Maps Corresponding to the Subgroups Gamma(0)(N) of the Modular Group

The normalizer Gamma(B)(N) of Gamma(0)(N) in PSL(2, R) is the triangle group (2, 3, infinity) for N = 2(alpha)3(beta) where alpha = 0, 2, 4, 6; beta = 0, 2. In this paper we examine relationship between the normalizer and the regular maps for these values. We define a family of subgroups of the normalizer and then we study maps with triangular faces using these subgroups and calculating the associated arithmetic structure.