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In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving the edges and faces of the crystal to the positions of features of the same size and shape. For a periodic crystal (as opposed to a quasicrystal), the group must also maintain the three-dimensional translational symmetry that defines crystallinity. The geometric properties of a crystal must look exactly the same before and after applying any of the operations in its point group. In the classification of crystals, each point group defines a so-called crystal class.

There are infinitely many three-dimensional point groups. However, the crystallographic restriction on the 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 determines, among other things, the directional variation of physical properties that arise from its structure, including optical properties such as birefringency, or electro-optical features such as the Pockels effect.

Notation

The point groups are named according to their component symmetries. There are several 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_n (for cyclic) indicates that the group has an n-fold rotation axis. C_nh is C_n with the addition of a mirror (reflection) plane perpendicular to the axis of rotation. C_nv is C_n with the addition of n mirror planes parallel to the axis of rotation.


• 
  S_2n (for Spiegel, German for mirror) denotes a group with only a 2n-fold rotation-reflection axis.


• 
  D_n (for dihedral, or two-sided) indicates that the group has an n-fold rotation axis plus n twofold axes perpendicular to that axis. D_nh has, in addition, a mirror plane perpendicular to the n-fold axis. D_nd has, in addition to the elements of D_n, mirror planes parallel to the n-fold axis.


• The letter T (for tetrahedron) indicates that the group has the symmetry of a tetrahedron. T_d includes improper rotation operations, T excludes improper rotation operations, and T_h is 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_h) or without (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.

D_4d and D_6d are actually forbidden because they contain improper rotations with n=8 and 12 respectively. The 27 point groups in the table plus T, T_d, T_h, O and O_h constitute 32 crystallographic point groups.

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	6m2	6⁄mmm
Trigonal	3	3		32	3m	3m
Tetragonal	4	4	4⁄m	422	4mm	42m	4⁄mmm
Orthorhombic				222		mm2	mmm
Monoclinic	2		2⁄m		m
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	1 displaystyle 1	C1	11	[ ]+	1
	1	1	2~displaystyle tilde 2	Ci = S2	×	[2+,2+]	2
Monoclinic	2	2	2 displaystyle 2	C2	22	[2]+	2
	m	m	m displaystyle m	Cs = C1h	*	[ ]	2
	2mdisplaystyle tfrac 2m	2/m	2:m displaystyle 2:m	C2h	2*	[2,2+]	4
Orthorhombic	222	222	2:2 displaystyle 2:2	D2 = V	222	[2,2]+	4
	mm2	mm2	2⋅m displaystyle 2cdot m	C2v	*22	[2]	4
	2m2m2mdisplaystyle tfrac 2mtfrac 2mtfrac 2m	mmm	m⋅2:m displaystyle mcdot 2:m	D2h = Vh	*222	[2,2]	8
Tetragonal	4	4	4 displaystyle 4	C4	44	[4]+	4
	4	4	4~displaystyle tilde 4	S4	2×	[2+,4+]	4
	4mdisplaystyle tfrac 4m	4/m	4:m displaystyle 4:m	C4h	4*	[2,4+]	8
	422	422	4:2 displaystyle 4:2	D4	422	[4,2]+	8
	4mm	4mm	4⋅m displaystyle 4cdot m	C4v	*44	[4]	8
	42m	42m	4~⋅mdisplaystyle tilde 4cdot m	D2d = Vd	2*2	[2+,4]	8
	4m2m2mdisplaystyle tfrac 4mtfrac 2mtfrac 2m	4/mmm	m⋅4:m displaystyle mcdot 4:m	D4h	*422	[4,2]	16
Trigonal	3	3	3 displaystyle 3	C3	33	[3]+	3
	3	3	6~displaystyle tilde 6	C3i = S6	3×	[2+,6+]	6
	32	32	3:2 displaystyle 3:2	D3	322	[3,2]+	6
	3m	3m	3⋅m displaystyle 3cdot m	C3v	*33	[3]	6
	32mdisplaystyle tfrac 2m	3m	6~⋅mdisplaystyle tilde 6cdot m	D3d	2*3	[2+,6]	12
Hexagonal	6	6	6 displaystyle 6	C6	66	[6]+	6
	6	6	3:m displaystyle 3:m	C3h	3*	[2,3+]	6
	6mdisplaystyle tfrac 6m	6/m	6:m displaystyle 6:m	C6h	6*	[2,6+]	12
	622	622	6:2 displaystyle 6:2	D6	622	[6,2]+	12
	6mm	6mm	6⋅m displaystyle 6cdot m	C6v	*66	[6]	12
	6m2	6m2	m⋅3:m displaystyle mcdot 3:m	D3h	*322	[3,2]	12
	6m2m2mdisplaystyle tfrac 6mtfrac 2mtfrac 2m	6/mmm	m⋅6:m displaystyle mcdot 6:m	D6h	*622	[6,2]	24
Cubic	23	23	3/2 displaystyle 3/2	T	332	[3,3]+	12
	2mdisplaystyle tfrac 2m3	m3	6~/2displaystyle tilde 6/2	Th	3*2	[3+,4]	24
	432	432	3/4 displaystyle 3/4	O	432	[4,3]+	24
	43m	43m	3/4~displaystyle 3/tilde 4	Td	*332	[3,3]	24
	4mdisplaystyle tfrac 4m32mdisplaystyle tfrac 2m	m3m	6~/4displaystyle tilde 6/4	Oh	*432	[4,3]	48

See also

• Molecular symmetry


• Point group


• Space group


• Point groups in three dimensions


• Crystal system

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