  Let $\phi:\Z/p\to GL_{n}(\Z)$ denote an integral representation of the cyclic
group of prime order $p$. This induces a $\Z/p$-action on the torus
$X=\R^{n}/\Z^{n}$. The goal of this paper is to explicitly compute the
cohomology groups $H^{*}(X/\Z/p;\Z)$ for any such representation. As a
consequence we obtain an explicit calculation of the integral cohomology of the
classifying space associated to the family of finite subgroups for any
crystallographic group $\Gamma =\Z^n\rtimes\Z/p$ with prime holonomy.
