Point Group - Three Dimensions

Three Dimensions

Point groups in three dimensions, sometimes called molecular point groups, after their wide use in studying the symmetries of small molecules.

They come in 7 infinite families of axial or prismatic groups, and 7 additional polyhedral or Platonic groups. In Schönflies notation,*

Axial groups: C_n, S_2n, C_nh, C_nv, D_n, D_nd, D_nh
Polyhedral groups: T, T_d, T_h, O, O_h, I, I_h

Applying the crystallographic restriction theorem to these groups yields 32 Crystallographic point groups.

Intl*	Geo	Orbifold	Schönflies	Conway	Coxeter	Order
1	1	1	C₁	C₁	+	1
1	22	×1	C_i = S₂	CC₂		2
2 = m	1	*1	C_s = C_1v = C_1h	±C₁ = CD₂		2
2 3 4 5 6 n	2 3 4 5 6 n	22 33 44 55 66 nn	C₂ C₃ C₄ C₅ C₆ C_n	C₂ C₃ C₄ C₅ C₆ C_n	+ + + + + +	2 3 4 5 6 n
2mm 3m 4mm 5m 6mm nmm nm	2 3 4 5 6 n	22 33 44 55 66 nn	C_2v C_3v C_4v C_5v C_6v C_nv	CD₄ CD₆ CD₈ CD₁₀ CD₁₂ CD_2n		4 6 8 10 12 2n
2/m 3/m 4/m 5/m 6/m n/m	2 2 3 2 4 2 5 2 6 2 n 2	2* 3* 4* 5* 6* n*	C_2h C_3h C_4h C_5h C_6h C_nh	±C₂ CC₆ ±C₄ CC₁₀ ±C₆ ±C_n / CC_2n		4 6 8 10 12 2n
4 3 8 5 12 2n n	4 2 6 2 8 2 10 2 12 2 2n 2	2× 3× 4× 5× 6× n×	S₄ S₆ S₈ S₁₀ S₁₂ S_2n	CC₄ ±C₃ CC₈ ±C₅ CC₁₂ CC_2n / ±C_n		4 6 8 10 12 2n

Intl	Geo	Orbifold	Schönflies	Conway	Coxeter	Order
222 32 422 52 622 n22 n2	2 2 3 2 4 2 5 2 6 2 n 2	222 223 224 225 226 22n	D₂ D₃ D₄ D₅ D₆ D_n	D₄ D₆ D₈ D₁₀ D₁₂ D_2n	+ + + + + +	4 6 8 10 12 2n
mmm 6m2 4/mmm 10m2 6/mmm n/mmm 2nm2	2 2 3 2 4 2 5 2 6 2 n 2	222 223 224 225 226 22n	D_2h D_3h D_4h D_5h D_6h D_nh	±D₄ DD₁₂ ±D₈ DD₂₀ ±D₁₂ ±D_2n / DD_4n		8 12 16 20 24 4n
42m 3m 82m 5m 122m 2n2m nm	4 2 6 2 8 2 10 2 12 2 n 2	22 23 24 25 26 2n	D_2d D_3d D_4d D_5d D_6d D_nd	±D₄ ±D₆ DD₁₆ ±D₁₀ DD₂₄ DD_4n / ±D_2n		8 12 16 20 24 4n
23	3 3	332	T	T	+	12
m3	4 3	3*2	T_h	±T		24
43m	3 3	*332	T_d	TO		24
432	4 3	432	O	O	+	24
m3m	4 3	*432	O_h	±O		48
532	5 3	532	I	I	+	60
53m	5 3	*532	I_h	±I		120

(*) When the Intl entries are duplicated, the first is for even n, the second for odd n.

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