Is there an example of a manifold with fundamental group $\mathbb Z/3 \mathbb Z$?

Question

I feel a little confused because I was told that there exist some manifolds with fundamental group $\mathbb Z/3 \mathbb Z$, but I can’t find an example, On the other hand, since any manifold $M$ has an orientable $2$-cover, which means $\pi_1(M)$ has a subgroup of index $2$ which seems to be a contradiction with $\pi_1(M)$ can be $\mathbb Z/3 \mathbb Z$.

Answer 1

There are infinitely many $3$-manifolds whose fundamental groups isomorphic to $\mathbb Z / 3\mathbb Z$.

Indeed, for $\mathrm gcd(3,q)=1$, take any Lens space $L(3,q)$.