Let $V$ be an $n$-dimensional vector space over $\mathbb{C}$, with basis $e_1, e_2, \ldots, e_n$. Let $GL(V)$ be the group of invertible linear transformations $\phi: V \to V$. Define $\Phi: S_n \to GL(V)$ as follows. For each $g\in S_n$, let $\Phi(g)$ be the (unique) linear transformation that maps $e_i$ to $e_j$ if and only if $g(i)=j$.
I proved that $\Phi$ is a group homomorphism. But I am having trouble proving followings
Does $\Phi$ make $V$ into a simple $\mathbb{C}[S_n]$-module?
For $g\in S_n$, what is the minimal polynomial of $\Phi(g)$?
For the first part of the question, I need to show every cyclic submodule generated by a nonzero element of $V$ equals $V$. And for the second part, I only have experience in calculating minimal polynomial on concrete examples like matrix with number entries. So it would be nice if someone can illustrate how to calculate minimal polynomial in more abstract setting.