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.