Regular Maps Constructed from Linear Groups

For an integer m > 2, let L(-1)2 (Zm) be the group of 2 × 2 matrices over Zm with determinants ±1. Then for each subgroup H with {±1} ⩽ H ⩽ centre (L(-1)2(Zm)) there exists a regular map of type {3, m} whose group is L(-1)2 (Zm)/H. Ways are determined in which one such map can cover another, and in which a map decomposes as a blend of simpler maps. In all this, the prime factorization of m plays the key role. In some cases the structural results enable a simple description of the corresponding automorphism groups. Also described are realizations for the maps based on the ʹplaneʹ Z2m.