The symmetric genus of metacyclic groups

Let G be a finite group. The symmetric genus of G is the minimum genus of any Riemann surface on which G acts faithfully. Here we determine a useful lower bound for the symmetric genus of a finite group with a cyclic quotient group. The lower bound is attained for the family of K-metacyclic groups, and we determine the symmetric genus of each nonabelian subgroup of a K-metacyclic group. We also provide some examples of other groups for which the lower bound is attained. We use the standard representation of a finite group as a quotient of a noneuclidean crystallographic (NEC) group by a Fuchsian surface group.