Edge-transitive cyclic regular covers of the Möbius–Kantor graph

A regular cover X˜ of a connected graph XX is called elementary abelian or cyclic if its group of covering transformations is elementary abelian or cyclic, respectively. Elementary abelian regular covers of the Möbius–Kantor graph whose fiber preserving groups are edge- but not vertex-transitive were considered by Malnič et al. [A. Malnič, D. Marušič, S. Miklavič, P. Potočnik, Semisymmetric elementary abelian covers of the Möbius–Kantor graph, Discrete Math. 307 (2007) 2156–2175]. In this paper, cyclic regular covers of the Möbius–Kantor graph whose fiber-preserving groups are edge-transitive are classified. As an application, cubic edge-transitive graphs of order 16p16p for each prime pp are classified. Also, it is shown that with the exception of the Ljubljana graph on 112 vertices, all cubic edge-transitive graphs of order 16p16p are arc-transitive.