Semisymmetric elementary abelian covers of the M\"obius-Kantor graph

Let ${\text{p}}_N\text{\hspace{0.17em}⁣}:\text{tX}\to X$ 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 ${\text{p}}_N$ is called $p$-elementary abelian. The projection ${\text{p}}_N$ is vertex-transitive (edge-transitive) if some vertex-transitive (edge-transitive) subgroup of $\Aut X$liftsalong$\p_N$, and semisymmetric if it is edge- but not vertex-transitive. The projection ${\text{p}}_N$ is minimal semisymmetric if $p_N$ cannot be written as a composition ${\text{p}}_N=\text{p}\circ {\text{p}}_M$ of two (nontrivial) regular covering projections, where ${\text{p}}_M$ 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 {\em J. Algebr. Combin.}, {\bf 20} (2004), 71–97). In this paper, all pairwise nonisomorphic minimal semisymmetric elementary abelian regular covering projections of the M"{o}bius-Kantor graph, the Generalized Petersen graph $\text{GP}(8,3)$, are constructed. No such covers exist for $p=2$. Otherwise, the number of such covering projections is equal to $(p-1)\mathrm{/}4$ and $1+(p-1)\mathrm{/}4$ in cases $p\equiv 5,9,13,17,21(\mathrm{m}\mathrm{o}\mathrm{d}24)$ and $p \equiv 1 (\mod 24)$,respectively,andto$(p+1)/4$and$1+ (p+1)/4$incases$p \equiv 3,7,11,15,23 (\mod 24)$and$p \equiv 19 (\mod 24)$, respectively. For each such covering projection the voltage rules generating the corresponding covers are displayed explicitly.