The co-prime order graph associated with a finite group

Let $G$ be a finite group. The co-prime order graph of $G$ is the graph whose vertex set is $G$, and two distinct vertices $x,y$ are adjacent if gcd$(o(x),o(y))$ is either $1$ or a prime, where $o(x)$ and $o(y)$ are the orders of $x$ and $y$, respectively. In this paper, we characterize all finite groups whose co-prime order graphs are complete and classify all finite groups whose co-prime order graphs are planar. Also, we compute the vertex-connectivity of the co-prime order graph of a cyclic group, a dihedral group and a generalized quaternion group, which answers a question by Banerjee (2019). Finally, we prove that, for a fixed positive integer $k$, there are finitely many finite groups whose co-prime order graphs have (non)orientable genus $k$. As applications, we classify all finite groups whose co-prime order graphs have (non)orientable genus one and two.