# Constructing Character Tables of Central Extensions in GAP

### THOMAS BREUER
Lehrstuhl D für Mathematik
RWTH, 52056 Aachen, Germany

### February 19th, 2004

This note has three aims. First it shows how the GAP system [GAP04] can be utilized to construct character tables of certain central extensions from known character tables; the GAP functions used for that are part of the GAP Character Table Library [Bre04]. 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 GAP functions.

# Contents

1 Coprime Central Extensions
1.1 The Character Table Head
1.2 The Irreducible Characters
1.3 Ordering of Conjugacy Classes
1.4 Compatibility with Smaller Factor Groups
2 Examples
2.1 Central Extensions of Simple ATLAS Groups
2.2 Compatible Central Extensions of Maximal Subgroups
2.3 The `2B` Centralizer in 3.Fi24′ (January 2004)

## 1 Coprime Central Extensions

In this section, we will deal with the following situation. Let H be a group, Z be a cyclic central subgroup in H, and Z = Z1 Z2 for subgroups Z1 and Z2 of coprime orders m and n, say. For the sake of simplicity, suppose that both m and n are primes; the general case is then obtained by iterating the construction process.
Our aim is to compute the character table of H from the character tables of H/Z1 and H/Z2. We assume that the factor fusions from these tables to that of the common factor group H/Z are known. Again for the sake of simplicity, we will take the character table of H/Z as an input. (See Section 2.3 for an example where two different orderings of classes and characters of H/Z arise from the tables of H/Z1 and H/Z2.)
For example, the character table of H = 12.M22 can be computed from those of 6.M22 and 4.M22, and the character table of 6.M22 can be computed from those of 3.M22 and 2.M22 (see Section 2.1).

### 1.1 The Character Table Head

The conjugacy classes and power maps of H are uniquely determined by the input data specified above.
Suppose that a class C of elements of H/Z has nC preimage classes in H/Z1 and mC preimage classes in H/Z2; then nC is either 1 or n, and mC is either 1 or m. The preimage classes of C in H/Z1 and H/Z2 are parametrized by { j; 0 ≤ j < nC } and { i; 0 ≤ i < mC }, respectively, and the preimage classes in H are parametrized by the pairs { (i,j); 0 ≤ i < mC, 0 ≤ j < nC }.
The centralizer orders of these classes in H are mC nC times the centralizer order of C in H/Z.
The factor fusion onto H/Z1 is then given by mapping the class with the parameter (i,j) to the class with the parameter j; analogously, the factor fusion onto H/Z2 maps this class to the class with the parameter i. To see this, let Z = 〈z 〉, and set z1 = zn and z2 = zm. Now take an element g ∈ H for which g Z lies in C. Then the elements g z1i z2j, 1 ≤ i ≤ mC, 1 ≤ j ≤ nC form a set of representatives of the preimage classes of C in H. In H/Z1 and H/Z2, these elements map to g z2j Z1, 1 ≤ j ≤ nC and g z1i Z2, 1 ≤ i ≤ mC, respectively, which are sets of representatives of the classes in question in these groups.
For each prime p, the factor fusions determine the p-th power map of H from the p-th power maps of H/Z1 and H/Z2. To see this, take a class C0 in H that is a preimage of the class C of H/Z, and let K be the class of p-th powers of the elements in C. Then the image of C0 under the p-th power map is one of the preimages of K. We know the images of C0 under the factor fusions to H/Z1 and H/Z2, and thus also their images K1 and K2 under the p-th power maps of these groups. So the class of p-th powers of the elements in C0 is the unique class that is mapped to K1 and K2 under the factor fusions.
The construction of the character table head of H from the input data specified above is implemented by the GAP function `CharacterTableOfCommonCentralExtension`.

### 1.2 The Irreducible Characters

First of all, it should be said that it is not obvious how the irreducible characters of H can be computed from the irreducible characters of H/Z1 and H/Z2. Clearly the irreducible characters of the two factor groups can be inflated to H via the factor fusions, so we have to find those irreducibles that have neither Z1 nor Z2 in their kernels.
For that, we use the following heuristic. Let εz be a complex primitive |z|-th root of unity. For integers i, set Irrz,i(H) = { χ ∈ Irr(H); χ(z) = εzi χ(1) }. Then Irr(H) = ∪i=0|z|−1 Irrz,i(H), as a disjoint union. If i is a multiple of m or n, respectively, then Irrz,i(H) consists of the inflations of certain irreducible characters of H/Z1 or H/Z2, respectively. The remaining irreducible characters of H lie in Irrz,i(H) with i coprime to |z|. These characters are algebraic conjugates of Irrz,1(H), so it suffices to compute this subset; the conjugates are then derived as the last step.
Since Irrz,i(H) ⊗Irrz,j(H) ⊂ Z[ Irrz,i+j(H) ] holds, we start with the tensor products of the known irreducible characters in Irrz,i(H) and Irrz,j(H) with the property i+j ≡ 1 mod m n.
For example, if we have m = 2 and n = 3 then Irrz,3(H) consists of the inflations of those characters in Irr(H/Z2) that are not characters of H/Z, and Irrz,4(H) consists of the inflations of certain characters in Irr(H/Z1) that are not characters of H/Z. The tensor products of these sets of characters lie in the span of Irrz,1(H).
In general these tensor products are reducible, but some of them may be in fact irreducible, so we first take these irreducibles, and reduce the other tensor products with them. (If H is a direct product of Z and H/Z then all missing irreducibles are obtained this way.)
Then we tensor algebraic conjugates of the known characters in the span of Irrz,1(H) with characters in suitable sets Irrz,i(H), in order to get more characters in Irrz,1(H); for example, Irrz,1(H) ⊗Irrz,0(H) is a subset of Z[Irrz,1(H)].
In the case m = 2 and n = 3, also Irrz,5(H) ⊗Irrz,2(H) yields linear combinations of Irrz,1(H). Note that Irrz,5(H) consists of the complex conjugates of Irrz,1(H).
In the next step, we apply the LLL algorithm (implemented via the GAP function `LLL`) to the set of reducible characters in Z[Irrz,1(H)] which we got from the tensor products, and hope to find irreducibles. In the examples shown below, this step yields all desired irreducible characters.
The GAP function `IrreduciblesForCharacterTableOfCommonCentralExtension` implements the strategy sketched above.

### 1.3 Ordering of Conjugacy Classes

One "natural" choice for the ordering of the columns in the character table of H is given by respecting the ordering of columns in the character table of H/Z, and taking the preimage of the class C corresponding to the parameter (k mod mC, k mod nC) as the k-th class for C.
If the preimages of C in H/Z1 and H/Z2 have class representatives g Z1, g z2 Z1, g z22 Z1, … and g Z2, g z1 Z2, g z12 Z2, …, respectively (in this ordering), then the above rule yields representatives of preimages in H in the ordering g, g (z1 z2), g (z1 z2)2, ….
In the case m = 2, n = 3, the following pattern arises for classes of H/Z that have m and n preimages in H/Z1 and H/Z2, respectively. The vertices are labelled by the roots of unity with which the values of the characters in the set Irrz,1(H) on the first preimage must be multiplied in order to obtain the values on the given class; we have ω = exp(2 πi/3).

### 1.4 Compatibility with Smaller Factor Groups

It may happen that a cyclic central subgroup Z0 of H contains Z properly. Then we choose a class ordering relative to that in the factor group H/Z0, mainly because the ATLAS tables of this type are sorted this way.
The typical case is the character table of a central extension of the type 12.G that shall be constructed from the character tables of the groups of the types 4.G and 6.G; here we prefer to order the preimages of a class in the smaller factor group of the type G according to the above rule. This results in the following pattern, where ε = exp(2 πi/12) holds (cf. Section "ATLAS Tables" in the manual of the GAP Character Table Library).
A more important aspect concerns the computation of the irreducible characters. Let Z0 = 〈z0 〉. Instead of computing Irrz,1(H), we compute the set Irrz0,1(H).
In the computation of the character table of a central extension of the type 12.G as mentioned above, with |z0| = 12, we start with the characters Irrz0,3(H) ⊗Irrz0,10(H) ∪ Irrz0,4(H) ⊗Irrz0,9(H) ⊆ Z[Irrz0,1(H)], and later form tensor products involving algebraic conjugates of the characters in the span of Irrz0,1(H), using that Irrz0,1(H) ⊗Irrz0,0(H) ∪ Irrz0,2(H) ⊗Irrz0,11(H) ∪ Irrz0,5(H) ⊗Irrz0,8(H) ∪ Irrz0,6(H) ⊗Irrz0,7(H) is a subset of Z[Irrz0,1(H)].
Without that modification, the computation of irreducibles is significantly more involved.
The GAP function `CharacterTableOfCommonCentralExtension` chooses the class ordering relative to larger cyclic factor groups, as in the above picture, and also uses the above refinement for the computation of irreducible characters.

## 2 Examples

The following examples use the GAP Character Table Library, so we first load this package.

        gap> LoadPackage( "ctbllib" );
        true

### 2.1 Central Extensions of Simple ATLAS Groups

For the following groups, the ATLAS contains the character tables of central extensions M.G of simple groups G with |M| divisible by two different primes; in all these cases, we have M ∈ { 6, 12 }.

        gap> list:= [
        >     #         G          m.G          n.G           mn.G
        >
        >     [      "A6",      "2.A6",      "3.A6",        "6.A6" ],
        >     [      "A7",      "2.A7",      "3.A7",        "6.A7" ],
        >     [   "L3(4)",   "2.L3(4)",   "3.L3(4)",     "6.L3(4)" ],
        >     [ "2.L3(4)", "4_1.L3(4)",   "6.L3(4)",  "12_1.L3(4)" ],
        >     [ "2.L3(4)", "4_2.L3(4)",   "6.L3(4)",  "12_2.L3(4)" ],
        >     [     "M22",     "2.M22",     "3.M22",       "6.M22" ],
        >     [   "2.M22",     "4.M22",     "6.M22",      "12.M22" ],
        >     [   "U4(3)",   "2.U4(3)", "3_1.U4(3)",   "6_1.U4(3)" ],
        >     [   "U4(3)",   "2.U4(3)", "3_2.U4(3)",   "6_2.U4(3)" ],
        >     [ "2.U4(3)",   "4.U4(3)", "6_1.U4(3)",  "12_1.U4(3)" ],
        >     [ "2.U4(3)",   "4.U4(3)", "6_2.U4(3)",  "12_2.U4(3)" ],
        >     [   "O7(3)",   "2.O7(3)",   "3.O7(3)",     "6.O7(3)" ],
        >     [   "U6(2)",   "2.U6(2)",   "3.U6(2)",     "6.U6(2)" ],
        >     [     "Suz",     "2.Suz",     "3.Suz",       "6.Suz" ],
        >     [    "Fi22",    "2.Fi22",    "3.Fi22",      "6.Fi22" ],
        >   ];;

As was discussed in the sections 1.3 and 1.4, the class ordering of the result tables is the same as that in the GAP library tables, so it is enough to check whether the set of characters in the computed table coincides with the set of characters in the library table.
In order to list information about the progress, we set the relevant info level to 1.

        gap> SetInfoLevel( InfoCharacterTable, 1 );
        gap> for entry in list do
        >   id    := entry[4];
        >   tblG  := CharacterTable( entry[1] );
        >   tblmG := CharacterTable( entry[2] );
        >   tblnG := CharacterTable( entry[3] );
        >   lib   := CharacterTable( id );
        >   res:= CharacterTableOfCommonCentralExtension( tblG, tblmG, tblnG, id );
        >   if not res.IsComplete then
        >     Print( "#E  not complete: ", id, "\n" );
        >   fi;
        >   if not IsSubset( Irr( lib ), res.irreducibles ) then
        >     Print( "#E  inconsistent: ", id, "\n" );
        >   fi;
        > od;
        #I  6.A6: need 4 faithful irreducibles
        #I  6.A6: 4 found by tensoring
        #I  6.A7: need 5 faithful irreducibles
        #I  6.A7: 5 found by tensoring
        #I  6.L3(4): need 7 faithful irreducibles
        #I  6.L3(4): 7 found by LLL
        #I  12_1.L3(4): need 5 faithful irreducibles
        #I  12_1.L3(4): 2 found by tensoring
        #I  12_1.L3(4): 3 found by tensoring
        #I  12_2.L3(4): need 6 faithful irreducibles
        #I  12_2.L3(4): 6 found by LLL
        #I  6.M22: need 10 faithful irreducibles
        #I  6.M22: 1 found by tensoring
        #I  6.M22: 9 found by LLL
        #I  12.M22: need 7 faithful irreducibles
        #I  12.M22: 7 found by LLL
        #I  6_1.U4(3): need 15 faithful irreducibles
        #I  6_1.U4(3): 1 found by tensoring
        #I  6_1.U4(3): 14 found by LLL
        #I  6_2.U4(3): need 12 faithful irreducibles
        #I  6_2.U4(3): 12 found by LLL
        #I  12_1.U4(3): need 12 faithful irreducibles
        #I  12_1.U4(3): 4 found by tensoring
        #I  12_1.U4(3): 8 found by tensoring
        #I  12_2.U4(3): need 9 faithful irreducibles
        #I  12_2.U4(3): 9 found by LLL
        #I  6.O7(3): need 12 faithful irreducibles
        #I  6.O7(3): 2 found by tensoring
        #I  6.O7(3): 10 found by LLL
        #I  6.U6(2): need 28 faithful irreducibles
        #I  6.U6(2): 2 found by tensoring
        #I  6.U6(2): 26 found by LLL
        #I  6.Suz: need 29 faithful irreducibles
        #I  6.Suz: 29 found by LLL
        #I  6.Fi22: need 34 faithful irreducibles
        #I  6.Fi22: 4 found by tensoring
        #I  6.Fi22: 30 found by LLL
        gap> SetInfoLevel( InfoCharacterTable, 0 );

We see that in all cases, the irreducible characters of the groups M.G are obtained by reducing tensor products and applying the LLL algorithm.

### 2.2 Compatible Central Extensions of Maximal Subgroups

The GAP Character Table Library contains the character tables of all maximal subgroups of the groups 4.M22, 3.M22, 2.Suz, and 3.Suz. So we can use the approach from Section 1 for computing the character tables of the maximal subgroups of 6.M22, 12.M22, and 6.Suz.
These tables are contained in the GAP Character Table Library. Several of the groups are direct products, and the library tables of direct products are usually stored in the form of Kronecker products of the tables of the factors, so the class ordering of the result tables does not necessarily coincide with the class ordering in the library tables.

        gap> sublist:= list{ [ 6, 7, 14 ] };
        [ [ "M22", "2.M22", "3.M22", "6.M22" ],
          [ "2.M22", "4.M22", "6.M22", "12.M22" ],
          [ "Suz", "2.Suz", "3.Suz", "6.Suz" ] ]
        gap> for entry in sublist do
        >   tblG  := CharacterTable( entry[1] );
        >   tblmG := CharacterTable( entry[2] );
        >   tblnG := CharacterTable( entry[3] );
        >   lib   := CharacterTable( entry[4] );
        >
        >   maxesG   := List( Maxes( tblG ), CharacterTable );
        >   maxesmG  := List( Maxes( tblmG ), CharacterTable );
        >   maxesnG  := List( Maxes( tblnG ), CharacterTable );
        >   maxeslib := List( Maxes( lib ), CharacterTable );
        >
        >   for i in [ 1 .. Length( maxesG ) ] do
        >     id:= Identifier( maxeslib[i] );
        >     res:= CharacterTableOfCommonCentralExtension( maxesG[i], maxesmG[i],
        >                                                   maxesnG[i], id );
        >     if not res.IsComplete then
        >       Print( "#E  not complete: ", id, "\n" );
        >     fi;
        >     if not IsSubset( Irr( maxeslib[i] ), res.irreducibles ) then
        >       trans:= TransformingPermutationsCharacterTables( maxeslib[i],
        >                                                        res.tblmnG );
        >       if not IsRecord( trans ) then
        >         Print( "#E  not transformable: ", id, "\n" );
        >       fi;
        >     fi;
        >   od;
        > od;

Since we get no output, all tables in question can be computed with the GAP functions, and coincide (up to permutations of rows and columns) with the library tables.

### 2.3 The `2B` Centralizer in 3.Fi24′ (January 2004)

As is stated in [CCN+85,p. 207], the `2B` centralizer N0 in the sporadic simple Fischer group Fi24′ has the structure 21+12+.3U4(3).22. The character table of N0 is contained in the GAP Character Table Library since the year 2000.
Our aim is to compute the character table of the preimage N of N0 in the central extension 3.Fi24′ of Fi24′; let Z1 denote the centre of 3.Fi24′.
Using the "dihedral group method" in the faithful permutation representation of degree 920 808 for 3.Fi24′, we first compute a generating set of N. This group has three orbits of the lengths 774 144, 145 152, and 1 512; the actions on the first two orbits are faithful, and the action on the orbit of length 1 512 (which consists of the fixed points of the central involution of N) has kernel exactly the central subgroup Z2, say, of order 2 in N.
Since the permutation representation on 1 512 points is so small, it is straightforward to compute the character table of N/Z2 using the implementation of Dixon's algorithm in GAP; now this table is part of the GAP Character Table Library.
Now we note that N is a central extension of N0/Z(N0) by the cyclic group Z = Z1 Z2 of order 6, and that we know the character tables of the groups N/Z1 and N/Z2. So we can apply the method described in Section 1 for computing the character table of N.
First we fetch the input data.

        gap> tblmG := CharacterTable( "F3+N2B" );;
        gap> tblG  := tblmG / ClassPositionsOfCentre( tblmG );;
        gap> tblnG := CharacterTable( "2^12.3^2.U4(3).2_2'" );;

The character tables of the library table of N0 and the character table of N/Z2 obtained from the permutation group are not compatible in the sense that the tables of the factor groups modulo the centres are not sorted compatibly, so we have to compute and store the fusion from `tblnG` to `tblG`.

        gap> f2:= tblnG / ClassPositionsOfCentre( tblnG );;
        gap> trans:= TransformingPermutationsCharacterTables( f2, tblG );;
        gap> tblnGfustblG:= OnTuples( GetFusionMap( tblnG, f2 ),
        >                             trans.columns );;
        gap> StoreFusion( tblnG, tblnGfustblG, tblG );
        gap> IsSubset( Irr( tblnG ), List( Irr( tblG ), x -> x{ tblnGfustblG } ) );
        true

Now we apply `CharacterTableOfCommonCentralExtension`.

        gap> SetInfoLevel( InfoCharacterTable, 1 );
        gap> id:= "3.2^(1+12).3U4(3).2";;
        gap> res:= CharacterTableOfCommonCentralExtension( tblG, tblmG, tblnG, id );;
        #I  3.2^(1+12).3U4(3).2: need 36 faithful irreducibles
        #I  3.2^(1+12).3U4(3).2: 16 found by tensoring
        #I  3.2^(1+12).3U4(3).2: 20 found by LLL
        gap> SetInfoLevel( InfoCharacterTable, 0 );

So we have found all missing irreducibles of N. Let us check whether the result table coincides with the table in the GAP Character Table Library.

        gap> lib:= CharacterTable( "3.F3+N2B" );;
        gap> IsRecord( TransformingPermutationsCharacterTables( res.tblmnG, lib ) );
        true

We were interested in the character table because N is a maximal subgroup of 3.Fi24′. So the class fusion into the table of this group is an interesting information. We assume that the class fusion of N0 into Fi24′ is known, and compute only those possible class fusions that are compatible with this map.

        gap> 3f3p:= CharacterTable( "3.F3+" );;
        gap> f3p:= CharacterTable( "F3+" );;
        gap> approxfus:= CompositionMaps( InverseMap( GetFusionMap( 3f3p, f3p ) ),
        >                    CompositionMaps( GetFusionMap( tblmG, f3p ),
        >                        GetFusionMap( lib, tblmG ) ) );;
        gap> poss:= PossibleClassFusions( lib, 3f3p, rec( fusionmap:= approxfus ) );;
        gap> Length( poss );
        1

It turns out that only one map has this property. (Without the condition on the compatibility, we would have got 128 possibilities, which form one orbit under table automorphisms.)

## References

[Bre04]
  Thomas Breuer, Manual for the GAP Character Table Library, Version 1.1, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 2004.
[CCN+85]
  J[ohn] H. Conway, R[obert] T. Curtis, S[imon] P. Norton, R[ichard] A. Parker, and R[obert] A. Wilson, Atlas of finite groups, Oxford University Press, 1985.
[GAP04]
  The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.4, 2004, `http://www.gap-system.org`.

File translated from TEX by TTH, version 3.55.
On 31 Mar 2004, 10:54.