  For a torsion-free virtually polycyclic group $\Gamma$, we give a canonical
homomorphism form certain finite-dimensional cochain complex to the
$\Q$-polynomial de Rham complex of the simplicial classifying space $B\Gamma$
which induces a cohomology isomorphism. By this result, we obtain the
Sullivan's minimal model of certain differential graded algebra defined on
$B\Gamma$ and we obtain new examples of hard Lefschetz symplectic manifolds and
hard Lefschetz contact manifolds.
