simple sections of $G$ are isomorphic to $A_5$ or $A_6$.

In this paper we attack the problem stated above using some more recent construction methods that were developed to study the rank conjecture. In particular, we use some ideas from our earlier papers [@unlu-yalcin1] and [@unlu-yalcin2] where we constructed free actions on products of spheres for finite groups which have representations with small fixity and for $p$-groups with small rank. Fixity of a representation $V$ of a finite group $G$ over a field $F$ is defined as the maximum of dimensions $\dim_{F} V^g $ over all elements $g \in G$. If $\rho : G \to U(n)$ is a faithful complex representation with fixity $f$, then $G$ acts freely on the space $X=U(n)/U(n-f-1)$. For small values of $f$, one can modify the space $X$ to obtain a free action on a product of $f+1$ spheres (see [@adem-davis-unlu] and [@unlu-yalcin1]). In this paper we improve this result to all values of fixity for actions on finite CW-complexes homotopy equivalent to product of spheres.

\[thm:main\] Let $G$ be a finite group. If $G$ has a faithful complex representation with fixity $f$, then $G$ acts freely on a finite complex $X$ homotopy equivalent to 