Suppose a group $G$ acts properly on a simplicial complex $Γ$. Let $l$ be the number of $G$-invariant vertices and $p_1, p_2, ... p_m$ be the sizes of the $G$-orbits having size greater than 1. Then $Γ$ must be a subcomplex of $Λ= Δ^{l-1}* \partial Δ^{p_1-1}*... * \partial Δ^{p_m-1}$. A result of Novik gives necessary conditions on the face numbers of Cohen-Macaulay subcomplexes of $Λ$. We show that these conditions are also sufficient, and thus provide a complete characterization of the face numbers of these comp…