In general, a group may be metacyclic according to the second definition and fail the first one. For example, the quaternion group has a normal cyclic subgroup of order
4, thus it satisfies definition (2). On the other hand, the commutant consists of two elements , the quotient is isomorphic to the finite
group C2×C2, and thus the group is not cyclic.

The first definition is more classical, but nowadays essentially all algebraists use the second definition, which is the one used in the remainder of this article.

A complete classification of finite metacyclic groups has been given by Hempel (2000). The metacyclic groups are all generated by two elements which are subject to three
relations depending on several numerical parameters. As a special, consider the groups
generated by two elements and such that

where .
For
and
this is the definition of the dihedral group , where is the rotation of around the center and is the reflection around the axis of symmetry of a regular
-gon.

A group is always metacyclic if its Sylow p-subgroups are all cyclic (Scott 1987, p. 356; Rose 1994,
pp. 246-247). In particular, it follows that every group of squarefree
order is metacyclic.

Alonso, J. "Groups of Square-Free Order, an Algorithm." Math. Comput.30, 632-637, 1976.Hall, M. The
Theory of Groups. Providence, RI: Chelsea, p. 146, 1976.Hempel,
C. E. "Metacyclic Groups." Commun. Algebra28, 3865-3897,
2000.Mac Lane, S. and Birkhoff, G. Algebra,
rd ed. New York: Macmillan, p. 462, 1967.Rose, J. S.
A
Course on Group Theory. New York: Dover, 1994.Scott, W. R.
Group
Theory. New York: Dover, 1987.