Metabelian group (reviewed)

A finite group $G$ is metabelian if its commutator subgroup is abelian. Equivalently, $G$ is an extension of an abelian group by an abelian group. Metabelian groups are solvable, and they include all abelian, dihedral, quaternion, metacyclic, extra​special groups and groups of $p$-rank one.

Every metacylic group is metabelian, but the converse does not hold, as shown by these examples.