Consistency relations of an extension polycyclic free abelian lattice group by quaternion point group

Abstract

An extension of a free abelian lattice group by finite group is a torsion free crystallographic group. It expounds its symmetrical properties or known as homological invariants. One of the methods to compute its homological invariants is by determining the polycyclic presentation of the group. These polycyclic presentations are first shown to satisfy its consistency relations. Therefore, our focus is to show that this extension polycyclic free abelian lattice group by quaternion point group satisfy its consistency relations.