Point group

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Groupe ponctuel (Fr); Punktgruppe (Ge); Gruppo punto (It); 点群 (Ja).


A point group is a group of symmetry operations all of which leave at least one point unmoved. A crystallographic point group is a point group that maps a point lattice onto itself: in three dimensions rotations and rotoinversions are restricted to 1, 2, 3, 4, 6 and \bar 1, \bar 2 (= m), \bar 3, \bar 4, \bar 6 respectively.


Crystallographic point groups occur:

  • in vector space, as symmetries of the external shapes of crystals (morphological symmetry), as well as symmetry of the physical properties of the crystal ("vector point group");
  • in point space, as site-symmetry groups of points in lattices or in crystal structures, and as symmetries of atomic groups and coordination polyedra ("point point group").

See also

Chapter 10 in International Tables for Crystallography, Volume A