irreducible characters of a group extension

Question

Let $G$ be a group and $M$ is a minimal normal subgroup of $G$. Suppose that $M$ is an elementary abelian $2$-group and $M=C_G(M)$, so that $G/M$ acts on $M$. The action is faithful and assume that is also irreducible. Suppose that furthermore $G/M$ is a non abelian simple group (for my purposes, $G/M$ can be $A_5$ or $PSL_2(8)$, by the way).

I want to know some information about the irreducible complex characters of $G$, for example the degree of real characters.

My approach is check all the irreducible representations of $G/M$ over $GF(2)$ (that are known), build the semidirect product of $G/M$ by $M$ on GAP and print the character table. Unfortunately, this may not be the case since the group $M$ may not have a complement in $G$. It would be possible to calculate the second cohomology group $H^2(G/M,M)$ with GAP, check that is trivial (if it is true) and conclude that $M$ has a complement in $G$ (the extension splits, namely). There is a package in GAP, cohomolo, maintained by D. Holt, that does this kind of calculation, but I'm struggling to load it (I'm having some technical problems) and I wandering if there a more theoretical approach.

Answer 1

Well, if you really only interested in $G/M = A_5$ or ${\rm PSL}(2,8)$, then I can tell you what you want to know.

$A_5$ has just two nontrivial irreducible modules over ${\rm GF}(2)$, both of dimension $4$, and they both have trivial $2$-cohomology.

${\rm PSL}(2,8)$ has four such modules, of dimensions $6,8$ and $12$. Of these, only the one of dimension $6$ (which is the natural module) has a nonsplit extension, and that one is unique.

You can access all of these groups except for the split extension $2^{12}:L_2(8)$ from the perfect groups database in GAP.

For example, the nonsplit extension $2^6\!\cdot\! L_2(8)$ can be accessed by:

> gap> G := PerfectGroup( IsPermGroup,32256,2);
  L2(8) N 2^6

and then you can just compute its character table directly.