Let $M$ be a $m$-dimensional manifold whose cohomology ring and cell structure are well-understood, such that there is a free action of the symmetric group $S\_n$ on $M$. Then we have a $n!$-sheeted covering space
$$
\pi:M\to M/S\_n.
$$
Let $S\_n$ act on the Euclidean spaces $\mathbb{R}^n$ and $\mathbb{C}^n$ by permuting the order of coordinates.
We obtain vector bundles by attaching Euclidean spaces as fibres to the covering $\pi$:
$$
\xi:\mathbb{R}^n\to M\times\_{S\_n}\mathbb{R}^n\to M/S\_n,
$$
$$
\xi^\mathbb{C}: \mathbb{C}^n\to M\times\_{S\_n}\mathbb{C}^n\to M/S\_n.
$$
**Question:** (1). Any methods to know the Stiefel-Whitney classes $$
w(\xi)?
$$
**Question:** (2). Any methods to know the Chern classes $$
c(\xi^\mathbb{C})?
$$


Note: if we take coefficients to be $\mathbb{Q}$, then all the Chern class vanish. Hence we take the coefficients to be $\mathbb{Z}/p$ for $p$ prime.



vector bundles associated to a covering space
When you have an Euclidean vector space of type:
$$\mathbb{R}^n\rightarrow M\times\_{S\_n}\mathbb{R}^n\rightarrow M/S\_n$$
Then its classifying map $\phi:M/S\_n\rightarrow BO(n)$ factors as:
$$M/S\_n\rightarrow BS\_n\stackrel{\rho\_n}{\rightarrow} BO(n)$$
where $\rho\_n$ is induced by the regular reprsentation $S\_n\rightarrow O(n)$.


Thus what you want to compute is the morphism 
$$\rho\_n^\*:H^\*(BO(n),\mathbb{F}\_2)\rightarrow H^\*(BS\_n,\mathbb{F}\_2)$$
A nice way to formulate such a result is to put all $n$ together. Then you have a map of Hopf rings 
$$\rho:H^\*(BO(\bullet),\mathbb{F}\_2)\rightarrow H^\*(BS\_{\bullet},\mathbb{F}\_2).$$
Then set $w\_{i,n}=\rho^\*\_n(w\_i)\in H^i(BS\_n,\mathbb{F}\_2)$ where $w\_i$ is the i-th Stiefel-whitney class and $w(k,l)=w\_{2^k(2^l-1),2^{k+l}}$. Then Giusti, Salvatore and Sinha proved in their paper "Mod-2 cohomology of symmetric groups as a Hopf ring" (theorem $10.7$):


"The classes $w(k,l)$ generate $H^\*(BS\_{\bullet},\mathbb{F}\_2)$ as a Hopf ring."


