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 $|x|=2$ 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. 


**Question 1:** Is something similar true for the integral cohomology of $S\_4$? Namely, is there 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 $S\_4$? 


**Question 2:** Is there an analogous situation for $H^\*(S\_4; \mathbb{Z}\_2)$ in terms of Stiefel-Whitney classes of the corresponding **real** representations?



Characteristic classes of symmetric group $S_4$
For Q2, my collaborators and I show that all mod-two cohomology of symmetric groups is generated by Stiefel-Whitney classes of standard representations, if you allow both cup product and transfer (induction) product. It is better, however, to take other Hopf ring generators, in which case all cohomology of symmetric groups is a free divided powers Hopf ring on classes in $H^{2^k - 1}(S\_{2^k})$. See <https://arxiv.org/abs/0909.3292>, in particular Section 10 for discussion of Stiefel-Whitney classes.


For Q1, the integral groups are known, but not ring structure. (I happen to be working out the Bocksteins now, at the prime two.) I haven't considered the characteristic class question.


