  Let $n\geq 3$. 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
$\Gamma_{G}$ with holonomy group $G$. Using this approach we obtain an explicit
description of the holonomy representation of the Bieberbach group
$\Gamma_{G}$. As an application, when the holonomy group is cyclic of odd
order, we study the holonomy representation of $\Gamma_{G}$ and determine the
existence of Anosov diffeomorphisms and K\"ahler geometry of the flat manifold
${\cal X}_{\Gamma_{G}}$ with fundamental group the Bieberbach group
$\Gamma_{G}\leq B_n/[P_n,P_n]$.
