Let n ≥ 3. In this paper we show that for any finite abelian subgroup G of S n the crystallographic group B n /[P n , P n ] has Bieberbach subgroups Γ G with holonomy group G. Using this approach we obtain an explicit description of the holonomy representation of the Bieberbach group Γ G . As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of Γ G and determine the existence of Anosov diffeomorphisms and Kähler geometry of the flat manifold X Γ G with fundamental group the Bieberbach group Γ G ≤ B n /[P n , P n ].