For the symmetric group $S_3$, it is classically known that \begin{equation} H^*(S_3;\mathbb{Z})\cong \mathbb{Z}[x,y]/(2x,6y,x^2-3y), \end{equation} where $|x|=1$ and $|y|=4$. Moreover, $x$ can be identified with the first (and top) Chern class of the 1-dimensional sign representation of $S_3$, and likewise $y$ can be identified with the top Chern class of the standard representation.

My Question: Is something similar true for $S_4$? Namely, is there are an explicitly computed presentation of $H^*(S_4;\mathbb{Z})$ whose generators $x,y,z$ are the respective top Chern classes of (1) the sign representation, (2) the 2-dimensional representation $S_4$ given by composing the projection $S_4\rightarrow S_3$ with the standard representation of $S_3$, and (3) the standard representation of $S_4$?