Semisymmetric elementary abelian covers of the Möbius–Kantor graph

Let ℘N:X˜→X be a regular covering projection of connected graphs with the group of covering transformations isomorphic to N. If N is an elementary abelian p  -group, then the projection ℘N℘N is called p  -elementary abelian. The projection ℘N℘N is vertex-transitive (edge-transitive) if some vertex-transitive (edge-transitive) subgroup of Aut XX lifts along ℘N℘N, and semisymmetric if it is edge- but not vertex-transitive. The projection ℘N℘N is minimal semisymmetric if ℘N℘N cannot be written as a composition ℘N=℘∘℘M℘N=℘∘℘M of two (nontrivial) regular covering projections, where ℘M℘M is semisymmetric.Finding elementary abelian covering projections can be grasped combinatorially via a linear representation of automorphisms acting on the first homology group of the graph. The method essentially reduces to finding invariant subspaces of matrix groups over prime fields (see [A. Malnič, D. Marušič, P. Potočnik, Elementary abelian covers of graphs, J. Algebraic Combin. 20 (2004) 71–97]).In this paper, all pairwise nonisomorphic minimal semisymmetric elementary abelian regular covering projections of the Möbius–Kantor graph, the Generalized Petersen graph GP(8,3)GP(8,3), are constructed. No such covers exist for p=2p=2. Otherwise, the number of such covering projections is equal to (p-1)/4(p-1)/4 and 1+(p-1)/41+(p-1)/4 in cases p≡5,9,13,17,21(mod24) and p≡1(mod24), respectively, and to (p+1)/4(p+1)/4 and 1+(p+1)/41+(p+1)/4 in cases p≡3,7,11,15,23(mod24) and p≡19(mod24), respectively. For each such covering projection the voltage rules generating the corresponding covers are displayed explicitly.