On cohomology of crystallographic groups with cyclic holonomy of split type

We disprove a conjecture stating that the integral cohomology of any n  -dimensional crystallographic group Zn⋊ZmZn⋊Zm admits a decomposition:H⁎(Zn⋊Zm)≅⊕i+j=⁎Hi(Zm,Hj(Zn)) by providing a complete list of counterexamples up to dimension 6. This finishes the computations of the cohomology of 6-dimensional crystallographic groups arising as orbifold fundamental groups of certain Calabi–Yau toroidal orbifolds. We also find a counterexample with odd order holonomy, m=9m=9, in dimension 8.