Extended simplicial rational Nomizu's Theorem

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.