Space-group algorithms. I. The space group and its symmetry elements

A well-defined and efficient algorithm, which enables the derivation of any crystallographic space group and full characterization of its symmetry operations and elements, is described and illustrated. The algorithm is based on a representation of crystallographic point groups in terms of cyclic groups, and on isomorphism relations between the point groups and the corresponding factor-group representations of the space groups. The characterization of the symmetry operations and the corresponding symmetry elements is also presented in an algorithmic manner, with particular emphasis on the orientation of the axes of rotation and their location vectors. The above algorithms have been implemented in a computer program, an application of which to the space group Pa is shown and some relevant programming considerations are given. The input to this general program can be fully adapted to the space-group tables in Vol. A of International Tables for Crystallography [(1983). Dordrecht: Reidel].