Crystallographic point group

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, 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

The correspondence between different notations

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Crystal family	Crystal system	Hermann-Mauguin		Shubnikov[2]	Schoenflies	Orbifold	Coxeter	Order
		(full)	(short)
Triclinic		1	1	${\displaystyle 1\ }$	C1	11	[ ]+	1
		1	1	${\displaystyle {\tilde {2}}}$	Ci = S2	×	[2+,2+]	2
Monoclinic		2	2	${\displaystyle 2\ }$	C2	22	[2]+	2
		m	m	${\displaystyle m\ }$	Cs = C1h	*	[ ]	2
		${\displaystyle {\tfrac {2}{m}}}$	2/m	${\displaystyle 2:m\ }$	C2h	2*	[2,2+]	4
Orthorhombic		222	222	${\displaystyle 2:2\ }$	D2 = V	222	[2,2]+	4
		mm2	mm2	${\displaystyle 2\cdot m\ }$	C2v	*22	[2]	4
		${\displaystyle {\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}$	mmm	${\displaystyle m\cdot 2:m\ }$	D2h = Vh	*222	[2,2]	8
Tetragonal		4	4	${\displaystyle 4\ }$	C4	44	[4]+	4
		4	4	${\displaystyle {\tilde {4}}}$	S4	2×	[2+,4+]	4
		${\displaystyle {\tfrac {4}{m}}}$	4/m	${\displaystyle 4:m\ }$	C4h	4*	[2,4+]	8
		422	422	${\displaystyle 4:2\ }$	D4	422	[4,2]+	8
		4mm	4mm	${\displaystyle 4\cdot m\ }$	C4v	*44	[4]	8
		42m	42m	${\displaystyle {\tilde {4}}\cdot m}$	D2d = Vd	2*2	[2+,4]	8
		${\displaystyle {\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}$	4/mmm	${\displaystyle m\cdot 4:m\ }$	D4h	*422	[4,2]	16
Hexagonal	Trigonal	3	3	${\displaystyle 3\ }$	C3	33	[3]+	3
		3	3	${\displaystyle {\tilde {6}}}$	C3i = S6	3×	[2+,6+]	6
		32	32	${\displaystyle 3:2\ }$	D3	322	[3,2]+	6
		3m	3m	${\displaystyle 3\cdot m\ }$	C3v	*33	[3]	6
		3${\displaystyle {\tfrac {2}{m}}}$	3m	${\displaystyle {\tilde {6}}\cdot m}$	D3d	2*3	[2+,6]	12
	Hexagonal	6	6	${\displaystyle 6\ }$	C6	66	[6]+	6
		6	6	${\displaystyle 3:m\ }$	C3h	3*	[2,3+]	6
		${\displaystyle {\tfrac {6}{m}}}$	6/m	${\displaystyle 6:m\ }$	C6h	6*	[2,6+]	12
		622	622	${\displaystyle 6:2\ }$	D6	622	[6,2]+	12
		6mm	6mm	${\displaystyle 6\cdot m\ }$	C6v	*66	[6]	12
		6m2	6m2	${\displaystyle m\cdot 3:m\ }$	D3h	*322	[3,2]	12
		${\displaystyle {\tfrac {6}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}$	6/mmm	${\displaystyle m\cdot 6:m\ }$	D6h	*622	[6,2]	24
Cubic		23	23	${\displaystyle 3/2\ }$	T	332	[3,3]+	12
		${\displaystyle {\tfrac {2}{m}}}$3	m3	${\displaystyle {\tilde {6}}/2}$	Th	3*2	[3+,4]	24
		432	432	${\displaystyle 3/4\ }$	O	432	[4,3]+	24
		43m	43m	${\displaystyle 3/{\tilde {4}}}$	Td	*332	[3,3]	24
		${\displaystyle {\tfrac {4}{m}}}$3${\displaystyle {\tfrac {2}{m}}}$	m3m	${\displaystyle {\tilde {6}}/4}$	Oh	*432	[4,3]	48

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