Mathematics

Subgroups used during the computation of the 2-modular character table of Fi₂₃

During the computation of the 2-modular character table of the sporadic simple group Fi₂₃ Gerhard Hiß, Felix Noeske and myself used several subgroups. This page is to document which groups were used exactly. The groups are described by GAP straight line programs in terms of standard generators for Fi₂₃ in the sense of Rob Wilson's Atlas of Group Representations.

From this page one can download a GAP program that contains those straight line programs together with instructions for use.

GAP program fi23slps.g

Use for example as follows in GAP:

gap> LoadPackage("atlasrep");
gap> Read("fi23slps.g");
gap> gens := AtlasGenerators("Fi23",1).generators;;
gap> ngens := ResultOfStraightLineProgram(fi23_m7,gens);;
gap> kgens := ResultOfStraightLineProgram(m7_k,ngens);;
gap> sygens := ResultOfStraightLineProgram(m7_sy,ngens);;
gap> ugens := ResultOfStraightLineProgram(sy_u,sygens);;
gap> coreps := ResultOfStraightLineProgram(reps,gens);;

After that,

• gens are generators for Fi₂₃
• ngens are generators for the 7th maximal subgroup N of type 3¹⁺⁸.2¹⁺⁶.3¹⁺².2S₄
• kgens are generators for the condensation subgroup of type 3¹⁺⁸
• sygens are generators for the 3-Sylow subgroup of N
• ugens are generators for the helper subgroup U of order 6561
• coreps are N-N-double-coset representatives in Fi₂₃.