Regular Cyclic Coverings of the Platonic Maps

The Möbius-Kantor map {4 + 4, 3} [CMo, §8.8, 8.9] is a regular orientable map of type {8, 3} and genus 2. It is a 2-sheeted covering of the cube {4, 3}, branched over the centers of its six faces, each of which lifts to an octagonal face. Its (orientation-preserving) automorphism group is isomorphic to GL2(3), a double covering of the automorphism group PGL2(3) ∼= S4 of the cube. The aim of this note is to describe all the regular maps and hypermaps which can be obtained in a similar manner as cyclic branched coverings of the Platonic maps M, with the branching at the face-centers, vertices, or midpoints of edges. The method used is to consider the action of AutM on certain homology modules; in a companion paper [SJ] we use cohomological techniques to give explicit constructions of these coverings in terms of voltage assignments.