The orbit polytope for a finite group G acting linearly and freely on a sphere S is used to construct a cellularized fundamental domain for the action. A resolution of Z over G results from the associated G-equivariant cellularization of S. This technique is applied to the generalized binary tetrahedral group family; the homology groups, the cohomology rings and the Reidemeister torsions of the related spherical space forms are determined
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