Computation of five- and six-dimensional Bieberbach groups

This paper deals with the computation and classification of 5-and 6-dimensi onal torsion-free crystallographic groups, known as Bieberbach groups. We d escribe the basis of an algorithm that decides torsion-freeness of a crysta llographic group as well as the triviality of its centre. The computations were done using the computer package CARAT, which handles enumeration, cons truction, recognition and comparison problems for crystallographic groups u p to dimension 6. The complete list of isomorphism types of Bieberbach groups up to dimension 6 can be found online.