  A compact manifold is called Bieberbach if it carries a flat Riemannian
metric. Bieberbach manifolds are aspherical, therefore the supremum of their
systolic ratio, over the set of Riemannian metrics, is finite by a fundamental
result of M. Gromov.
  We study the optimal systolic ratio of compact of $3$-dimensional orientable
Bieberbach manifolds which are not tori, and prove that it cannot be realized
by a flat metric. We also highlight a metric that we construct on one type of
such manifolds ($C_2$) which has interesting geometric properties : it is
extremal in its conformal class and the systole is realized by "very many"
geodesics.
