It is well-known that Bieberbach proved that any closed flat Riemannian manifold is a quotient of a torus. Let $(M,g)$ be a flat Riemannian $3$-manifold, we know that $(M,g)\cong T^3/\Gamma$, where $\Gamma$ is a finite group.

Q

• Suppose that $M$ is orientable and $b_1(M)=0$ or $H_1(M,\mathbb Z)=0$, can we have a classification for group $\Gamma$ acting on $T^3=\mathbb R^3/\mathbb Z^3$?

• On the other hand: On Wiki(https://en.wikipedia.org/wiki/Flat_manifold), it writes that complete list of the 6 orientable and 4 non-orientable compact examples is related to Seifert fiber space. I think that it means that closed oriented flat(Riemannian) Seifert fiber spaces are classified, could anyone give a hint?

I think above questions were already solved, any reference is welcome!