Categorisation of point groups

This table lists point group symmetries along with their symmetry operations, the order of the group (i.e. the number of symmetry operations) and common notations. maps to
symbol

links to a correlation table, and Irreducible representation product

links to tables of products of irreducible representations. The group produced by combination with inversion is listed under "x i". This, in the case of crystolographic point groups, is the Laue class which corresponds to the symmetry of reciprocal space. Isomorphic groups are also listed where character tables are available.

Operations Order Schönflies
Symbol International
Symbol Full
Symmetry
Symbol Correlation
Table Irred. Rep.
products x i Isomorph.
with

Cubic

E, 4C₃, 4C₃², 3C₂ 12 T 23 23 T_h

E, 8C₃, 3C₂, 3 sigma _v, i, 8S₆ 24 T_h m3 T_h

E, 6C₄, 8C₃, 3C₂, 6C₂' 24 O 432 432 O_h T_d

E, 8C₃, 3C₂, 6S₄, 6 sigma _d 24 T_d 3m 3m O_h O

E, 8C₃, 6C₂, 6C₄, 3C₂', i, 6S₄, 8S₆, 3 sigma _h, 6 sigma _d 48 O_h m3m O_h

Tetragonal

E, C₄, C₂, C₄³ 4 C₄ 4 4 C_4h S₄

E, S₄, C₂, S₄³ 4 S₄ C_4h C₄

E, C₄, C₂, C₄³, i, S₄³, sigma _h, S₄ 8 C_4h 4/m C_4h

E, 2C₄, C₂, 2C₂', 2C₂'' 8 D₄ 422 422 D_4h C_4v, D_2d

E, 2C₄, C₂, 2 sigma _v, 2 sigma _d 8 C_4v 4mm 4mm D_4h D₄, D_2d

E, 2S₄, C₂, 2C₂', 2 sigma _d 8 D_2d (V_d) 2m 2m D_4h D₄, C_4v

E, 2C₄, C₂, 2C₂', 2C₂'', i, 2S₄, sigma _h, 2 sigma _v, 2 sigma _d 16 D_4h 4/mmm D_4h

Orthorhombic

E, C₂, C₂', C₂'' 4 D₂ (V) 222 222 D_2h C_2v, C_2h

E, C₂, sigma _v, sigma _v' 4 C_2v mm2 mm2 D_2h D₂, C_2h

E, C₂, C₂', C₂'', i, sigma , sigma ', sigma '' 8 D_2h (V_h) mmm D_2h

Rhombic symmetry for defects in crystals is often divided into two types:
Type I: C₂ coincides with the [110] direction and C₂' and C₂'' with [001] and [1-10] directions respectively (or, sigma _v and sigma _v' coincide with the planes (1-10) and (001)). Also belonging to type I are centres for which C₂ coincides with [001] and sigma _v and sigma _v' with (110) and (1-10).
Type II: C₂ axis coincides with [001] and the axes C₂' and C₂'' with [100] and [010], (or, alternatively, sigma _v and sigma _v' coincide with (010) and (100)).

Monoclinic

E, C₂ 2 C₂ 2 2 C_2h C_s, C_i

E, sigma _h 2 C_s (C_1h) m m C_2h C₂, C_i

E, C₂, i, sigma _h 4 C_2h 2/m C_2h D₂, C_2v

Triclinic

E 1 C₁ 1 1 C_i

E, i 2 C_i (S₂) C_i C_s, C₂

Trigonal

E, C₃, C₃² 3 C₃ 3 3 S₆

E, C₃, C₃², i, S₆⁵, S₆ 6 S₆ (C_3i) S₆ C₆, C_3h

E, 2C₃, 3C₂ 6 D₃ 32 32 D_3d C_3v

E, 2C₃, 3 sigma _v 6 C_3v 3m 3m D_3d D₃

E, 2C₃, 3C₂, i, 2S₆, 3 sigma _d 12 D_3d m D_3d C_6v, D₆, D_3h

Hexagonal

E, C₆, C₃, C₂, C₃², C₆⁵ 6 C₆ 6 6 C_6h S₆, C_3h

E, C₃, C₃², sigma _h, S₃, S₃² 6 C_3h (S₃) C_6h S₆, C₆

E, C₆, C₃, C₂, C₃², C₆⁵, i, S₃², S₆⁵, sigma _h, S₆, S₃ 12 C_6h 6/m C_6h

E, 2C₆, 2C₃, C₂, 3C₂', 3C₂'' 12 D₆ 622 622 D_6h C_6v, D_3d, D_3h

E, 2C₆, 2C₃, C₂, 3 sigma _v, 3 sigma _d 12 C_6v 6mm 6mm D_6h D₆, D_3d, D_3h

E, 2C₃, 3C₂, sigma _h, 2S₃, 3 sigma _v 12 D_3h m2 m2 D_6h D₆, D_3d, C_6v

E, 2C₆, 2C₅, C₂, 3C₂', 3C₂'', i, 2S₃, 2S₆, sigma _h, 3 sigma _d, 3 sigma _v 24 D_6h 6/mmm D_6h

Non-Crystallographic

Operations Order Schönflies
Symbol International
Symbol Full
Symmetry
Symbol Correlation
Table Irred. Rep.
products x i Isomorph.
with

E, 2 C infinity C - - C_h

E, 2, i, 2 S infinity C_h /m - - C_h

E, 2, sigma _v C_v m - - D_h

E, 2 C infinity , sigma _v, i, 2, C₂ D_h /mm - - D_h

E, C₅, C₅², C₅³, C₅⁴ 5 C₅ 5 - S₁₀

E, S₈, C₄, S₈³, C₂, S₈⁵, C₄³, S₈⁷ 8 S₈ - - C_8h

E, 2C₅, 2C₅², 5C₂ 10 D₅ - - D_5d C_5v

E, 2C₅, 2C₅², 5 sigma _v 10 C_5v - - D_5d D₅

E, C₅, C₅², C₅³, C₅⁴, sigma _h, S₅, S₅⁷, S₅³, S₅⁹ 10 C_5h - - C_10h

E, 2S₈, 2C₄, 2S₈³, C₂, 4C₂', 4 sigma _d 16 D_4d - - D_8h

E, 2C₅, 2C₅², 5C₂, i, 2S₁₀, 2S₁₀³, 5 sigma _d 20 D_5d - - D_5d D_5h

E, 2C₅, 2C₅², 5C₂, sigma _h, 2S₅, 2S₅², 5 sigma _d 20 D_5h - - D_10h D_5d

E 2S₁₂ 2C₆ 2S₄ 2C₃ 2S₁₂⁵ C₂ 6C₂' 6 sigma _d 24 D_6d - - D_12h

E, 12C₅, 12C₅², 20C₃, 15C₂ 60 I - - I_h

E, 12C₅, 12C₅², 20C₃, 15C₂, i, 12S₁₀, 12S₁₀³, 20S₆, 15 sigma 120 I_h - - I_h

Categorisation of point groups by symmetry operations

Last modified: Thu Nov 23 12:07:15 GMT 2000 by JPG (goss@excc.ex.ac.uk)
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Operations	Order	Schönflies Symbol	International Symbol	Full Symmetry Symbol	Correlation Table	Irred. Rep. products	x i	Isomorph. with
Cubic
E, 4C₃, 4C₃², 3C₂	12	T	23	23			T_h
E, 8C₃, 3C₂, 3_v, i, 8S₆	24	T_h	m3				T_h
E, 6C₄, 8C₃, 3C₂, 6C₂'	24	O	432	432			O_h	T_d
E, 8C₃, 3C₂, 6S₄, 6_d	24	T_d	3m	3m			O_h	O
E, 8C₃, 6C₂, 6C₄, 3C₂', i, 6S₄, 8S₆, 3_h, 6_d	48	O_h	m3m				O_h
Tetragonal
E, C₄, C₂, C₄³	4	C₄	4	4			C_4h	S₄
E, S₄, C₂, S₄³	4	S₄					C_4h	C₄
E, C₄, C₂, C₄³, i, S₄³, _h, S₄	8	C_4h	4/m				C_4h
E, 2C₄, C₂, 2C₂', 2C₂''	8	D₄	422	422			D_4h	C_4v, D_2d
E, 2C₄, C₂, 2_v, 2_d	8	C_4v	4mm	4mm			D_4h	D₄, D_2d
E, 2S₄, C₂, 2C₂', 2_d	8	D_2d (V_d)	2m	2m			D_4h	D₄, C_4v
E, 2C₄, C₂, 2C₂', 2C₂'', i, 2S₄, _h, 2_v, 2_d	16	D_4h	4/mmm				D_4h
Orthorhombic
E, C₂, C₂', C₂''	4	D₂ (V)	222	222			D_2h	C_2v, C_2h
E, C₂, _v, _v'	4	C_2v	mm2	mm2			D_2h	D₂, C_2h
E, C₂, C₂', C₂'', i, , ', ''	8	D_2h (V_h)	mmm				D_2h
Rhombic symmetry for defects in crystals is often divided into two types: Type I: C₂ coincides with the [110] direction and C₂' and C₂'' with [001] and [1-10] directions respectively (or, _v and _v' coincide with the planes (1-10) and (001)). Also belonging to type I are centres for which C₂ coincides with [001] and _v and _v' with (110) and (1-10). Type II: C₂ axis coincides with [001] and the axes C₂' and C₂'' with [100] and [010], (or, alternatively, _v and _v' coincide with (010) and (100)).
Monoclinic
E, C₂	2	C₂	2	2			C_2h	C_s, C_i
E, _h	2	C_s (C_1h)	m	m			C_2h	C₂, C_i
E, C₂, i, _h	4	C_2h	2/m				C_2h	D₂, C_2v
Triclinic
E	1	C₁	1	1			C_i
E, i	2	C_i (S₂)					C_i	C_s, C₂
Trigonal
E, C₃, C₃²	3	C₃	3	3			S₆
E, C₃, C₃², i, S₆⁵, S₆	6	S₆ (C_3i)					S₆	C₆, C_3h
E, 2C₃, 3C₂	6	D₃	32	32			D_3d	C_3v
E, 2C₃, 3_v	6	C_3v	3m	3m			D_3d	D₃
E, 2C₃, 3C₂, i, 2S₆, 3_d	12	D_3d	m				D_3d	C_6v, D₆, D_3h
Hexagonal
E, C₆, C₃, C₂, C₃², C₆⁵	6	C₆	6	6			C_6h	S₆, C_3h
E, C₃, C₃², _h, S₃, S₃²	6	C_3h (S₃)					C_6h	S₆, C₆
E, C₆, C₃, C₂, C₃², C₆⁵, i, S₃², S₆⁵, _h, S₆, S₃	12	C_6h	6/m				C_6h
E, 2C₆, 2C₃, C₂, 3C₂', 3C₂''	12	D₆	622	622			D_6h	C_6v, D_3d, D_3h
E, 2C₆, 2C₃, C₂, 3_v, 3_d	12	C_6v	6mm	6mm			D_6h	D₆, D_3d, D_3h
E, 2C₃, 3C₂, _h, 2S₃, 3_v	12	D_3h	m2	m2			D_6h	D₆, D_3d, C_6v
E, 2C₆, 2C₅, C₂, 3C₂', 3C₂'', i, 2S₃, 2S₆, _h, 3_d, 3_v	24	D_6h	6/mmm				D_6h
Non-Crystallographic
Operations	Order	Schönflies Symbol	International Symbol	Full Symmetry Symbol	Correlation Table	Irred. Rep. products	x i	Isomorph. with
E, 2		C		-		-	C_h
E, 2, i, 2		C_h	/m	-		-	C_h
E, 2, _v		C_v	m	-		-	D_h
E, 2, _v, i, 2, C₂		D_h	/mm	-		-	D_h
E, C₅, C₅², C₅³, C₅⁴	5	C₅	5	-			S₁₀
E, S₈, C₄, S₈³, C₂, S₈⁵, C₄³, S₈⁷	8	S₈	-	-			C_8h
E, 2C₅, 2C₅², 5C₂	10	D₅	-	-			D_5d	C_5v
E, 2C₅, 2C₅², 5_v	10	C_5v	-	-			D_5d	D₅
E, C₅, C₅², C₅³, C₅⁴, _h, S₅, S₅⁷, S₅³, S₅⁹	10	C_5h	-	-			C_10h
E, 2S₈, 2C₄, 2S₈³, C₂, 4C₂', 4_d	16	D_4d	-	-			D_8h
E, 2C₅, 2C₅², 5C₂, i, 2S₁₀, 2S₁₀³, 5_d	20	D_5d	-	-			D_5d	D_5h
E, 2C₅, 2C₅², 5C₂, _h, 2S₅, 2S₅², 5_d	20	D_5h	-	-			D_10h	D_5d
E 2S₁₂ 2C₆ 2S₄ 2C₃ 2S₁₂⁵ C₂ 6C₂' 6_d	24	D_6d	-	-			D_12h
E, 12C₅, 12C₅², 20C₃, 15C₂	60	I	-	-			I_h
E, 12C₅, 12C₅², 20C₃, 15C₂, i, 12S₁₀, 12S₁₀³, 20S₆, 15	120	I_h	-	-			I_h

Categorisation of point groups by symmetry operations

Last modified: Thu Nov 23 12:07:15 GMT 2000 by JPG (goss@excc.ex.ac.uk) V A

Last modified: Thu Nov 23 12:07:15 GMT 2000 by JPG (goss@excc.ex.ac.uk)
V A