This question already has an answer here:
Let $G$ a group and its order is $255$. Prove that $G$ is cyclic.
I easily demonstrated that the group has only one $17$-Sylow subgroup $P$ that is normal in $G$ and it's cyclic since it is of a prime order. Then $G/P$ is also cyclic since a group of order $15$ is cyclic. Then $G$ can be seen as $G=P(G/P)$ since the orders are coprime and then $G$ is cyclic. Is it correct?