Is $(X_G, d_G)$ a compact manifold?

Question

Let $G$ a compact topological group act on $(X,d)$ by isometries. We define a relation $\sim$ on $(X, d)$ as follows: for $x,y\in X$: $$x\sim y \Leftrightarrow x=gy \ \text{ for some } g\in G.$$ It is clear that $\sim$ is an equivalence relation on $X$. Denote the equivalence class of $\sim$ which contains $x$ by $[x]$ and denote the set of all equivalence classes of $\sim$ by $X_G$ i.e. $X_G=\{[x]\mid x\in X \}$ . We define a metric on $X_G$ as follows: $$d_G([x], [y])=\inf _{g, g'\in G}d(gx, g'y).$$ Now I want to know if $(X,d)$ is a compact manifold, is $(X_G, d_G)$ a compact manifold too?