Groups with cyclic Commutator subgroup Is anything known about class of groups with cyclic commutator subgroup?
 A: It is perhaps worth remarking that (as must be well-known), (non-trivial) finite groups $G$ with $G^{\prime}$ cyclic are supersolvable- that is, all of their chief factors are cyclic of prime order. Let $G$ be such a group. Then whenever $N \lhd G$, the derived subgroup of $G/N$ is also cyclic. Now let $M$ be a minimal normal subgroup of $G.$ Then $[G,M] \lhd G$ and $[G,M] \leq M.$ If $[G,M] = 1,$ then $M \leq Z(G),$ and $M$ is cyclic by minimality. If $[G,M] = M,$ then $M \leq G^{\prime},$ so $M$ is cyclic (and of prime order). By induction, $G/M$ is supersolvable, so now $G$ is.
A: Such groups are a special class of metabelian groups. In general, a classification is not possible, but several special cases have been treated. As an example,
the $2$-generated finite $p$-groups with this property (i.e., with cyclic commutator subgroup) have been classified by R. J. Miech in the article On $p$-groups with a cyclic commutator. 
Some other examples have been studied, e.g., see Commutator subgroup of group generated by two elements of order 2, etc.
