I'm reading Dummit & Foote, Sec. 6.1.
My question is the following. If $G$ is a finite nilpotent group with a cyclic normal subgroup $N$ such that $G/N$ is also cyclic, when is $G$ abelian?
I know that dihedral groups are not abelian, and I think the question is equivalent to every Sylow subgroup being abelian.
EDIT: So, the real question is about finding all metacyclic finite p-groups that are not abelian.