Abstract. Space groups of a crystal expound its symmetry properties. One of the symmetry properties is the central subgroup of the nonabelian tensor square of a group. It is a normal subgroup of the group which can be ascertained by finding the abelianisation of the group and the nonabelian tensor square of the abelianisation group. In this research, our focus is to explicate the central subgroup of the nonabelian tensor square of the torsion free space groups of a crystal which are called the Bieberbach groups.

A Bieberbach group is defined to be a torsion free crystallographic group which is an extension of a free abelian lattice group by a finite point group. This paper aims to determine a mathematical representation of a Bieberbach group with quaternion point group of order eight. Such mathematical representation is the exterior square. Mathematical method from representation theory is used to find the exterior square of this group. The exterior square of this group is found to be nonabelian. Keywords: mathematical structure; exterior square; Bieberbach group; quaternion point group

A space group of a crystal describes its symmetrical properties. Many mathematical approaches have been explored to study these properties. One of the properties is on exploration of the nonabelian tensor square of the group. Determining the polycyclic presentation of the group before computing the nonabelian tensor square is the method used in this research. Therefore, this research focuses on computing the polycyclic presentations of the torsion free space group named Bieberbach group with a quaternion point group of order eight.

A Bieberbach group is a crystallographic group which is an extension of a free abelian group of finite rank by a finite extension. Meanwhile, research on homological invariants has been on interest of many authors since it is related to the study of the properties of the crystal using mathematical approach. One of the homological invariants is the exterior square. In this paper, the exterior square of a Bieberbach group of dimension four with dihedral extension is computed theoretically.

Exploration of a group's properties is vital for better understanding about the group.  Amongst other properties, the homological invariants including the nonabelian tensor square of a group can be explicated by showing that the group is polycyclic.  In this paper, the polycyclic presentations of certain crystallographic groups with quaternion point group of order eight are shown to be consistent; which implies that these groups are polycyclic.