On chirality groups and regular coverings of regular oriented hypermaps

We prove that if the Walsh bipartite map M = W (ℋ) of a regular oriented hypermap ℋ is also orientably regular then both M and ℋ have the same chirality group, the covering core of M (the smallest regular map covering M) is the Walsh bipartite map of the covering core of ℋ and the closure cover of M (the greatest regular map covered by M) is the Walsh bipartite map of the closure cover of ℋ. We apply these results to the family of toroidal chiral hypermaps (3, 3, 3) _b,c = W ⁻¹{6, 3} _b,c induced by the family of toroidal bipartite maps {6, 3} _b,c .