  This paper is the first part in a 2 part study of an elementary functorial
construction from the category of finite non-abelian groups to a category of
singular compact, oriented 2-manifolds. After a desingularization process this
construction results in a collection of compact, connected, oriented tesselated
smooth surfaces equipped with a closed-cell structure which is face and edge
transitive and which has at most 2 orbits of vertices. These tesselated
surfaces can also be viewed as abstract 3-polytopes (or as graph embeddings in
the corresponding surface) which are either equivar or dual to abstract
quasiregular polytopes. This construction generally results in a large
collection of tesselated surfaces per group, for example when the construction
is applied to \Sigma_6 it yields 4477 tesselated surfaces of 27 distinct genus
and even more varieties of tesselation cell structure. We study the
distribution of these surfaces in various groups and some interesting resulting
tesselations. In a second paper, we show that extensions of groups result in
branched coverings between the component surfaces in their decompositions. We
also exploit functoriality to obtain interesting faithful, orientation
preserving actions of subquotients of these groups and their automorphism
groups on these surfaces and in the corresponding mapping class groups.
