The strong symmetric genus and generalized symmetric groups G(n, 3)

Abstract

The generalized symmetric groups are defined to be the groups G(n, m) = ℤm ≀ Σm where n, m ∈ ℤ+. The strong symmetric genus of a finite group G is the smallest genus of a closed orientable topological surface on which G acts faithfully as a group of orientation-preserving automorphisms. The present paper extends work on the strong symmetric genus by Conder, who studied the symmetric groups, which are the groups G(n, 1), and the author, who studied the hyperoctahedral groups, which are the groups G(n, 2). We determine the strong symmetric genus of the groups G(n, 3).