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 $S\_n$](http://en.wikipedia.org/wiki/Representation_theory_of_the_symmetric_group) 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. 



Constructing inequivalent irreps of finite groups
I think, the article by Vahid Dabbaghian-Abdoly, Journal of Symbolic Computation
Volume 39, Issue 6, June 2005, Pages 671-688, entitled "An algorithm for constructing representations of finite groups", [doi:10.1016/j.jsc.2005.01.002](http://dx.doi.org/10.1016/j.jsc.2005.01.002), and the references in the introduction give you what you are looking for. I think, this is the state of the art to this day. Basically, finding all the irreducible representations can be done in polynomial time.


I am not sure off the top of my head what the complexity of your suggested algorithm will be, but the bottleneck will likely be the fact that $S\_n$ always has an irreducible representation of degree $n$. Generically, decomposing a representation of degree $|G|$ is much more work than you should have to do.


