In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. Equivalently, a metacyclic group is a group having a cyclic normal subgroup , such that the quotient is also cyclic.
Metacyclic groups are metabelian and supersolvable. In particular, they are solvable.
A group is metacyclic if it has a normal subgroup such that and are both cyclic. [1]