The Exterior Square of a Bieberbach Group with Quaternion Point Group of Order Eight

Abstract

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