Question: Suppose that $X$ is a path-connected space with $\pi_1(X)=S_3$, which is the 3-symmetric group. I just wonder that whether $X$ has a universal cover.
Try: Based on Hatcher, $X$ has a universal cover iff $X$ is path-connected, locally path-connected, and semilocally simply-connected. However, to prove that $X$ meets the latter two conditions is not easy.
I know that $X$ with $\pi_1(X)=S_3$ can be realized by a CW complex and any CW complex meets these three conditions so that has a universal cover. But this is not the way to show that any such $X$ has a universal cover.