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| In [[crystallography]], a '''crystallographic point group''' is a set of [[symmetry]] operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the [[crystal]] to the positions of features of the same kind. For a periodic crystal (as opposed to a [[quasicrystal]]), the group must also be consistent with maintenance of the three-dimensional [[translational symmetry]] that defines crystallinity. The macroscopic properties of a crystal would look exactly the same before and after any of the operations in its point group. In the classification of crystals, each point group is also known as a '''crystal class'''.
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| There are infinitely many three-dimensional point groups. However, the [[crystallographic restriction theorem|crystallographic restriction]] of the infinite families of general [[point group]]s results in there being only 32 crystallographic point groups. These 32 point groups are one-and-the same as the 32 types of morphological (external) crystalline symmetries derived in 1830 by [[Johann F. C. Hessel|Johann Friedrich Christian Hessel]] from a consideration of observed crystal forms.
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| The point group of a crystal, among other things, determines directional variation of the physical properties that arise from its structure, including [[crystal optics|optical properties]] such as whether it is [[birefringence|birefringent]], or whether it shows the [[Pockels effect]].
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| ==Notation==
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| The point groups are denoted by their component symmetries. There are a few standard notations used by crystallographers, [[mineralogist]]s, and [[physicists]].
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| For the correspondence of the two systems below, see '''[[crystal system]]'''.
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| ===Schoenflies notation===
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| {{main|Schoenflies notation}}
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| {{details|Point groups in three dimensions}}
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| 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:
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| *''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 a mirror plane parallel to the axis of rotation.
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| *''S<sub>2n</sub>'' (for ''Spiegel'', German for [[mirror]]) denotes a group that contains only a ''2n''-fold [[rotation-reflection axis]].
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| *''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.
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| *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.
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| *The letter ''O'' (for [[octahedron]]) indicates that the group has the symmetry of an octahedron (or [[cube]]), with (''O<sub>h</sub>'') or without (''O'') improper operations (those that change handedness).
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| Due to the [[crystallographic restriction theorem]], ''n'' = 1, 2, 3, 4, or 6 in 2- or 3-dimensional space.
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| {| class="wikitable" border="1"
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| |-
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| ! n
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| ! 1
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| ! 2
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| ! 3
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| ! 4
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| ! 6
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| |-
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| | ''C<sub>n</sub>''
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| | ''C<sub>1</sub>''
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| | ''C<sub>2</sub>''
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| | ''C<sub>3</sub>''
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| | ''C<sub>4</sub>''
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| | ''C<sub>6</sub>''
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| |-
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| | ''C<sub>nv</sub>''
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| | ''C<sub>1v</sub>''=''C<sub>1h</sub>''
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| | ''C<sub>2v</sub>''
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| | ''C<sub>3v</sub>''
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| | ''C<sub>4v</sub>''
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| | ''C<sub>6v</sub>''
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| |-
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| | ''C<sub>nh</sub>''
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| | ''C<sub>1h</sub>''
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| | ''C<sub>2h</sub>''
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| | ''C<sub>3h</sub>''
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| | ''C<sub>4h</sub>''
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| | ''C<sub>6h</sub>''
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| |-
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| | ''D<sub>n</sub>''
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| | ''D<sub>1</sub>''=''C<sub>2</sub>''
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| | ''D<sub>2</sub>''
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| | ''D<sub>3</sub>''
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| | ''D<sub>4</sub>''
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| | ''D<sub>6</sub>''
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| |-
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| | ''D<sub>nh</sub>''
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| | ''D<sub>1h</sub>''=''C<sub>2v</sub>''
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| | ''D<sub>2h</sub>''
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| | ''D<sub>3h</sub>''
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| | ''D<sub>4h</sub>''
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| | ''D<sub>6h</sub>''
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| |-
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| | ''D<sub>nd</sub>''
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| | ''D<sub>1d</sub>''=''C<sub>2h</sub>''
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| | ''D<sub>2d</sub>''
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| | ''D<sub>3d</sub>''
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| |style="background:silver"| ''D<sub>4d</sub>''
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| |style="background:silver"| ''D<sub>6d</sub>''
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| |-
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| | ''S<sub>2n</sub>''
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| | ''S<sub>2</sub>''
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| | ''S<sub>4</sub>''
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| | ''S<sub>6</sub>''
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| |style="background:silver"| ''S<sub>8</sub>''
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| |style="background:silver"| ''S<sub>12</sub>''
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| |}
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| ''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.
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| === Hermann–Mauguin notation===
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| {{main|Hermann–Mauguin notation}}
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| An abbreviated form of the [[Hermann–Mauguin notation]] commonly used for [[space group]]s also serves to describe crystallographic point groups. Group names are
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| {| class=wikitable
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| !Class
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| !colspan=7|Group names
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| !rowspan=6 width=480|[[File:Group-subgroup relationship (3D).png|480px]]
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| |-
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| ![[cubic crystal system|Cubic]]
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| |23|| m{{overline|3}}|| || 432|| {{overline|4}}3m|| m{{overline|3}}m ||
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| |-
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| ![[hexagonal crystal system|Hexagonal]]
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| |6|| {{overline|6}}|| <sup>6</sup>⁄<sub>m</sub>|| 622|| 6mm|| {{overline|6}}2m|| <sup>6</sup>⁄<sub>m</sub>mm
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| |-
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| ![[trigonal crystal system|Trigonal]]
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| |3|| {{overline|3}}|| || 32|| 3m|| {{overline|3}}m ||
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| |-
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| ![[tetragonal crystal system|Tetragonal]]
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| |4||{{overline|4}}|| <sup>4</sup>⁄<sub>m</sub>|| 422|| 4mm|| {{overline|4}}2m|| <sup>4</sup>⁄<sub>m</sub>mm
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| |-
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| ![[monoclinic crystal system|Monoclinic]]<BR>[[orthorhombic crystal system|Orthorhombic]]
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| |2|| || <sup>2</sup>⁄<sub>m</sub>||222|| m|| mm2|| mmm
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| |-
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| ![[triclinic crystal system|Triclinic]]
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| |1|| {{overline|1}} || || || || ||
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| |align=center|Subgroup relations of the 32 crystallographic point groups<BR>(rows represent group orders from bottom to top as: 1,2,3,4,6,8,12,16,24, and 48.)
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| |}
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| ===The correspondence between different notations===
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| {| class="wikitable"
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| |-
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| !rowspan=2|[[Crystal system]]
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| !colspan=2|[[Hermann-Mauguin notation|Hermann-Mauguin]]
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| !rowspan=2|Shubnikov<ref>http://it.iucr.org/Ab/ch12o1v0001/sec12o1o3/</ref>
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| !rowspan=2|[[Schoenflies notation|Schoenflies]]
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| !rowspan=2|[[Orbifold notation|Orbifold]]
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| !rowspan=2|[[Coxeter notation|Coxeter]]
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| !rowspan=2|Order
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| |- align=center
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| !(full)
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| !(short)
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| |- align=center
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| ! rowspan="2"|[[triclinic crystal system|Triclinic]]
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| || 1 || 1 || <math>1\ </math>||''C<sub>1</sub>'' || 11 || [ ]<sup>+</sup> || 1
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| |- align=center
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| | {{overline|1}} || {{overline|1}} || <math>\tilde{2}</math> ||''C<sub>i</sub> = S<sub>2</sub>'' || x || [2<sup>+</sup>,2<sup>+</sup>] ||2
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| |- align=center
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| !rowspan="3"| [[monoclinic crystal system|Monoclinic]]
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| || 2 || 2 || <math>2\ </math> ||''C<sub>2</sub>'' || 22 || [2]<sup>+</sup> || 2
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| |- align=center
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| | m || m || <math>m\ </math> ||''C<sub>s</sub> = C<sub>1h</sub>'' || * || [ ] || 2
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| |- align=center
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| | <math>\color{Black}\tfrac{2}{m}</math> || 2/m || <math>2:m\ </math> || ''C<sub>2h</sub>'' || 2* || [2,2<sup>+</sup>] || 4
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| |- align=center
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| !rowspan="3"| [[orthorhombic crystal system|Orthorhombic]]
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| || 222 ||222 ||<math>2:2\ </math> ||''D<sub>2</sub> = V'' || 222 || [2,2]<sup>+</sup> || 4
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| |- align=center
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| | mm2 || mm2 || <math>2 \cdot m\ </math> ||''C<sub>2v</sub>'' || *22 || [2] || 4
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| |- align=center
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| | <math>\color{Black}\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
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| |- align=center
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| ! rowspan="7"|[[tetragonal crystal system|Tetragonal]]
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| || 4 || 4 || <math>4\ </math> ||''C<sub>4</sub>'' || 44 || [4]<sup>+</sup> || 4
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| |- align=center
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| | {{overline|4}} ||{{overline|4}} || <math>\tilde{4}</math>
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| || ''S<sub>4</sub>'' || 2x || [2<sup>+</sup>,4<sup>+</sup>] ||4
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| |- align=center
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| | <math>\color{Black}\tfrac{4}{m}</math> || 4/m || <math>4:m\ </math>|| ''C<sub>4h</sub>'' || 4* || [2,4<sup>+</sup>] || 8
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| |- align=center
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| |422 || 422 || <math>4:2\ </math> || ''D<sub>4</sub>'' || 422 || [4,2]<sup>+</sup> || 8
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| |- align=center
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| |4mm || 4mm ||<math>4 \cdot m\ </math> || ''C<sub>4v</sub>'' || *44 || [4] || 8
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| |- align=center
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| | {{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
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| |- align=center
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| | <math>\color{Black}\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
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| |- align=center
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| !rowspan=5|[[trigonal crystal system|Trigonal]]
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| || 3 || 3 || <math>3\ </math> || ''C<sub>3</sub>'' || 33 || [3]<sup>+</sup> || 3
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| |- align=center
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| |{{overline|3}} || {{overline|3}} ||<math>\tilde{6}</math> || ''S<sub>6</sub> = C<sub>3i</sub>'' || 3x || [2<sup>+</sup>,6<sup>+</sup>] ||6
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| |- align=center
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| | 32 || 32 || <math>3:2\ </math> || ''D<sub>3</sub>'' || 322 || [3,2]<sup>+</sup> || 6
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| |- align=center
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| | 3m || 3m || <math>3 \cdot m\ </math> || ''C<sub>3v</sub>'' || *33 || [3] || 6
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| |- align=center
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| | {{overline|3}}<math>\color{Black}\tfrac{2}{m}</math> ||{{overline|3}}m || <math>\tilde{6}\cdot m</math> || ''D<sub>3d</sub>'' || 2*3 || [2<sup>+</sup>,6] || 12
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| |- align=center
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| ! rowspan="7"|[[hexagonal crystal system|Hexagonal]]
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| ||6 || 6 || <math>6\ </math> || ''C<sub>6</sub>'' || 66 || [6]<sup>+</sup> || 6
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| |- align=center
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| | {{overline|6}} || {{overline|6}} || <math>3:m\ </math> || ''C<sub>3h</sub>'' || 3* || [2,3<sup>+</sup>] || 6
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| |- align=center
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| | <math>\color{Black}\tfrac{6}{m}</math> || 6/m || <math>6:m\ </math> || ''C<sub>6h</sub>'' || 6* || [2,6<sup>+</sup>] || 12
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| |- align=center
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| | 622 || 622 || <math>6:2\ </math> || ''D<sub>6</sub>'' || 622 || [6,2]<sup>+</sup> || 12
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| |- align=center
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| | 6mm || 6mm ||<math>6 \cdot m\ </math> || ''C<sub>6v</sub>'' || *66 || [6] || 12
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| |- align=center
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| | {{overline|6}}m2 || {{overline|6}}m2 || <math>m \cdot 3:m\ </math> || ''D<sub>3h</sub>'' || *322 || [3,2] || 12
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| |- align=center
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| | <math>\color{Black}\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
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| |- align=center
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| !rowspan="5"|[[cubic crystal system|Cubic]]
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| || 23 || 23 || <math>3/2\ </math> || ''T'' || 332 || [3,3]<sup>+</sup> || 12
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| |- align=center
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| | <math>\color{Black}\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
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| |- align=center
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| | 432 || 432 || <math>3/4\ </math> || ''O'' || 432 || [4,3]<sup>+</sup> || 24
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| |- align=center
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| | {{overline|4}}3m || {{overline|4}}3m || <math>3/\tilde{4}</math> || ''T<sub>d</sub>'' || *332 || [3,3] || 24
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| |- align=center
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| | <math>\color{Black}\tfrac{4}{m}</math>{{overline|3}}<math>\color{Black}\tfrac{2}{m}</math> || m{{overline|3}}m || <math>\tilde{6}/4</math> || ''O<sub>h</sub>'' || *432 || [4,3] || 48
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| |}
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| ==See also==
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| * [[Molecular symmetry]]
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| * [[Point group]]
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| * [[Space group]]
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| * [[Point groups in three dimensions]]
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| * [[Crystal system]]
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| == References ==
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| <references />
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| ==External links==
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| {{commons category|Point groups}}
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| *[http://it.iucr.org/Ab/ch12o1v0001/ Point-group symbols in International Tables for Crystallography (2006). Vol. A, ch. 12.1, pp. 818-820]
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| *[http://it.iucr.org/Ab/ch10o1v0001/table10o1o2o4/ Names and symbols of the 32 crystal classes in International Tables for Crystallography (2006). Vol. A, ch. 10.1, p. 794]
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| *[http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/Solids.html Pictorial overview of the 32 groups]
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| *[http://webhost.bridgew.edu/shaefner/symmetry/pointgroup/tutorial.html#flowchart Point Groups - Flow Chart]
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| [[Category:Symmetry]]
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| [[Category:Crystallography]]
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| [[Category:Discrete groups]]
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