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: Cn, S2n, Cnh, Cnv, Dn, Dnd, Dnh
  • Polyhedral groups: T, Td, Th, O, Oh, I, Ih

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 Ci = S2 CC2 2
2 = m 1 *1 Cs = C1v = C1h ±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
Cn		 C2
C3
C4
C5
C6
Cn		 +
+
+
+
+
+
 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		 C2v
C3v
C4v
C5v
C6v
Cnv		 CD4
CD6
CD8
CD10
CD12
CD2n




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*		 C2h
C3h
C4h
C5h
C6h
Cnh		 ±C2
CC6
±C4
CC10
±C6
±Cn / CC2n




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
S2n		 CC4
±C3
CC8
±C5
CC12
CC2n / ±Cn




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
Dn		 D4
D6
D8
D10
D12
D2n		 +
+
+
+
+
+		 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		 D2h
D3h
D4h
D5h
D6h
Dnh		 ±D4
DD12
±D8
DD20
±D12
±D2n / DD4n




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
 2*2
2*3
2*4
2*5
2*6
2*n		 D2d
D3d
D4d
D5d
D6d
Dnd		 ±D4
±D6
DD16
±D10
DD24
DD4n / ±D2n




8
12
16
20
24
4n
23 3 3 332 T T + 12
m3 4 3 3*2 Th ±T 24
43m 3 3 *332 Td TO 24
432 4 3 432 O O + 24
m3m 4 3 *432 Oh ±O 48
532 5 3 532 I I + 60
53m 5 3 *532 Ih ±I 120
(*) When the Intl entries are duplicated, the first is for even n, the second for odd n.

Read more about this topic:  Point Group

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