Given a finite group $G$, let $n$ be the smallest integer s.t. $G \subset S_n$ à la Cayley. I guess that if I want to construct the complex irreps (not just the character table) of $G$ then I could take the irreps of $S_n$ and restrict them to $G$. It seems plausible that after decomposition this might yield all the irreps of $G$, but I'm not entirely sure of this. In any event it seems inefficient, even if it works.
So my question is: what is the (is there a?) general technique for constructing all the inequivalent complex irreps of a finite group?
Maybe this is better suited to the underflow site, but since it's come up in actual work (albeit of the documentary sort) I'm posting here.