Isometry dimension of a compact flat manifold ($\dim {\mathrm{Isom}}_0(M)=\dim ({\mathbb{R}}^n{)}^{\mathrm{Hol}(M)}$)

On a compact flat Riemannian manifold, the integrated Bochner identity for Killing vector fields makes every Killing vector field parallel. Parallel vector fields are exactly the fixed vectors of the linear holonomy. Thus the six orientable closed flat three-manifold types have identity-component isometry dimensions three for the torus, one for each cyclic nontrivial holonomy, and zero for the Hantzsche-Wendt manifold.