# Crystallographic point group

In crystallography, a **crystallographic point group** is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind. For a periodic crystal (as opposed to a quasicrystal), the group must also be consistent with maintenance of the three-dimensional translational symmetry that defines crystallinity. The macroscopic properties of a 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 is also known as a **crystal class**.

There are infinitely many three-dimensional point groups. However, the crystallographic restriction of the infinite families of general point groups results in there being only 32 crystallographic point groups. These 32 point groups are one-and-the same as the 32 types of morphological (external) crystalline symmetries derived in 1830 by Johann Friedrich Christian Hessel from a consideration of observed crystal forms.

The point group of a crystal, among other things, determines directional variation of the physical properties that arise from its structure, including optical properties such as whether it is birefringent, or whether it shows the Pockels effect.

## Notation

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**.

### Schoenflies notation

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

*C*(for cyclic) indicates that the group has an_{n}*n*-fold rotation axis.*C*is_{nh}*C*with the addition of a mirror (reflection) plane perpendicular to the axis of rotation._{n}*C*is_{nv}*C*with the addition of n mirror planes parallel to the axis of rotation._{n}*S*(for_{2n}*Spiegel*, German for mirror) denotes a group that contains only a*2n*-fold rotation-reflection axis.*D*(for dihedral, or two-sided) indicates that the group has an_{n}*n*-fold rotation axis plus*n*twofold axes perpendicular to that axis.*D*has, in addition, a mirror plane perpendicular to the_{nh}*n*-fold axis.*D*has, in addition to the elements of_{nd}*D*, mirror planes parallel to the_{n}*n*-fold axis.- The letter
*T*(for tetrahedron) indicates that the group has the symmetry of a tetrahedron.*T*includes improper rotation operations,_{d}*T*excludes improper rotation operations, and*T*is_{h}*T*with the addition of an inversion. - The letter
*O*(for octahedron) indicates that the group has the symmetry of an octahedron (or cube), with (*O*) or without (_{h}*O*) improper operations (those that change handedness).

Due to the crystallographic restriction theorem, *n* = 1, 2, 3, 4, or 6 in 2- or 3-dimensional space.

n | 1 | 2 | 3 | 4 | 6 |
---|---|---|---|---|---|

C_{n} |
C_{1} |
C_{2} |
C_{3} |
C_{4} |
C_{6} |

C_{nv} |
C=_{1v}C_{1h} |
C_{2v} |
C_{3v} |
C_{4v} |
C_{6v} |

C_{nh} |
C_{1h} |
C_{2h} |
C_{3h} |
C_{4h} |
C_{6h} |

D_{n} |
D=_{1}C_{2} |
D_{2} |
D_{3} |
D_{4} |
D_{6} |

D_{nh} |
D=_{1h}C_{2v} |
D_{2h} |
D_{3h} |
D_{4h} |
D_{6h} |

D_{nd} |
D=_{1d}C_{2h} |
D_{2d} |
D_{3d} |
D_{4d} |
D_{6d} |

S_{2n} |
S_{2} |
S_{4} |
S_{6} |
S_{8} |
S_{12} |

*D _{4d}* and

*D*are actually forbidden because they contain improper rotations with n=8 and 12 respectively. The 27 point groups in the table plus

_{6d}*T*,

*T*,

_{d}*T*,

_{h}*O*and

*O*constitute 32 crystallographic point groups.

_{h}### 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

Class | Group names | |||||||
---|---|---|---|---|---|---|---|---|

Cubic | 23 | m3 | 432 | 43m | m3m | |||

Hexagonal | 6 | 6 | ^{6}⁄_{m} | 622 | 6mm | 62m | ^{6}⁄_{m}mm | |

Trigonal | 3 | 3 | 32 | 3m | 3m | |||

Tetragonal | 4 | 4 | ^{4}⁄_{m} | 422 | 4mm | 42m | ^{4}⁄_{m}mm | |

Monoclinic Orthorhombic |
2 | ^{2}⁄_{m} | 222 | m | mm2 | mmm | ||

Triclinic | 1 | 1 | Subgroup relations of the 32 crystallographic point groups (rows represent group orders from bottom to top as: 1,2,3,4,6,8,12,16,24, and 48.) |

### The correspondence between different notations

Crystal system | Hermann-Mauguin | Shubnikov^{[1]} |
Schoenflies | Orbifold | Coxeter | Order | |
---|---|---|---|---|---|---|---|

(full) | (short) | ||||||

Triclinic | 1 | 1 | C _{1} | 11 | [ ]^{+} | 1 | |

1 | 1 | C _{i} = S_{2} | × | [2^{+},2^{+}] | 2 | ||

Monoclinic | 2 | 2 | C _{2} | 22 | [2]^{+} | 2 | |

m | m | C _{s} = C_{1h} | * | [ ] | 2 | ||

2/m | C _{2h} | 2* | [2,2^{+}] | 4 | |||

Orthorhombic | 222 | 222 | D _{2} = V | 222 | [2,2]^{+} | 4 | |

mm2 | mm2 | C _{2v} | *22 | [2] | 4 | ||

mmm | D = _{2h}V _{h} | *222 | [2,2] | 8 | |||

Tetragonal | 4 | 4 | C _{4} | 44 | [4]^{+} | 4 | |

4 | 4 | S _{4} | 2× | [2^{+},4^{+}] | 4 | ||

4/m | C _{4h} | 4* | [2,4^{+}] | 8 | |||

422 | 422 | D _{4} | 422 | [4,2]^{+} | 8 | ||

4mm | 4mm | C _{4v} | *44 | [4] | 8 | ||

42m | 42m | D = _{2d}V_{d} | 2*2 | [2^{+},4] | 8 | ||

4/mmm | D _{4h} | *422 | [4,2] | 16 | |||

Trigonal | 3 | 3 | C _{3} | 33 | [3]^{+} | 3 | |

3 | 3 | S _{6} = C_{3i} | 3× | [2^{+},6^{+}] | 6 | ||

32 | 32 | D _{3} | 322 | [3,2]^{+} | 6 | ||

3m | 3m | C _{3v} | *33 | [3] | 6 | ||

3 | 3m | D _{3d} | 2*3 | [2^{+},6] | 12 | ||

Hexagonal | 6 | 6 | C _{6} | 66 | [6]^{+} | 6 | |

6 | 6 | C _{3h} | 3* | [2,3^{+}] | 6 | ||

6/m | C _{6h} | 6* | [2,6^{+}] | 12 | |||

622 | 622 | D _{6} | 622 | [6,2]^{+} | 12 | ||

6mm | 6mm | C _{6v} | *66 | [6] | 12 | ||

6m2 | 6m2 | D _{3h} | *322 | [3,2] | 12 | ||

6/mmm | D _{6h} | *622 | [6,2] | 24 | |||

Cubic | 23 | 23 | T | 332 | [3,3]^{+} | 12 | |

3 | m3 | T _{h} | 3*2 | [3^{+},4] | 24 | ||

432 | 432 | O | 432 | [4,3]^{+} | 24 | ||

43m | 43m | T _{d} | *332 | [3,3] | 24 | ||

3 | m3m | O _{h} | *432 | [4,3] | 48 |

## See also

## References

## External links

Wikimedia Commons has media related to .Point groups |

- Point-group symbols in International Tables for Crystallography (2006). Vol. A, ch. 12.1, pp. 818-820
- Names and symbols of the 32 crystal classes in International Tables for Crystallography (2006). Vol. A, ch. 10.1, p. 794
- Pictorial overview of the 32 groups
- Point Groups - Flow Chart
- Inorganic Chemistry Group Theory Practice Problems