Arc-transitive abelian regular covers of the Heawood graph

Elementary abelian regular coverings of Platonic maps

“…The character table is as shown in Table 3, with λ, μ = (1 ± √ 5)/2. In addition to the principal character χ 1 , there are algebraically conjugate characters χ 2 and χ 3 obtained from the natural representation ρ n of G as a rotation group, while the irreducible characters χ 4 and χ 5 are the non-principal summands of the permutation characters corresponding to the doubly transitive natural permutations representations of G as A 5 and as PSL 2 (5).…”

“…The prime p = 7 has order e = 4 in Z * 5 ; since ϕ(5) = 4, the Galois group C = Gal F 7 4 ∼ = C 4 has ϕ(5)/e = 1 orbit Γ on the set Π 5 ⊂ F 7 4 ⊂ F 7 of primitive 5th roots of 1. The cyclotomic polynomial Φ 5 …”

“…The one-dimensional rational summands χ 1 and χ 2 lead to both central and non-central cyclic coverings N of M. As in the case of the tetrahedron, the coefficient 2 of χ 5 means that the number of coverings is unbounded as a function of p: in fact, since ρ 5 ⊕ ρ 5 has p + 1 submodules isomorphic to ρ 5 …”

“…In recent years it has also been applied, often using the language of voltage assignments, to abelian coverings of graphs. For instance Malnič et al [19] have developed the theory and applied it to elementary abelian coverings of dipoles and of the Heawood graph, while Kwak and Oh [17] and Conder and Ma [4,5] have respectively considered elementary abelian coverings of the octahedral graph and abelian coverings of various cubic graphs. In fact, the present work could be restated in terms of coverings of the Platonic graphs, restricted to those coverings which respect the surface-embeddings.…”

On 3-arc-transitive covers of the dodecahedron graph

“…As an application, all arc-transitive abelian regular covers of several small order symmetric cubic graphs, such as the complete graph K 4 , the complete bipartite graph K 3,3 , the cube Q 3 , the Petersen graph (see [4] for more details), and the Heawood graph (see [5] for more details) were classified. An investigation of the results in [4,5] suggests that for the 2-arc-transitive graph K 4 , there exist 1-arc and 2-arc-transitive abelian regular covering graphs; for the 3-arc-transitive graph K 3,3 , there exist 1-arc, 2-arc and 3-arc-transitive abelian regular covering graphs; for the 2-arc-transitive graph Q 3 , there exist 1-arc and 2-arc-transitive abelian regular covering graphs; and for the 3-arc-transitive Petersen graph, there exist 2-arc and 3-arctransitive abelian regular covering graphs.…”