Irreducible Representations of $S4$

Question

so ive been trying to get all the irreducible representations of $S4$ and their dimensions, I know that there are 5 because they are 5 conjugacy classes in $S4$ and that 2 have dimension 1 because $S4/[S4,S4] \cong \mathbb{Z}_2$, but i need to find the dimension of the other tree, where i tried using the formula $|S4|=1+1+n_3^2+n_4^2+n_5^2$. My question is is there any other way to find out the dimensions of these representations without having to find the right 3 numbers that squared give 22? Is there another way to do it ? Thanks.