Let $G$ be a finite group which has exactly eight Sylow $7$-subgroups. Prove that there exist a normal subgroup $N$ of $G$ such that its index is divisible by $56$ but not by $49$.
Give me some hints.
Thanks in advance.
HINT: $G$ acts transitively by conjugation on the set of Sylow 7-subgroups, which gives us a homomorphism $G\to S_8$. What can bve said about its kernel?