Restudying Marty Isaacs' book Finite Group Theory, Chapter 5 - Transfer, I thought of the following by working through some easy examples and I am wondering if it is true.
Suppose $G$ is finite and metacyclic in the sense that $G'$ and $G/G'$ are both cyclic. Let $P \in Syl_p(G)$, where $p$ is the smallest prime divisor of $|G|$. Does it follow that $P$ is cyclic?
I tried to prove that $P/P'$ is cyclic, since then we are done ($P' \subseteq \Phi(P)$). Tried to work through $P \cap G' \unlhd G$ and one shows that in fact $G=PC_G(P \cap G')$ and $G/C_G(P \cap G')$ is a cyclic $p$-group. But my analysis does not lead to anything useful. Any thoughts?