Does every perfect finite group have a fixed point free representation?

Question

Fixed point free representations of finite group are important for the spherical space form problem and also show up in other contexts for example Perfect semi direct products

A representation $ \pi $ is a called "fixed point free" if the only $ \pi(g) $ that has $ 1 $ as an eigenvalue is $ \pi(1) $. So a fixed point free representation must be faithful.

Given the character table of a finite group $ G $, what is the best way to find all it fixed point free irreps?

It is easy to check if an irrep is faithful, but beyond that I'm not sure what to do.

It would be especially nice if there was any easy way to implement this, say in GAP

I would like to learn relevant facts so that I can answer the question:

Does every perfect finite group have a fixed point free representation?

For example, the two degree $ 2 $ irreps of $ SL(2,5) $ are fixed point free. And I think the standard $ n-1 $ dim irrep of $ A_n $ is fixed point free for all $ n\geq 3 $.

Answer 1

To test in GAP, one can use the function EigenvaluesChar -- it returns a list and you want the entry at position element order as this corresponds to 1. So for example, given a group G, you could do

c:=CharacterTable(G);o:=OrdersClassRepresentatives(c);

and then

Filtered(Irr(c),x->IsZero(List([2..Length(x)],y->EigenvaluesChar(x,y)[o[y]])));

gives you the characters in which classes apart of the identity (starting at 2) have zero occurrences of Eigenvalue $1$.

With this we find that PerfectGroup(960,1); is a counterexample to the conjecture.