I am looking for a finite group $G$ and an irreducible projective representation $\rho: G \to PGL(\mathbb C^n)$ such that for *any* abelian subgroup $A\subset G$, the restricted representation $\rho|\_A$ is reducible.


(EDIT: Moreover $\rho$ should be a `genuine' projective representation, in the sense that it represents a non-trivial cohomology class. In other words I am not allowing $\rho$ to be equivalent to a linear representation.)


More prosaically, I'm looking for a projective representation that is truly dependent on the non-abelian structure of the original group.


Of course it would be nice if the example were not a horribly complicated group; the 'simpler' the better. Alternatively I would of course greatly appreciate if someone could get me on the way to derive such an example myself.



(ir)reducibility of projective representation when restricted to abelian subgroup
The group ${\rm SL}(2,7)$ has a faithful complex irreducible representation of degree $4$. Viewing this as a projective representation of $G = {\rm PSL}(2,7)$ gives an example where every Abelian subgroup of $G$ acts reducibly, but $G$ acts irreducibly. This is because the only non-cyclic Abelian subgroups of $G$ are Klein $4$-groups, and the largest degree of a complex irreducible projective representation of a Klein $4$-group is $2$.


