Poly-(cyclic or finite) group

A group is poly-(cyclic or finite) when it has a finite subnormal series $1=G_0◃G_1◃\cdots ◃G_r=G$ in which each quotient $G_i/G_{i-1}$ is a cyclic group or a finite group. Each term need only be normal in the next term. The polycyclic group condition instead requires every factor to be cyclic.