  This paper is an exploration of the faithful transitive permutation
representations of the orientation-preserving automorphisms groups of highly
symmetric toroidal maps and hypermaps. The main theorems of this paper give a
list of all possible degrees of these specific groups. This extends prior
accomplishments of the authors, wherein their focus was confined to the study
of the automorphisms groups of toroidal regular maps and hypermaps.
  In addition the authors bring out the recently developed {\sc GAP} package
{\sc corefreesub} that can be used to find faithful transitive permutation
representations of any group. With the aid of this powerful tool, the authors
show how Schreier coset graphs of the automorphism groups of toroidal maps and
hypermaps can be easily constructed.
