Chapter 1.3 Computer checking of the subgroup data

Symmetry relations between space groups

Second Online Edition (2011)

Part 1. Space groups and their subgroups

  1. Franz Gähler

Published Online: 15 DEC 2011

DOI: 10.1107/97809553602060000792

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Gähler, F. 2011. Computer checking of the subgroup data. International Tables for Crystallography. A1:1:1.3:25–26.

Author Information

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany

Publication History

  1. Published Online: 15 DEC 2011


Most of the data in Part 2 of this volume were checked by a computer-algebra program based upon the package Cryst, which in turn is an extension of the computer-algebra program GAP. The basic capabilities of Cryst are described. A space group is represented as a group of augmented matrices. The implementation of a membership test for such groups is explained, along with how the test can be used to determine whether two space groups are equal or whether one is a subgroup of the other. The algorithm for computing the maximal subgroups of space groups is also outlined. The tests that were performed are then described. The program parsed the LaTeX source files that were used to produce the tables of data. All the data tabulated for the low-index subgroups were checked for correctness and consistency. The low-index subgroups were recomputed from scratch and compared with the tabulated subgroups. Unfortunately, such tests could not be applied to the infinite series of maximal isomorphic subgroups of arbitrarily high index.


  • computation of maximal subgroups;
  • Cryst;
  • GAP;
  • translationengleiche subgroups;
  • klassengleiche subgroups