A classification of the finite 2-generator cyclic-by-abelian groups of prime-power order

We classify the finite 2-generator cyclic-by-abelian groups of prime-power order. We associate to each such group G a list inv(G) of numerical group invariants which determines the isomorphism type of G. Then we describe the set formed by all the possible values of inv(G). This allows us to develop practical algorithms to construct all finite non-Abelian 2-generator cyclic-by-abelian groups of a given prime-power order, to compute the invariants of such a group, and to decide whether two such groups are isomorphic.