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

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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 that contains 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.

n	1	2	3	4	6
C_n	C₁	C₂	C₃	C₄	C₆
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	D₁=C₂	D₂	D₃	D₄	D₆
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	S₂	S₄	S₆	S₈	S₁₂

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	62m	⁶⁄_mmm
Trigonal	3	3		32	3m	3m
Tetragonal	4	4	⁴⁄_m	422	4mm	42m	⁴⁄_mmm
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
Crystal system	(full)	(short)	Shubnikov^[1]	Schoenflies	Orbifold	Coxeter	Order
Triclinic	1	1	$1\$	C₁	11	[ ]⁺	1
Triclinic	1	1	$\tilde{2}$	C_i = S₂	×	[2⁺,2⁺]	2
Monoclinic	2	2	$2\$	C₂	22	[2]⁺	2
	m	m	$m\$	C_s = C_1h	*	[ ]	2
	$\tfrac{2}{m}$	2/m	$2:m\$	C_2h	2*	[2,2⁺]	4
Orthorhombic	222	222	$2:2\$	D₂ = V	222	[2,2]⁺	4
	mm2	mm2	$2 \cdot m\$	C_2v	*22	[2]	4
	$\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}$	mmm	$m \cdot 2:m\$	D_2h = V_h	*222	[2,2]	8
Tetragonal	4	4	$4\$	C₄	44	[4]⁺	4
	4	4	$\tilde{4}$	S₄	2×	[2⁺,4⁺]	4
	$\tfrac{4}{m}$	4/m	$4:m\$	C_4h	4*	[2,4⁺]	8
	422	422	$4:2\$	D₄	422	[4,2]⁺	8
	4mm	4mm	$4 \cdot m\$	C_4v	*44	[4]	8
	42m	42m	$\tilde{4}\cdot m$	D_2d = V_d	2*2	[2⁺,4]	8
	$\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}$	4/mmm	$m \cdot 4:m\$	D_4h	*422	[4,2]	16
Trigonal	3	3	$3\$	C₃	33	[3]⁺	3
	3	3	$\tilde{6}$	S₆ = C_3i	3×	[2⁺,6⁺]	6
	32	32	$3:2\$	D₃	322	[3,2]⁺	6
	3m	3m	$3 \cdot m\$	C_3v	*33	[3]	6
	3 $\tfrac{2}{m}$	3m	$\tilde{6}\cdot m$	D_3d	2*3	[2⁺,6]	12
Hexagonal	6	6	$6\$	C₆	66	[6]⁺	6
	6	6	$3:m\$	C_3h	3*	[2,3⁺]	6
	$\tfrac{6}{m}$	6/m	$6:m\$	C_6h	6*	[2,6⁺]	12
	622	622	$6:2\$	D₆	622	[6,2]⁺	12
	6mm	6mm	$6 \cdot m\$	C_6v	*66	[6]	12
	6m2	6m2	$m \cdot 3:m\$	D_3h	*322	[3,2]	12
	$\tfrac{6}{m}\tfrac{2}{m}\tfrac{2}{m}$	6/mmm	$m \cdot 6:m\$	D_6h	*622	[6,2]	24
Cubic	23	23	$3/2\$	T	332	[3,3]⁺	12
	$\tfrac{2}{m}$ 3	m3	$\tilde{6}/2$	T_h	3*2	[3⁺,4]	24
	432	432	$3/4\$	O	432	[4,3]⁺	24
	43m	43m	$3/\tilde{4}$	T_d	*332	[3,3]	24
	$\tfrac{4}{m}$ 3 $\tfrac{2}{m}$	m3m	$\tilde{6}/4$	O_h	*432	[4,3]	48