Representation of $S_4$ Is there a general method to work out all irreducible complex representation of a group?

Describe all the the irreducible complex representation of the group $S_4$.

$S_4$ is the symmetric group on four letter.
 A: Just in case Peter Crooks' excellent answer is more general than you really want, here is a specific argument for $S_4$.
Presumably you can find the representations of $S_3$, or you wouldn't be attempting $S_4$.
Since $S_3$ is a quotient of $S_4$, you have the representations of degrees $1,1$ and $2$ of $S_3$.
Then you have the $3$-dimensional representation $\rho$ which is a constituent of the standard permutation representation, so that is easily calculated. (The permutation representation of any 2-transitive permutation group decomposes as the trivial rep plus an irreducible).
Finally, you get $\rho \otimes -1$, which equals $\rho$ on $A_4$ and $-\rho$ on $S_4 \setminus A_4$. Since $1^2+1^2+2^2+3^2+3^2=24$ that's the lot!
A: In some generality, the irreducible complex representations of $S_n$ are naturally indexed by the partitions of $n$. The irreducible representation associated to a partition $\lambda$ is called the Specht module $S^{\lambda}$. It has a basis indexed by the standard Young tableaux of shape $\lambda$. 
In principle, the Specht modules of $S_n$ can be described explicitly. I think the case of $n=4$ should be reasonably straightforward. I would suggest that you look in "Young Tableaux" by Fulton. 
