In this paper we list all possible degrees of a faithful transitive permutation representation of the group of symmetries of a regular map of types {4, 4} and {3, 6} and we give examples of graphs, called CPR-graphs, representing some of these permutation representations.
In [1], we claim that we give a complete list of the possible degrees of a faithful transitive permutation representation of the groups of the toroidal regular maps. Indeed, the list given for type {4, 4} is complete; however, recently we were surprised with the existence of exceptional degrees for the map {3, 6}. After, we struggled to find the reason why there were some degrees missing in our classification. The fact is that there is a gap in one proof, having consequences in two of our main theorems. Our goal is to fill in that gap. In what follows, let G be the group of symmetries of a toroidal regular map of type {4, 4} or {3, 6}, and suppose G is represented as a faithful transitive permutation representation group of degree n. Let T be the translation group generated by unitary independent translations of order s defining the groups of {4, 4} s or {3, 6} s for s ∈ {(s, 0), (s, s)} (see [1]). We recall the following results. Lemma 1.1 [1] [Lemma 1.1] If T is transitive, then n = s 2. An immediate consequence of Lemma 1.1 is that if n = s 2 , T is intransitive. Note that T is a normal subgroup of G that is a direct product of two cyclic groups of order s. Consider that α and β are the actions of the generators of T on a block and let K := α, β. In [1], we consider the cases where K is a cyclic group or a direct product, but these cases do not cover all possibilities. In what follows, we complete all the proofs of [1] where this has any effect.
Recently the classification of all possible faithful transitive permutation representations of the group of symmetries of a regular toroidal map was accomplished. In this paper we complete this investigation on a surface of genus 1 considering the group of a regular toroidal hypermap of type (3, 3, 3).
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