Chapter 10.1 Crystallographic and noncrystallographic point groups

Space‐group symmetry

First Online Edition (2006)

Part 10. Point groups and crystal classes

  1. Th. Hahn,
  2. H. Klapper

Published Online: 1 JAN 2006

DOI: 10.1107/97809553602060000520

International Tables for Crystallography

International Tables for Crystallography

How to Cite

Hahn, T. and Klapper, H. 2006. Crystallographic and noncrystallographic point groups. International Tables for Crystallography. A:10:10.1:762–803.

Author Information

  1. Institut für Kristallographie, Rheinisch‐Westfälische Technische Hochschule, Aachen, Germany

Publication History

  1. Published Online: 1 JAN 2006



Chapter 10.1 treats the geometric and group‐theoretical aspects of both crystallographic and noncrystallographic point groups. The central part of the chapter is an extensive tabulation of the 10 two‐dimensional and the 32 three‐dimensional crystallographic point groups, containing for each group the stereographic projections of the symmetry elements and the face poles of the general crystal form and a table with the names and Miller indices {hkl} of the general and special face and point forms. Each point‐group table concludes with the three major projection symmetries. The section continues with descriptions and diagrams of the sub‐ and supergroups of the crystallographic point groups and with tabulations and diagrams of the (infinitely many) noncrystallographic point groups and their subgroups, with special emphasis on the two icosahedral point groups. All entries and terms are thoroughly described in the text.


  • crystallographic point groups;
  • noncrystallographic point groups;
  • point groups;
  • crystal systems;
  • stereographic projections;
  • crystal forms;
  • point forms;
  • Wyckoff positions;
  • eigensymmetries;
  • crystal classes;
  • subgroups;
  • supergroups;
  • icosahedral point groups