  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 the integers 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.
