Genus of commuting conjugacy class graph of certain finite groups

For a non-abelian group , its commuting conjugacy class graph is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of and two distinct vertices and are adjacent if there exists some elements and such that . In this paper we compute the genus of for six well-known classes of non-abelian two-generated groups (viz. and ) and determine whether for these groups are planar, toroidal, double-toroidal or triple-toroidal.