See metacyclic group for more on the general, modern definition which includes non-cyclic p-groups; see (Hall 1959, Th.
Every group whose Sylow subgroups are cyclic is itself metacyclic, so supersolvable.
9.4.3): The character theory of Z-groups is well understood (Çelik 1976), as they are monomial groups.
A generalization due to Hall are the A-groups, those groups with abelian Sylow subgroups.
These groups behave similarly to Z-groups, but can have arbitrarily large derived length (Hall 1940).