  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 $(3,3,3)$
that is a subgroup of index $2$ of the group of symmetries of a toroidal map of
type $\{6,3\}$.
