Semisymmetric elementary abelian covers of the Heawood graph

A regular covering projection ℘ : X ˜ → X of connected graphs is said to be elementary abelian if the group of covering transformations is elementary abelian, and semisymmetric if the fibre-preserving group acts edge- but not vertex-transitively. Malnič et al. [A. Malnič, D. Marušič, P. Potočnik, Elementary abelian covers of graphs, J. Algebraic Combin. 20 (2004) 71–97] determined all pairwise nonisomorphic semisymmetric elementary abelian covering projections of the Heawood graph. However, the semisymmetry of covering graphs arising from these covering projections has not been yet verified, which was also pointed out by Conder et al. in [M.D.E. Conder, A. Malnič, P. Potočnik, A census of cubic semisymmetric graphs on up to 768 vertices, J. Algebraic Combin. 23 (2006) 255–294]. In this paper, it is shown that all these covering graphs are indeed semisymmetric, namely, their full automorphism groups are edge- but not vertex-transitive.