Abelian regular coverings of the quaternion hypermap

Abstract: A hypermap is an embedding of a connected hypergraph into an orientable closed surface. A covering between hypermaps is a homomorphism between the embedded hypergraphs which extends to an orientation-preserving covering of the supporting surfaces. A covering of a hypermap onto itself is an automorphism, and a hypermap is regular if its automorphism group acts transitively on the brins. Depending on the algebraic theory of regular hypermaps and hypermap operations, the abelian regular coverings over the quaternion hypermap are investigated. We define normalized multicyclic coverings between regular hypermaps, generalizing almost totally branched coverings studied in [K. Hu, R. Nedela, N.-E Wang, Branched cyclic regular coverings over platonic maps, European J. Combin. 36 (2014) 531--549]. It is shown that the covering transformation group of a normalized multicyclic covering is a nilpotent group with bounded class. As an application the abelian normalized bicyclic coverings over the quaternion hypemap are classified. In particular, those coverings which possess various level of external symmetry or fulfil certain smoothness conditions are explicitly determined.