A map is considered to be semi-equivelar if all of the face-cycles at its vertices are of the same type. In this article, we show that a surface with Euler characteristic of \(-2\) admits 39 distinct types of semi-equivelar maps. Moreover, we classify (up to isomorphism) a specific category of semi-equivelar maps on this surface. This approach can be extended to classify the other types of maps. For every orientable surface of genus \(g> 2\) , there is a regular covering of an orientable surface with genus 2 by the cyclic group of covering transformations. This result was algebraically demonstrated by Harvey (Q J Math 17:86–97, 1966). In this article, we provide a combinatorial explanation of a similar result for a class of semi-equivelar maps using the concept of a connected sum. For semi-equivelar maps on surfaces with Euler characteristic \(-2\) , we demonstrate the existence of m-th covering maps on higher genus surfaces. We also show that the symmetry groups of the m-th covering maps are either \(\mathbb {D}_m\) (dihedral group of order 2m) or \(\mathbb {Z}_m\) (cyclic group of order m).
