Genus and crosscap of normal subgroup based power graphs of finite groups

Let H be a normal subgroup of a group G. The normal subgroup based power graph \(\Gamma _H(G)\) ${\mathrm{\Gamma }}_H(G)$ of G is the simple undirected graph with vertex set \(V(\Gamma _H(G))= (G\setminus H)\cup \{e\}\) $V({\mathrm{\Gamma }}_H(G))=(G\backslash H)\cup \{e\}$ and two distinct vertices a and b are adjacent if either \(aH = b^m H\) $aH=b^{m}H$ or \(bH=a^nH\) $bH=a^{n}H$ for some \(m,n \in \mathbb {N}\) $m,n\in \mathbb{N}$ . In this paper, we continue the study of normal subgroup based power graph and characterize all the pairs (G, H), where H is a non-trivial normal subgroup of G, such that the genus of \(\Gamma _H(G)\) ${\mathrm{\Gamma }}_H(G)$ is at most 2. Moreover, we determine all the subgroups H and the quotient groups \(\frac{G}{H}\) $\frac{G}{H}$ such that the cross-cap of \(\Gamma _H(G)\) ${\mathrm{\Gamma }}_H(G)$ is at most three.