[GAP Forum] Computation of Character Table

[Alexander Hulpke]

Dear Forum, Dear Shunsuke Tsuchioka,

> I want to get a character table of a group of order 57600.
> 
> Let $g$ be a Schur Cover of the symmetric group of degree 10
> and $h: g \twoheadrightarrow S_{10}$ be a natural surjection.
> A wreath product $W:=S_{5}\wr S_{2}$ is naturally identified
> with a subgroup of $S_{10}$. I want to get the character table
> of the preimage subgroup $h^{-1}(W)$ of $g$.
> As shown below, it seems intractable by a naive approach.

You're almost there. The problem is that more involved calculations tend to be inefficient for finitely presented groups. (This is a fundamental issue with such groups, not just a lack of methods.)

Thus the best way to proceed is to convert to a permutation group, via converting Pr first to a finitely presented group on its own:

hom:=IsomorphismFpGroup(Pr);
iso:=IsomorphismPermGroup(Image(hom));
P:=Image(iso);

At this point, the built-in permutation degree reduction in 4.4  does not seem to work well (the next release will be much better in this area). A naive approach is to act on the cosets of a random cyclic subgroup 

repeat u:=Subgroup(P,[Random(P)]);until 1=Size(Core(P,u));
Q:=FactorCosetAction(P,u);
R:=Image(Q);

Now
CharacterTable(R);
will work quickly.

Best,

   Alexander Hulpke




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