This feature will be used in the construction of the character table of \(9.U_3(8).3_3\), in order to construct the table of the subgroup \(3.(3 \times U_3(8))\) and of the factor group \((3 \times U_3(8)).3_3\), see Section <a href="chap2_mj.html#X7AF324AF7A54798F">2.4-16</a>. These constructions are a straightforward generalization of those described in detail in Section <a href="chap2_mj.html#X80C37276851D5E39">2.2-4</a>.

There are several examples of Atlas groups of the structure \(3.G.3\). The character table of one such group is shown in the Atlas, the tables of their isoclinic variants can now be obtained from <code class="func">CharacterTableIsoclinic</code> (<a href="../../../doc/ref/chap71_mj.html#X85BE46F784B83938">Reference: CharacterTableIsoclinic</a>).

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

<div class="example"><pre>
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
</pre></div>

The character table of GL\((3,4)\) is in fact the one which is shown in the Atlas.

<div class="example"><pre>
gap> IsRecord( TransformingPermutationsCharacterTables( t,
> CharacterTable( GL( 3, 4 ) ) ) );
true
</pre></div>

<a id="X78F41D2A78E70BEE" name="X78F41D2A78E70BEE"></a>

<h5>2.2-6 Isoclinic Double Covers of Almost Simple Groups</h5>

The function <code class="func">CharacterTableIsoclinic</code> (<a href="../../../doc/ref/chap71_mj.html#X85BE46F784B83938">Reference: CharacterTableIsoclinic</a>) can also be used to switch between the character tables of double covers of groups of the type \(G.2\), where \(G\) is a perfect group, see <a href="chapBib_mj.html#biBCCN85">[CCN+85, Section 6.7]</a>. Typical examples are the double covers of symmetric groups.

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

<div class="example"><pre>
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) )
</pre></div>

For groups of the type \(4.G.2\), two different situations can occur. Either the distinguished central cyclic subgroup of order four in \(4.G\) is inverted by the elements in \(4.G.2 \setminus 4.G\), or this subgroup is central in \(4.G.2\). In the first case, calling <code class="func">CharacterTableIsoclinic</code> (<a href="../../../doc/ref/chap71_mj.html#X85BE46F784B83938">Reference: CharacterTableIsoclinic</a>) with the character table of \(4.G.2\) yields a character table with the same set of irreducibles, only the \(2\)-power map will in general differ from that of the input table. In the second case, the one argument version of <code class="func">CharacterTableIsoclinic</code> (<a href="../../../doc/ref/chap71_mj.html#X85BE46F784B83938">Reference: CharacterTableIsoclinic</a>) 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 \(4.L_3(4).2\).

<div class="example"><pre>
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" ) ]
</pre></div>

The centres of the groups \(4_1.L_3(4).2_1\), \(4_1.L_3(4).2_2\), \(4_2.L_3(4).2_1\), and \(4_2.L_3(4).2_3\) 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.

<div class="example"><pre>
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 ]
</pre></div>

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

<div class="example"><pre>
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 ]
</pre></div>

<a id="X834B42A07E98FBC6" name="X834B42A07E98FBC6"></a>

<h5>2.2-7 Characters of Normal Subgroups</h5>

Let \(G\) be a group and \(N\) be a normal subgroup of \(G\). We will need the following well-known facts about the relation between the irreducible characters of \(G\) and \(N\).

For an irreducible (Brauer) character \(\chi\) of \(N\) and \(g \in G\), we define \(\chi^g\) by \(\chi^g(n) = \chi(n^g)\) for all \(n \in N\), and set \(I_G(\chi) = \{ g \in G; \chi^g = \chi \}\) (see <a href="chapBib_mj.html#biBFeit82">[Fei82, p. 86]</a>).

If \(I_G(\chi) = N\) then the induced character \(\chi^G\) is an irreducible (Brauer) character of \(G\) (see <a href="chapBib_mj.html#biBFeit82">[Fei82, Lemma III 2.11]</a> or <a href="chapBib_mj.html#biBNav98">[Nav98, Theorem 8.9]</a> or <a href="chapBib_mj.html#biBLP10">[LP10, Corollary 4.3.8]</a>).

If \(G/N\) is cyclic and if \(I_G(\chi) = G\) then \(\chi = \psi_N\) for an irreducible (Brauer) character \(\psi\) of \(G\), and each irreducible (Brauer) character \(\theta\) with the property \(\chi = \theta_N\) is of the form \(\theta = \psi \cdot \epsilon\), where \(\epsilon\) is an irreducible (Brauer) character of \(G/N\) (see <a href="chapBib_mj.html#biBFeit82">[Fei82, Theorem III 2.14]</a> or <a href="chapBib_mj.html#biBNav98">[Nav98, Theorem 8.12]</a> or <a href="chapBib_mj.html#biBLP10">[LP10, Theorem 3.6.13]</a>).

Clifford's theorem ([Fei82, Theorem III 2.12] or [Nav98, Corollary 8.7] or [LP10, Theorem 3.6.2]) states that the restriction of an irreducible (Brauer) character of \(G\) to \(N\) has the form \(e \sum_{i=1}^t \varphi_i\) for a positive integer \(e\) and irreducible (Brauer) characters \(\varphi_i\) of \(N\), where \(t\) is the index of \(I_G(\varphi_1)\) in \(G\).

Now assume that \(G\) is a normal subgroup in a larger group \(H\), that \(G/N\) is an abelian chief factor of \(H\) and that \(\psi\) is an ordinary irreducible character of \(G\) such that \(I_H(\psi) = H\). Then either \(t = 1\) and \(e^2\) is one of \(1\), \(|G/N|\), or \(t = |G/N|\) and \(e = 1\) (see <a href="chapBib_mj.html#biBIsa76">[Isa76, Theorem 6.18]</a>).

<a id="X787F430E7FDB8765" name="X787F430E7FDB8765"></a>

<h4>2.3 The Constructions</h4>

<a id="X82E75B6880EC9E6C" name="X82E75B6880EC9E6C"></a>

<h5>2.3-1 Character Tables of Groups of the Structure \(M.G.A\)</h5>

(This kind of table construction is described in <a href="chapBib_mj.html#biBBre11">[Bre11]</a>.)

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

<center> <img src="ctblcons02.png" alt="groups of the structure M.G.A"/> </center>

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

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

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

If the orders of \(M\) and \(A\) are coprime then also the power maps of \(H\) can be computed from the above data. For each prime \(p\) that divides the orders of both \(M\) and \(A\), the \(p\)-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 \(H\) 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 <a href="chap2_mj.html#X80C37276851D5E39">2.2-4</a>, see Section <a href="chap2_mj.html#X83724BCE86FCD77B">2.4-12</a> for an example).

With the GAP function <code class="func">PossibleActionsForTypeMGA</code> (<a href="../doc/chap5_mj.html#X7899AA12836EEF8F">CTblLib: PossibleActionsForTypeMGA</a>), one can compute the possible orbit structures induced by \(G.A\) on the classes of \(M.G\), and <code class="func">PossibleCharacterTablesOfTypeMGA</code> (<a href="../doc/chap5_mj.html#X78F82DD67E083B88">CTblLib: PossibleCharacterTablesOfTypeMGA</a>) computes the possible ordinary character tables for a given orbit structure. For constructing the \(p\)-modular Brauer table of a group \(H\) of the structure \(M.G.A\), the GAP function <code class="func">BrauerTableOfTypeMGA</code> (<a href="../doc/chap5_mj.html#X83BE977185ADC24B">CTblLib: BrauerTableOfTypeMGA</a>) takes the ordinary character table of \(H\) and the \(p\)-modular tables of the subgroup \(M.G\) and the factor group \(G.A\) as its input. The \(p\)-modular table of \(G\) is not explicitly needed in the construction, it is implicitly given by the class fusions from \(M.G\) into \(M.G.A\) and from \(M.G.A\) onto \(G.A\); these class fusions must of course be available.

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

For examples, see Section <a href="chap2_mj.html#X817D2134829FA8FA">2.4</a>.

<a id="X7CCABDDE864E6300" name="X7CCABDDE864E6300"></a>

<h5>2.3-2 Character Tables of Groups of the Structure \(G.S_3\)</h5>

Let \(G\) be a finite group, and \(H\) be an upward extension of \(G\) such that the factor group \(H / G\) is a Frobenius group \(F = K C\) with abelian kernel \(K\) and cyclic complement \(C\) of prime order \(c\). (Typical cases for \(F\) are the symmetric group \(S_3\) on three points and the alternating group \(A_4\) on four points.) Let \(N\) and \(U\) denote the preimages of \(K\) and \(C\) under the natural epimorphism from \(H\) onto \(F\).

<center> <img src="ctblcons03.png" alt="Groups of the structure G.3.2"/> </center>

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

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

Concerning the computation of the ordinary irreducible characters of \(H\), we could induce the irreducible characters of \(U\) and \(N\) to \(H\), and then take the union of the irreducible characters among those and the irreducible differences of those. (For the case \(F = S_3\), this approach has been described in the Appendix of <a href="chapBib_mj.html#biBHL94">[HL94]</a>.)

The GAP function <code class="func">CharacterTableOfTypeGS3</code> (<a href="../doc/chap5_mj.html#X7E06095E7CB3316D">CTblLib: CharacterTableOfTypeGS3</a>) proceeds in a different way, which is suitable also for the construction of \(p\)-modular character tables of \(H\).

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