  In this paper, we prove that a normal subgroup N of an n-dimensional
crystallographic group G determines a geometric fibered orbifold structure on
the flat orbifold E^n/G, and conversely every geometric fibered orbifold
structure on E^n/G is determined by a normal subgroup N of G, which is maximal
in its commensurability class of normal subgroups of G. In particular, we prove
that E^n/G is a fiber bundle, with totally geodesic fibers, over a
b-dimensional torus, where b is the first Betti number of G.
  Let N be a normal subgroup of G which is maximal in its commensurability
class. We study the relationship between the exact sequence 1 -> N -> G -> G/N
-> 1 splitting and the corresponding fibration projection having an affine
section. If N is torsion-free, we prove that the exact sequence splits if and
only if the fibration projection has an affine section. If the generic fiber F
= Span(N)/N has an ordinary point that is fixed by every isometry of F, we
prove that the exact sequence always splits. Finally, we describe all the
geometric fibrations of the orbit spaces of all 2- and 3-dimensional
crystallographic groups building on the work of Conway and Thurston.
