On a finite nilpotent group with a cyclic normal subgroup 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.
Thanks.

Rod, you are right when you say this can be brought back to every Sylow group being abelian. Since $G$ is nilpotent you can reduce to $G$ being a $p$-group. However, a counterexample is easily found, take the quaternion group $G$ of order 8, generated by $i$ and $j$ as usual. Let $N$ be the subgroup $<i>$ of index 2. $N$ satisfies your conditions but $G$ is not abelian.
