Editing Crystallographic point group (section)

== Isomorphisms ==
{{See also|Crystal structure#Crystal systems}}

Many of the crystallographic point groups share the same internal structure. For example, the point groups {{overline|1}}, 2, and m contain different geometric symmetry operations, (inversion, rotation, and reflection, respectively) but all share the structure of the [[cyclic group]] C<sub>2</sub>. All [[Group isomorphism|isomorphic]] groups are of the same [[Order (group theory)|order]], but not all groups of the same order are isomorphic. The point groups which are isomorphic are shown in the following table:<ref>{{cite journal | last=Novak | first=I | title=Molecular isomorphism | journal=European Journal of Physics | publisher=IOP Publishing | volume=16 | issue=4 | date=1995-07-18 | issn=0143-0807 | doi=10.1088/0143-0807/16/4/001 | pages=151–153| bibcode=1995EJPh...16..151N | s2cid=250887121 }}</ref>

{| class="wikitable"
|-
![[Hermann–Mauguin notation|Hermann–Mauguin]]
![[Schoenflies notation|Schoenflies]]
![[Order (group theory)|Order]]
!colspan=2|[[List of small groups|Abstract group]]
|- align=center
|| 1 || ''C<sub>1</sub>'' || 1 || [[Trivial group|C<sub>1</sub>]] ||<math>G_1^1</math>
|- align=center
| {{overline|1}} ||''C<sub>i</sub> = S<sub>2</sub>'' ||2 || rowspan="3"| [[Cyclic group|C<sub>2</sub>]] ||rowspan=3|<math>G_2^1</math>
|- align=center
|| 2 ||''C<sub>2</sub>'' || 2
|- align=center
| m ||''C<sub>s</sub> = C<sub>1h</sub>'' || 2
|- align=center
|| 3 || ''C<sub>3</sub>'' || 3 || [[Cyclic group|C<sub>3</sub>]] ||<math>G_3^1</math>
|- align=center
|| 4 || ''C<sub>4</sub>'' || 4 ||rowspan="2"| [[Cyclic group|C<sub>4</sub>]] ||rowspan=2|<math>G_4^1</math>
|- align=center
| {{overline|4}} || ''S<sub>4</sub>'' || 4
|- align=center
| 2/m ||  ''C<sub>2h</sub>'' || 4 || rowspan="3" | [[Klein four-group|D<sub>2</sub>]] = C<sub>2</sub> × C<sub>2</sub> ||rowspan=3|<math>G_4^2</math>
|- align=center
||  222 ||''D<sub>2</sub> = V'' || 4
|- align=center
| mm2 ||''C<sub>2v</sub>'' ||  4
|- align=center
|{{overline|3}} ||''C<sub>3i</sub> = S<sub>6</sub>'' || 6 ||rowspan="3"|[[Cyclic group|C<sub>6</sub>]]||rowspan=3|<math>G_6^1</math>
|- align=center
||6 || ''C<sub>6</sub>'' || 6
|- align=center
| {{overline|6}} || ''C<sub>3h</sub>'' || 6
|- align=center
| 32 || ''D<sub>3</sub>'' || 6 || rowspan="2"| [[Dihedral group of order 6|D<sub>3</sub>]]||rowspan=2|<math>G_6^2</math>
|- align=center
| 3m || ''C<sub>3v</sub>'' || 6
|- align=center
| mmm ||''D<sub>2h</sub>'' = ''V<sub>h</sub>'' || 8 || D<sub>2</sub> × C<sub>2</sub>||<math>G_8^3</math>
|- align=center
|  4/m || ''C<sub>4h</sub>'' || 8 || C<sub>4</sub> × C<sub>2</sub>||<math>G_8^2</math>
|- align=center
|422 || ''D<sub>4</sub>'' || 8 || rowspan="3"| [[Dihedral group of order 8|D<sub>4</sub>]]||rowspan=3|<math>G_8^4</math>
|- align=center
| 4mm || ''C<sub>4v</sub>'' || 8
|- align=center
| {{overline|4}}2m || ''D<sub>2d</sub>'' = ''V<sub>d</sub>''|| 8
|- align=center
| 6/m || ''C<sub>6h</sub>'' || 12 || C<sub>6</sub> × C<sub>2</sub>||<math>G_{12}^2</math>
|- align=center
|| 23 || ''T'' || 12 || [[Alternating group|A<sub>4</sub>]]||<math>G_{12}^5</math>
|- align=center
| {{overline|3}}m || ''D<sub>3d</sub>'' || 12 || rowspan="4" | [[Dihedral group|D<sub>6</sub>]]||rowspan=4|<math>G_{12}^3</math>
|- align=center
| 622 || ''D<sub>6</sub>'' || 12
|- align=center
| 6mm || ''C<sub>6v</sub>'' || 12
|- align=center
| {{overline|6}}m2 || ''D<sub>3h</sub>'' || 12
|- align=center
| 4/mmm || ''D<sub>4h</sub>'' || 16 || D<sub>4</sub> × C<sub>2</sub>||<math>G_{16}^9</math>
|- align=center
| 6/mmm || ''D<sub>6h</sub>'' || 24 || D<sub>6</sub> × C<sub>2</sub>||<math>G_{24}^5</math>
|- align=center
| m{{overline|3}} || ''T<sub>h</sub>'' || 24 || A<sub>4</sub> × C<sub>2</sub>||<math>G_{24}^{10}</math>
|- align=center
| 432 || ''O''  ||  24 || rowspan="2"| [[Symmetric group|S<sub>4</sub>]]||rowspan=2|<math>G_{24}^{7}</math>
|- align=center
| {{overline|4}}3m || ''T<sub>d</sub>'' || 24
|- align=center
|  m{{overline|3}}m || ''O<sub>h</sub>'' || 48 || S<sub>4</sub> × C<sub>2</sub>||<math>G_{48}^7</math>
|}

This table makes use of [[cyclic group]]s (C<sub>1</sub>, C<sub>2</sub>, C<sub>3</sub>, C<sub>4</sub>, C<sub>6</sub>), [[dihedral group]]s (D<sub>2</sub>, D<sub>3</sub>, D<sub>4</sub>, D<sub>6</sub>), one of the [[alternating group]]s (A<sub>4</sub>), and one of the [[symmetric group]]s (S<sub>4</sub>). Here the symbol " × " indicates a [[Direct product of groups|direct product]].