Find all complex irreducible representations of $S_4\times C_3$.
I know that $S_4\simeq\langle x,y,z\mid x^2=y^3=z^4=xyz=1\rangle$, $C_3\simeq\langle a\mid a^3\rangle$, hence, $$S_4\times C_3\simeq\langle a,x,y,z\mid a^3=x^2=y^3=z^4=xyz=1, ax=xa, ay=ya, az=za\rangle.$$ There are $15$ conjugacy classes in it, so, there are $15$ non-isomorphic complex irreps.
We can easily see that there are two one-dimensional reps: $(x,y,z,a)\mapsto(s,1,s,1)$ where $s\in\{\pm1\}$.
There are $13$ more irreps left.
Then... It gets harder. Can you please help me to find the other ones?