Regular Oriented Hypermaps up to Five Hyperfaces

The chiral hypermaps with at most four hyperfaces were classified in [A. Breda d’Azevedo and R. Nedela,Chiral hypermaps with few hyperfaces, Math. Slovaca, 53 (2003), n.2, 107–128]. It arises from this classification that all chiral hypermaps are “canonical metacyclic”, that is, the one-step rotation about a hypervertex, or about a hyperedge, or about a hyperface, generates a normal subgroup in the orientation-preserving automorphism group. In this paper we complete the above classification by classifying the reflexible regular oriented hypermaps with three and four hyperfaces, and extend the classification to five hyperfaces. The chiral hypermaps arising in this work will be either canonical metacyclic or coverings of canonical metacyclic hypermaps. All have metacyclic monodromy groups and cyclic chirality groups.