Given a field $k$ with characteristic $p$, let $G$ be a transitive permutation group on $4p$ points. Let $P$ be a Sylow $p$-subgroup of $G$ and $Q\leq P$ is a $p$-subgroup of $P$ of index $p$. Now denote $H:=N\_G(Q)/Q$. Could anyone provide me with an counterexample suth that the dimension of the projective cover $P\_k$ of the trivial $kH$-module $k$ does not divide $p(p-1)$?



Dimension of projective cover of trivial $kG$-module
Here is a Magma calculation that shows that the group ${\rm PSL}(2,11)$ is a counterexample to your question



```
> G := PSL(2,11);
> I := AbsolutelyIrreducibleModules(G,GF(3));
> I;
[
    GModule of dimension 1 over GF(3),
    GModule of dimension 5 over GF(3),
    GModule of dimension 5 over GF(3),
    GModule of dimension 10 over GF(3),
    GModule of dimension 12 over GF(3^2),
    GModule of dimension 12 over GF(3^2)
]
//So GF(9) is a splitting field
> P := ProjectiveCover(TrivialModule(G,GF(9)));
> Dimension(P);
12
> CompositionFactors(P);
[
    GModule of dimension 1 over GF(3^2),
    GModule of dimension 10 over GF(3^2),
    GModule of dimension 1 over GF(3^2)
]

```

