Cohomological periodicities of crystallographic groups

We observe that an n-dimensional crystallographic group G has periodic cohomology in degrees greater than n   if it contains a torsion free finite index normal subgroup S⊴GS⊴G whose quotient G/SG/S has periodic cohomology. We then consider a different type of periodicity. Namely, we provide hypotheses on a crystallographic group G   that imply isomorphisms Hi(G/γcT,F)≅Hi(G/γc+dT,F)Hi(G/γcT,F)≅Hi(G/γc+dT,F) for FF the field of p   elements and γcTγcT a term in the relative lower central series of the translation subgroup T≤GT≤G. The latter periodicity provides a means of calculating the mod-p homology of certain infinite families of finite p-groups using a finite (machine) computation.