Let $P$ be a 3-Sylow group of the simple group $G$ of order 360, which is of order 9. And let $N_G(P)$ be the normalizer of $P$. Then $\frac{N_G(P)}{P}$ acts on $P$. (by conjugation) I want to know why the action is faithful. (Or $P$ is self-centralizing; $P=C_G(P)$.)
I know that there are totally ten of 3-Sylow subgroups in $G$ and $G$ can be embedded in the alternating group $A_{10}$. So I tried to deal with $A_{10}$ by figuring out the centralizer of a subgroup of order 9 (For example, the subgroup of 9-cycles), but this seems not to be easy. I want some good idea to solve this problem.