I have posted this question on mathstack echange but did not get any answer. It mam trying my luck here.

The only simple finite groups admitting an irreducible character of degree 3 are $\mathfrak{A}_5$ and $PSL(2,7)$. That seems to be a result coming from Blichfeldt's work on $GL(3,\mathbb{C})$, which I cannot find. Is there a proof available somewhere ?