The derived subgroup of the second Bieberbach group of dimension six with the quaternion point group of order eight

Crystallography is the study of the configuration and properties of a crystalline state. With the aid of mathematical approach, the crystal can be classified into different types of space groups, one of which is called the Bieberbach group. A mathematical approach has been used to solve problems regarding crystal properties and can provide a piece of information on the group structure such as the homological invariant. In computing the algebraic properties of a group, the group must first be transformed into a polycyclic presentation, such that the presentation consists of generators that describe the group. Based on the polycyclic presentation, the computation of the derived subgroup, G', is done to further explicate the homological invariants. The computation of G' is vital as it will be used to satisfy several definitions, theorems, and propositions in explicating the homological invariants of a group such as the nonabelian tensor square. The derived subgroup is written in the form of commutator. It is found that the derived subgroup for the second Bieberbach group of dimension six with the quaternion point group of order eight consists of 72 commutators in which 42 of the commutators are the identity elements, and it is further simplified to only consisting of 5 commutators.