Metabelian group
In mathematics, a metabelian group is a group whose commutator subgroup is abelian. Equivalently, a group G is metabelian if and only if there is an abelian normal subgroup A such that the quotient group G/A is abelian.
Subgroups of metabelian groups are metabelian, as are images of metabelian groups over group homomorphisms.
Metabelian groups are solvable. In fact, they are precisely the solvable groups of derived length at most 2.
In contrast to this last example, the symmetric group S4 of order 24 is not metabelian, as its commutator subgroup is the non-abelian alternating group A4.
...
Wikipedia