On the standard $n$-simplex $\Delta[n]$ in $\mathbb{R}^{n+1}$ there is the usual action of the symmetric group $S_n$, by permuting the vertices. Any element $g\in S_n$ then acts on $\Delta[n]$ linearly, by a permutation matrix.
By definition of group action, a group element $g$ acts on the set $\Delta[n]$ by a bijection. Yet, without further assumptions, it is not clear that the action should be linear.
Question: Are there simple examples of a finite group action on $\Delta[n]$, which is not linear? Or linear, but not by permutations?