There is a canonical faithful orthogonal representation of the symmetric group $S_{n+1}$, for $n\geq 1$: $$ S_{n+1}\to O(n) $$ given as follows.

*(1). I regard $O(n)$ as the isometry group of the unit sphere $S^{n-1}$ in $\mathbb{R}^{n}$.*

*(2). Let $\Delta^n$ be a regular $n$-simplex embedded in $\mathbb{R}^{n}$ such that*

```
(i). all its edges are of the same length;
(ii). all its $(n+1)$-vertices are in $S^n$;
(iii). its center is the 0.
```

*(3). I observe that any permutation on the $(n+1)$-vertices of $\Delta^n$ can be uniquely extended to an isometry of $S^n$. Hence I get an embedding of $S_{n+1}$ into $O(n)$.*

Regarding $O(n)$ as a manifold, we have a canonical action of $S_{n+1}$ on $O(n)$. Hence we have a covering map $$ O(n)\to O(n)/S_{n+1}. $$ Let the vector bundle (the action of $S_{n+1}$ on $\mathbb{R}^{n+1}$ is given by permutation of coordinates of $\mathbb{R}^{n+1}$) $$ \eta: \mathbb{R}^{n+1}\to O(n)\times _{S_{n+1}}\mathbb{R}^{n+1}\to O(n)/S_{n+1}. $$

**Question:** I want to know the Stiefel-Whitney class of $w(\eta)$. How to compute it?