point group

Crystallographic point group

In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a point fixed while moving each atom of the crystal to the position of an atom of the same kind. That is, an infinite crystal would look exactly the same before and after any of the operations in its point group. In the classification of crystals, each point group corresponds to a crystal class.

There are infinitely many 3D point groups; in crystallography, however, they are restricted to be compatible with the discrete translation symmetries of a crystal lattice. This crystallographic restriction of the infinite families of general point groups results in 32 crystallographic point groups.

The point group of a crystal, among other things, determines some of the crystal's optical properties, such as whether it is birefringent, or whether it shows the Pockels effect.


The point groups are denoted by their component symmetries. There are a few standard notations used by crystallographers, mineralogists, and physicists.

For the correspondence of the two systems below, see crystal system.

Schönflies notation

In Schönflies notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following:

  • The letter O (for octahedron) indicates that the group has the symmetry of an octahedron (or cube), with (Oh) or without (O) improper operations (those that change handedness).
  • The letter T (for tetrahedron) indicates that the group has the symmetry of a tetrahedron. Td includes improper operations, T excludes improper operations, and Th is T with the addition of an inversion.
  • Cn (for cyclic) indicates that the group has an n-fold rotation axis. Cnh is Cn with the addition of a mirror (reflection) plane perpendicular to the axis of rotation. Cnv is Cn with the addition of a mirror plane parallel to the axis of rotation.
  • Sn (for Spiegel, German for mirror) denotes a group that contains only an n-fold rotation-reflection axis.
  • Dn (for dihedral, or two-sided) indicates that the group has an n-fold rotation axis plus a twofold axis perpendicular to that axis. Dnh has, in addition, a mirror plane perpendicular to the n-fold axis. Dnv has, in addition to the elements of Dn, mirror planes parallel to the n-fold axis.

Due to the crystallographic restriction theorem, n = 1, 2, 3, 4, or 6.

Hermann-Mauguin notation

An abbreviated form of the Hermann-Mauguin notation commonly used for space groups also serves to describe crystallographic point groups. Group names are

  • 1, 1
  • 2, m, 2m
  • 222, mm2, mmm
  • 4,4, 4m, 422, 4mm, 42m, 4mmm
  • 3, 3, 32, 3m, 3m
  • 6, 6, 6m, 622, 6mm, 62m, 6mmm
  • 23, m3, 432, 43m, m3m

The correspondence between the three notations is:

Hermann-Mauguin Schoenflies Orbifold
1 C1 11 2 C2 22 222 D2 222 4 C4 44 3 C3 33 6 C6 66 23 T 332
1 S2 1x m C1h 1* mm2 C2v *22 4 S4 2x 3 S6 3x 6 C3h 3* m3 Th 3*2
2/m C2h 2* mmm D2h *222 4/m C4h 4* 32 D3 223 6/m C6h 6* 432 O 432
422 D4 224 3m C3v *33 622 D6 226 43m Td *332
4mm C4v *44 3m D3v 2*3 6mm C6v *66 m3m Oh *432
42m D2v 2*2 62m D3h *223
4/m mm D4h *224 6/m mm D6h *226

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