Let $G$ be a group of order $385$. Prove that $Z(G)$, the center of $G$, contains an element of order $7$.
I used the Sylow theorem and realized that there are one Sylow $11$-subgroup, which is normal in $G$, and also one Sylow $7$-subgroup, which is normal as well.
I tried to solve it with the commutator of $G$ and concluded that $G'$ is trivial. Does it make sense? Anyway I will be glad for a clue.