# Flat manifolds and irreducible representations

Let $M$ be a compact Riemannian manifold with vanishing curvature of Levi-Civita connection. Such manifolds were classified by Bieberbach; sometimes they are called Bieberbach manifolds. According to this classification, a Bieberbach manifold is a quotient of a torus by a finite group freely acting on it by isometries. Crystallographic group is a fundamental group of a Bieberbach manifold. There is an exact sequence $$0 \rightarrow {\mathbb Z}^n \rightarrow G \rightarrow L\rightarrow 0,$$ where $G$ is a crystallographic group, ${\mathbb Z}^n$ a group of translations in $G$, and $L$ the holonomy group of the Levi-Civita connection. The group $L$ is naturally embedded to the group $GL(T_xM)$ of automorphisms of the tangent space of the Bieberbach manifold.

Could $T_x M$ be irreducible as a representation of $L$? I have checked for dimension $\leq 5$, and $T_xM$ is never irreducible for dimension $\leq 5$.