The power graph $P(Ω)$ of a group $Ω$ is a graph with the vertex set $Ω$ such that two distinct vertices form an edge if and only if one of them is an integral power of the other. In this article, we determine the power graph of the group $\mathcal{G} = \langle s,r \, : r^{2^kp} = s^2 = e,~ srs^{-1} = r^{2^{k-1}p-1}\rangle$. Further, we compute its characteristic polynomial for the adjacency, Laplacian, and signless Laplacian matrices associated with this power graph. In addition, we determine its spectrum, Laplacian spectrum, and Laplacian energy.