On Symmetry of Flat Manifolds

Abstract

We give an example of a Bieberbach group $\Gamma$ for which $\Out(\Gamma)$ is a cyclic group of order $3$. We also calculate the outer automorphism group of a direct product of $n$ copies of a Bieberbach group with trivial center, for $n \in \mathbb{N}$. As a corollary we get that every symmetric group can be realized as an outer automorphism group of some Bieberbach group.