In his thesis, Bieberbach solved Hilbert 18 problem and proved that any compact, flat Riemannian manifold is a quotient of a torus. I need a reference to an orbifold version of this result: any compact, flat Riemannian manifold $M$ is a quotient of a torus.

It should not be hard to prove: we should take the development map and it should give a local isometry from the orbifold universal cover of $M$ to ${\Bbb R}^n$. The corresponding monodromy action defines a homomorphism from the orbifold fundamental group of $M$ to the group of affine isometries. The rotational part of its image is finite by Margulis lemma.

However, I am pretty sure it's published somewhere, and it's always safer (and more ethical) to cite.

Thanks in advance.