Editing Crystallographic point group (section)

==Notation==

The point groups are named according to their component symmetries. There are several standard notations used by crystallographers, [[mineralogist]]s, and [[physicists]].

For the correspondence of the two systems below, see '''[[crystal system]]'''.

===Schoenflies notation===
{{main|Schoenflies notation}}
{{details|Point groups in three dimensions}}

In [[Arthur Moritz Schoenflies|Schoenflies]] notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following:

*''C<sub>n</sub>'' (for [[cyclic group|cyclic]]) indicates that the group has an ''n''-fold rotation axis. ''C<sub>nh</sub>'' is ''C<sub>n</sub>'' with the addition of a mirror (reflection) plane perpendicular to the [[axis of rotation]]. ''C<sub>nv</sub>'' is ''C<sub>n</sub>'' with the addition of n mirror planes parallel to the axis of rotation.
*''S<sub>2n</sub>'' (for ''Spiegel'', German for [[mirror]]) denotes a group with only a ''2n''-fold [[Improper rotation|rotation-reflection axis]].
*''D<sub>n</sub>'' (for [[dihedral group|dihedral]], or two-sided) indicates that the group has an ''n''-fold rotation axis plus ''n'' twofold axes perpendicular to that axis. ''D<sub>nh</sub>'' has, in addition, a mirror plane perpendicular to the ''n''-fold axis. ''D<sub>nd</sub>'' has, in addition to the elements of ''D<sub>n</sub>'', mirror planes parallel to the ''n''-fold axis.
*The letter ''T'' (for [[tetrahedron]]) indicates that the group has the symmetry of a tetrahedron. ''T<sub>d</sub>'' includes [[improper rotation]] operations, ''T'' excludes improper rotation operations, and ''T<sub>h</sub>'' 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<sub>h</sub>'') 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.

{| class="wikitable"
|-
! n
! 1
! 2
! 3
! 4
! 6
|-
| ''C<sub>n</sub>''
| ''C<sub>1</sub>''
| ''C<sub>2</sub>''
| ''C<sub>3</sub>''
| ''C<sub>4</sub>''
| ''C<sub>6</sub>''
|-
| ''C<sub>nv</sub>''
| ''C<sub>1v</sub>''=''C<sub>1h</sub>''
| ''C<sub>2v</sub>''
| ''C<sub>3v</sub>''
| ''C<sub>4v</sub>''
| ''C<sub>6v</sub>''
|-
| ''C<sub>nh</sub>''
| ''C<sub>1h</sub>''
| ''C<sub>2h</sub>''
| ''C<sub>3h</sub>''
| ''C<sub>4h</sub>''
| ''C<sub>6h</sub>''
|-
| ''D<sub>n</sub>''
| ''D<sub>1</sub>''=''C<sub>2</sub>''
| ''D<sub>2</sub>''
| ''D<sub>3</sub>''
| ''D<sub>4</sub>''
| ''D<sub>6</sub>''
|-
| ''D<sub>nh</sub>''
| ''D<sub>1h</sub>''=''C<sub>2v</sub>''
| ''D<sub>2h</sub>''
| ''D<sub>3h</sub>''
| ''D<sub>4h</sub>''
| ''D<sub>6h</sub>''
|-
| ''D<sub>nd</sub>''
| ''D<sub>1d</sub>''=''C<sub>2h</sub>''
| ''D<sub>2d</sub>''
| ''D<sub>3d</sub>''
|style="background:silver"| ''D<sub>4d</sub>''
|style="background:silver"| ''D<sub>6d</sub>''
|-
| ''S<sub>2n</sub>''
| ''S<sub>2</sub>''
| ''S<sub>4</sub>''
| ''S<sub>6</sub>''
|style="background:silver"|  ''S<sub>8</sub>''
|style="background:silver"|  ''S<sub>12</sub>''
|}
''D<sub>4d</sub>'' and ''D<sub>6d</sub>'' are actually forbidden because they contain [[improper rotation]]s with n=8 and 12 respectively. The 27 point groups in the table plus ''T'', ''T<sub>d</sub>'', ''T<sub>h</sub>'', ''O'' and ''O<sub>h</sub>'' constitute 32 crystallographic point groups.

=== Hermann–Mauguin notation===
{{main|Hermann–Mauguin notation}}

An abbreviated form of the [[Hermann–Mauguin notation]] commonly used for [[space group]]s also serves to describe crystallographic point groups. Group names are
{| class=wikitable
! Crystal family
! Crystal system
!colspan=7|Group names
|-
!colspan=2|[[cubic crystal system|Cubic]]
|23|| m{{overline|3}}|| || 432|| {{overline|4}}3m|| m{{overline|3}}m ||
|-
!rowspan=2|[[hexagonal crystal family|Hexagonal]]
!Hexagonal
|6|| {{overline|6}}|| <sup>6</sup>⁄<sub>m</sub>|| 622|| 6mm|| {{overline|6}}m2|| 6/mmm
|-
!Trigonal
|3|| {{overline|3}}|| || 32|| 3m|| {{overline|3}}m || 
|-
!colspan=2|[[tetragonal crystal system|Tetragonal]]
|4||{{overline|4}}|| <sup>4</sup>⁄<sub>m</sub>|| 422|| 4mm|| {{overline|4}}2m||4/mmm
|-
!colspan=2|[[orthorhombic crystal system|Orthorhombic]]
| || || ||222|| || mm2|| mmm
|-
!colspan=2|[[monoclinic crystal system|Monoclinic]]
|2|| || <sup>2</sup>⁄<sub>m</sub>|| || m|| || 
|-
!colspan=2|[[triclinic crystal system|Triclinic]]
|1|| {{overline|1}} || || || || ||
|}
{{clear}}

===The correspondence between different notations===

{| class="wikitable"
|-
!rowspan=2|Crystal family
!rowspan=2|[[Crystal system]] 
!colspan=2|[[Hermann-Mauguin notation|Hermann-Mauguin]]
!rowspan=2|Shubnikov<ref>{{cite web |url=http://it.iucr.org/Ab/ch12o1v0001/sec12o1o3/ |title=(International Tables) Abstract |access-date=2011-11-25 |url-status=dead |archive-url=https://archive.today/20130704042455/http://it.iucr.org/Ab/ch12o1v0001/sec12o1o3/ |archive-date=2013-07-04 }}</ref> 
!rowspan=2|[[Schoenflies notation|Schoenflies]]
!rowspan=2|[[Orbifold notation|Orbifold]]
!rowspan=2|[[Coxeter notation|Coxeter]] 
!rowspan=2|Order
|- align=center
!(full)
!(short)
|- align=center
! rowspan="2" colspan="2"|[[triclinic crystal system|Triclinic]]
|| 1 || 1 ||  <math>1\ </math>||''C<sub>1</sub>'' || 11 || [&nbsp;]<sup>+</sup> || 1
|- align=center
| {{overline|1}} || {{overline|1}} || <math>\tilde{2}</math> ||''C<sub>i</sub> = S<sub>2</sub>'' || × || [2<sup>+</sup>,2<sup>+</sup>] ||2
|- align=center
!rowspan="3" colspan="2"| [[monoclinic crystal system|Monoclinic]]
|| 2 || 2 || <math>2\ </math> ||''C<sub>2</sub>'' || 22 || [2]<sup>+</sup> || 2
|- align=center
| m || m || <math>m\ </math> ||''C<sub>s</sub> = C<sub>1h</sub>'' || * || [&nbsp;] || 2
|- align=center
| <math>\tfrac{2}{m}</math> || 2/m || <math>2:m\ </math> || ''C<sub>2h</sub>'' ||  2* || [2,2<sup>+</sup>] || 4
|- align=center
!rowspan="3" colspan="2"| [[orthorhombic crystal system|Orthorhombic]]
||  222 ||222 ||<math>2:2\ </math> ||''D<sub>2</sub> = V'' || 222 || [2,2]<sup>+</sup> || 4
|- align=center
| mm2 || mm2 || <math>2 \cdot m\ </math> ||''C<sub>2v</sub>'' || *22 || [2] || 4
|- align=center
| <math>\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}</math> || mmm || <math>m \cdot 2:m\ </math> ||''D<sub>2h</sub>'' = ''V<sub>h</sub>'' ||  *222 || [2,2] || 8
|- align=center
! rowspan="7" colspan="2"|[[tetragonal crystal system|Tetragonal]]
||  4 || 4 || <math>4\ </math> ||''C<sub>4</sub>'' ||  44 || [4]<sup>+</sup> || 4
|- align=center
| {{overline|4}} ||{{overline|4}} || <math>\tilde{4}</math> 
|| ''S<sub>4</sub>'' || 2× || [2<sup>+</sup>,4<sup>+</sup>] ||4
|- align=center
|  <math>\tfrac{4}{m}</math> || 4/m || <math>4:m\ </math>|| ''C<sub>4h</sub>'' ||  4* || [2,4<sup>+</sup>] || 8
|- align=center
|422 || 422 || <math>4:2\ </math> || ''D<sub>4</sub>'' || 422 || [4,2]<sup>+</sup> || 8
|- align=center
|4mm || 4mm ||<math>4 \cdot m\ </math> || ''C<sub>4v</sub>'' ||  *44 || [4] || 8
|- align=center
| {{overline|4}}2m || {{overline|4}}2m || <math>\tilde{4}\cdot m</math> || ''D<sub>2d</sub>'' = ''V<sub>d</sub>''||  2*2 || [2<sup>+</sup>,4] || 8
|- align=center
| <math>\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}</math> || 4/mmm || <math>m \cdot 4:m\ </math> || ''D<sub>4h</sub>'' || *422 || [4,2] || 16
|- align=center
!rowspan="12"|[[hexagonal crystal family|Hexagonal]]
!rowspan="5"|Trigonal
|| 3 || 3 || <math>3\ </math> || ''C<sub>3</sub>'' ||  33 || [3]<sup>+</sup> || 3
|- align=center
|{{overline|3}} || {{overline|3}} ||<math>\tilde{6}</math> || ''C<sub>3i</sub> = S<sub>6</sub>'' ||  3× || [2<sup>+</sup>,6<sup>+</sup>] ||6
|- align=center
| 32 || 32 || <math>3:2\ </math> || ''D<sub>3</sub>'' ||  322 || [3,2]<sup>+</sup> || 6
|- align=center
| 3m || 3m || <math>3 \cdot m\ </math> || ''C<sub>3v</sub>'' ||  *33 || [3] || 6
|- align=center
| {{overline|3}}<math>\tfrac{2}{m}</math> ||{{overline|3}}m || <math>\tilde{6}\cdot m</math> || ''D<sub>3d</sub>'' || 2*3 || [2<sup>+</sup>,6] || 12
|- align=center
! rowspan="7"|Hexagonal
||6 || 6 || <math>6\ </math> || ''C<sub>6</sub>'' ||  66 || [6]<sup>+</sup> || 6
|- align=center
| {{overline|6}} || {{overline|6}} || <math>3:m\ </math> || ''C<sub>3h</sub>'' ||  3* || [2,3<sup>+</sup>] || 6
|- align=center
| <math>\tfrac{6}{m}</math> || 6/m || <math>6:m\ </math> || ''C<sub>6h</sub>'' ||  6* || [2,6<sup>+</sup>] || 12
|- align=center
| 622 || 622 || <math>6:2\ </math> || ''D<sub>6</sub>'' ||  622 || [6,2]<sup>+</sup> || 12
|- align=center
| 6mm || 6mm ||<math>6 \cdot m\ </math> || ''C<sub>6v</sub>'' ||  *66 || [6] || 12
|- align=center
| {{overline|6}}m2 || {{overline|6}}m2 || <math>m \cdot 3:m\ </math> || ''D<sub>3h</sub>'' ||  *322 || [3,2] || 12
|- align=center
| <math>\tfrac{6}{m}\tfrac{2}{m}\tfrac{2}{m}</math> || 6/mmm || <math>m \cdot 6:m\ </math> || ''D<sub>6h</sub>'' ||  *622 || [6,2] || 24
|- align=center
!rowspan="5" colspan="2"|[[cubic crystal system|Cubic]]
|| 23 || 23 || <math>3/2\ </math> || ''T'' ||  332 || [3,3]<sup>+</sup> || 12
|- align=center
| <math>\tfrac{2}{m}</math>{{overline|3}} || m{{overline|3}} || <math>\tilde{6}/2</math> || ''T<sub>h</sub>'' ||  3*2 || [3<sup>+</sup>,4] || 24
|- align=center
| 432 || 432 ||  <math>3/4\ </math> || ''O''  ||  432 || [4,3]<sup>+</sup> || 24
|- align=center
| {{overline|4}}3m || {{overline|4}}3m || <math>3/\tilde{4}</math> || ''T<sub>d</sub>'' ||  *332 || [3,3] || 24
|- align=center
| <math>\tfrac{4}{m}</math>{{overline|3}}<math>\tfrac{2}{m}</math> || m{{overline|3}}m || <math>\tilde{6}/4</math> || ''O<sub>h</sub>'' ||  *432 || [4,3] || 48
|}