1.2 Representations of crystallographic groups

Physical properties of crystals

Second Online Edition (2013)

Part 1. Tensorial aspects of physical properties

  1. T. Janssen

Published Online: 19 DEC 2013

DOI: 10.1107/97809553602060000901

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Janssen, T. 2013. Representations of crystallographic groups. International Tables for Crystallography. D:1:1.2:34–71.

Author Information

  1. Institute for Theoretical Physics, University of Nijmegen, 6524 ED Nijmegen, The Netherlands

Publication History

  1. Published Online: 19 DEC 2013


This chapter defines the representations of finite groups and considers successively the cases of crystallographic point groups in one, two and three dimensions and of space groups. It introduces the notions of irreducible representations. The character tables and irreducible representations are given for the 32 crystallographic point groups in three dimensions. The transformation properties of tensors are also considered. In the last section, the accompanying software Tenχar is introduced. This software can be used to determine the irreducible representations of finite point groups in three dimensions (the 32 crystallographic groups and the groups of the quasicrystalline phases) and the independent components of a tensor of any rank for each of these groups.


  • Brillouin zone;
  • Cartesian tensors;
  • Clebsch–Gordan coefficients;
  • Euclidean transformations;
  • Schur's lemma;
  • Voronoi cell;
  • character tables;
  • characters;
  • class multiplication constants;
  • co-representations;
  • double groups;
  • double space groups;
  • equivalence classes;
  • invariant tensors;
  • invariants;
  • irreducible representations;
  • lattice translation subgroups;
  • magnetic point groups;
  • magnetic space groups;
  • orthogonality relations;
  • pseudotensors;
  • reducible representations;
  • star;
  • symmetric tensors;
  • symmorphic space groups;
  • tensor product;
  • tensor product space;
  • tensor representations;
  • tensors;
  • time-reversal operator