Regular embeddings of where is a power of 2. II: The non-metacyclic case

The aim of this paper is to complete a classification of regular orientable embeddings of complete bipartite graphs K n , n , where n = 2 e . The method involves groups G which factorise as a product G = X Y of two cyclic groups of order n such that the two cyclic factors are transposed by an involutory automorphism. In particular, we give a classification of such groups G in the case where G is not metacyclic. We prove that for each n = 2 e , e ≥ 3 , there are up to map isomorphism exactly four regular embeddings of K n , n such that the automorphism group G preserving the surface orientation and the bi-partition of vertices is a non-metacyclic group, and that there is one such embedding when n = 4 .