Crystallographic point group

<h2 id="notation">Notation</h2>
The point groups are named according to their component symmetries. There are several standard notations used by crystallographers, <a href="/facts/Mineralogist/MeIK9lx9">mineralogists</a>, and <a href="/facts/Physicists/cQcgvSIm">physicists</a>.
For the correspondence of the two systems below, see <a href="/facts/Crystal_system/dMQLk1q4">crystal system</a>.

<h3>Schoenflies notation</h3>
Main article: <a href="/facts/Schoenflies_notation/c4y0GtEt">Schoenflies notation</a>
Further information: <a href="/facts/Point_groups_in_three_dimensions/Uc4MRIyl">Point groups in three dimensions</a>
In <a href="/facts/Arthur_Moritz_Schoenflies/pRP7SGPY">Schoenflies</a> notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following:

<ul><li>Cn (for <a href="/facts/Cyclic_group/JL5zfFuL">cyclic</a>) indicates that the group has an n-fold rotation axis. Cnh is Cn with the addition of a mirror (reflection) plane perpendicular to the <a href="/facts/Axis_of_rotation/PGcg3IY5">axis of rotation</a>. Cnv is Cn with the addition of n mirror planes parallel to the axis of rotation.</li>
<li>S2n (for Spiegel, German for <a href="/facts/Mirror/wMu4u6BB">mirror</a>) denotes a group with only a 2n-fold <a href="/facts/Improper_rotation/kcLdCD9r">rotation-reflection axis</a>.</li>
<li>Dn (for <a href="/facts/Dihedral_group/1PUlRh4M">dihedral</a>, or two-sided) indicates that the group has an n-fold rotation axis plus n twofold axes perpendicular to that axis. Dnh has, in addition, a mirror plane perpendicular to the n-fold axis. Dnd has, in addition to the elements of Dn, mirror planes parallel to the n-fold axis.</li>
<li>The letter T (for <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedron</a>) indicates that the group has the symmetry of a tetrahedron. Td includes <a href="/facts/Improper_rotation/kcLdCD9r">improper rotation</a> operations, T excludes improper rotation operations, and Th is T with the addition of an inversion.</li>
<li>The letter O (for <a href="/facts/Octahedron/G38q92xW">octahedron</a>) indicates that the group has the symmetry of an octahedron, with (Oh) or without (O) improper operations (those that change handedness).</li></ul>
Due to the <a href="/facts/Crystallographic_restriction_theorem/tojdCRI8">crystallographic restriction theorem</a>, n = 1, 2, 3, 4, or 6 in 2- or 3-dimensional space.

<table><tbody><tr><th>n</th><th>1</th><th>2</th><th>3</th><th>4</th><th>6</th></tr><tr><td>Cn</td><td>C1</td><td>C2</td><td>C3</td><td>C4</td><td>C6</td></tr><tr><td>Cnv</td><td>C1v=C1h</td><td>C2v</td><td>C3v</td><td>C4v</td><td>C6v</td></tr><tr><td>Cnh</td><td>C1h</td><td>C2h</td><td>C3h</td><td>C4h</td><td>C6h</td></tr><tr><td>Dn</td><td>D1=C2</td><td>D2</td><td>D3</td><td>D4</td><td>D6</td></tr><tr><td>Dnh</td><td>D1h=C2v</td><td>D2h</td><td>D3h</td><td>D4h</td><td>D6h</td></tr><tr><td>Dnd</td><td>D1d=C2h</td><td>D2d</td><td>D3d</td><td>D4d</td><td>D6d</td></tr><tr><td>S2n</td><td>S2</td><td>S4</td><td>S6</td><td>S8</td><td>S12</td></tr></tbody></table>
D4d and D6d are actually forbidden because they contain <a href="/facts/Improper_rotation/kcLdCD9r">improper rotations</a> with n=8 and 12 respectively. The 27 point groups in the table plus T, Td, Th, O and Oh constitute 32 crystallographic point groups.

<h3>Hermann–Mauguin notation</h3>
Main article: <a href="/facts/Hermann%25E2%2580%2593Mauguin_notation/NGHDFXKC">Hermann–Mauguin notation</a>
An abbreviated form of the <a href="/facts/Hermann%25E2%2580%2593Mauguin_notation/NGHDFXKC">Hermann–Mauguin notation</a> commonly used for <a href="/facts/Space_group/2XQx0J7M">space groups</a> also serves to describe crystallographic point groups. Group names are

<table><tbody><tr><th>Crystal family</th><th>Crystal system</th><th colspan="7">Group names</th></tr><tr><th colspan="2"><a href="/facts/Cubic_crystal_system/92Co7BKM">Cubic</a></th><td>23</td><td>m3</td><td></td><td>432</td><td>43m</td><td>m3m</td><td></td></tr><tr><th rowspan="2"><a href="/facts/Hexagonal_crystal_family/sSSok8wo">Hexagonal</a></th><th>Hexagonal</th><td>6</td><td>6</td><td>6⁄m</td><td>622</td><td>6mm</td><td>6m2</td><td>6/mmm</td></tr><tr><th>Trigonal</th><td>3</td><td>3</td><td></td><td>32</td><td>3m</td><td>3m</td><td></td></tr><tr><th colspan="2"><a href="/facts/Tetragonal_crystal_system/12Aop255">Tetragonal</a></th><td>4</td><td>4</td><td>4⁄m</td><td>422</td><td>4mm</td><td>42m</td><td>4/mmm</td></tr><tr><th colspan="2"><a href="/facts/Orthorhombic_crystal_system/3zo4F8P0">Orthorhombic</a></th><td></td><td></td><td></td><td>222</td><td></td><td>mm2</td><td>mmm</td></tr><tr><th colspan="2"><a href="/facts/Monoclinic_crystal_system/5PCwXvzX">Monoclinic</a></th><td>2</td><td></td><td>2⁄m</td><td></td><td>m</td><td></td><td></td></tr><tr><th colspan="2"><a href="/facts/Triclinic_crystal_system/Y6yEdqfN">Triclinic</a></th><td>1</td><td>1</td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

<h3>The correspondence between different notations</h3>
<table><tbody><tr><th rowspan="2">Crystal family</th><th rowspan="2"><a href="/facts/Crystal_system/dMQLk1q4">Crystal system</a></th><th colspan="2"><a href="/facts/Hermann-Mauguin_notation/NGHDFXKC">Hermann-Mauguin</a></th><th rowspan="2">Shubnikov<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></th><th rowspan="2"><a href="/facts/Schoenflies_notation/c4y0GtEt">Schoenflies</a></th><th rowspan="2"><a href="/facts/Orbifold_notation/HCbBoa2J">Orbifold</a></th><th rowspan="2"><a href="/facts/Coxeter_notation/ArUw6goe">Coxeter</a></th><th rowspan="2">Order</th></tr><tr><th>(full)</th><th>(short)</th></tr><tr><th rowspan="2" colspan="2"><a href="/facts/Triclinic_crystal_system/Y6yEdqfN">Triclinic</a></th><td>1</td><td>1</td><td> 1   {\displaystyle 1\ } </td><td>C1</td><td>11</td><td>[ ]+</td><td>1</td></tr><tr><td>1</td><td>1</td><td> 2 ~ {\displaystyle {\tilde {2}}} </td><td>Ci = S2</td><td>×</td><td>[2+,2+]</td><td>2</td></tr><tr><th rowspan="3" colspan="2"><a href="/facts/Monoclinic_crystal_system/5PCwXvzX">Monoclinic</a></th><td>2</td><td>2</td><td> 2   {\displaystyle 2\ } </td><td>C2</td><td>22</td><td>[2]+</td><td>2</td></tr><tr><td>m</td><td>m</td><td> m   {\displaystyle m\ } </td><td>Cs = C1h</td><td>*</td><td>[ ]</td><td>2</td></tr><tr><td> 2 m {\displaystyle {\tfrac {2}{m}}} </td><td>2/m</td><td> 2 : m   {\displaystyle 2:m\ } </td><td>C2h</td><td>2*</td><td>[2,2+]</td><td>4</td></tr><tr><th rowspan="3" colspan="2"><a href="/facts/Orthorhombic_crystal_system/3zo4F8P0">Orthorhombic</a></th><td>222</td><td>222</td><td> 2 : 2   {\displaystyle 2:2\ } </td><td>D2 = V</td><td>222</td><td>[2,2]+</td><td>4</td></tr><tr><td>mm2</td><td>mm2</td><td> 2 ⋅ m   {\displaystyle 2\cdot m\ } </td><td>C2v</td><td>*22</td><td>[2]</td><td>4</td></tr><tr><td> 2 m 2 m 2 m {\displaystyle {\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}} </td><td>mmm</td><td> m ⋅ 2 : m   {\displaystyle m\cdot 2:m\ } </td><td>D2h = Vh</td><td>*222</td><td>[2,2]</td><td>8</td></tr><tr><th rowspan="7" colspan="2"><a href="/facts/Tetragonal_crystal_system/12Aop255">Tetragonal</a></th><td>4</td><td>4</td><td> 4   {\displaystyle 4\ } </td><td>C4</td><td>44</td><td>[4]+</td><td>4</td></tr><tr><td>4</td><td>4</td><td> 4 ~ {\displaystyle {\tilde {4}}} </td><td>S4</td><td>2×</td><td>[2+,4+]</td><td>4</td></tr><tr><td> 4 m {\displaystyle {\tfrac {4}{m}}} </td><td>4/m</td><td> 4 : m   {\displaystyle 4:m\ } </td><td>C4h</td><td>4*</td><td>[2,4+]</td><td>8</td></tr><tr><td>422</td><td>422</td><td> 4 : 2   {\displaystyle 4:2\ } </td><td>D4</td><td>422</td><td>[4,2]+</td><td>8</td></tr><tr><td>4mm</td><td>4mm</td><td> 4 ⋅ m   {\displaystyle 4\cdot m\ } </td><td>C4v</td><td>*44</td><td>[4]</td><td>8</td></tr><tr><td>42m</td><td>42m</td><td> 4 ~ ⋅ m {\displaystyle {\tilde {4}}\cdot m} </td><td>D2d = Vd</td><td>2*2</td><td>[2+,4]</td><td>8</td></tr><tr><td> 4 m 2 m 2 m {\displaystyle {\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}} </td><td>4/mmm</td><td> m ⋅ 4 : m   {\displaystyle m\cdot 4:m\ } </td><td>D4h</td><td>*422</td><td>[4,2]</td><td>16</td></tr><tr><th rowspan="12"><a href="/facts/Hexagonal_crystal_family/sSSok8wo">Hexagonal</a></th><th rowspan="5">Trigonal</th><td>3</td><td>3</td><td> 3   {\displaystyle 3\ } </td><td>C3</td><td>33</td><td>[3]+</td><td>3</td></tr><tr><td>3</td><td>3</td><td> 6 ~ {\displaystyle {\tilde {6}}} </td><td>C3i = S6</td><td>3×</td><td>[2+,6+]</td><td>6</td></tr><tr><td>32</td><td>32</td><td> 3 : 2   {\displaystyle 3:2\ } </td><td>D3</td><td>322</td><td>[3,2]+</td><td>6</td></tr><tr><td>3m</td><td>3m</td><td> 3 ⋅ m   {\displaystyle 3\cdot m\ } </td><td>C3v</td><td>*33</td><td>[3]</td><td>6</td></tr><tr><td>3 2 m {\displaystyle {\tfrac {2}{m}}} </td><td>3m</td><td> 6 ~ ⋅ m {\displaystyle {\tilde {6}}\cdot m} </td><td>D3d</td><td>2*3</td><td>[2+,6]</td><td>12</td></tr><tr><th rowspan="7">Hexagonal</th><td>6</td><td>6</td><td> 6   {\displaystyle 6\ } </td><td>C6</td><td>66</td><td>[6]+</td><td>6</td></tr><tr><td>6</td><td>6</td><td> 3 : m   {\displaystyle 3:m\ } </td><td>C3h</td><td>3*</td><td>[2,3+]</td><td>6</td></tr><tr><td> 6 m {\displaystyle {\tfrac {6}{m}}} </td><td>6/m</td><td> 6 : m   {\displaystyle 6:m\ } </td><td>C6h</td><td>6*</td><td>[2,6+]</td><td>12</td></tr><tr><td>622</td><td>622</td><td> 6 : 2   {\displaystyle 6:2\ } </td><td>D6</td><td>622</td><td>[6,2]+</td><td>12</td></tr><tr><td>6mm</td><td>6mm</td><td> 6 ⋅ m   {\displaystyle 6\cdot m\ } </td><td>C6v</td><td>*66</td><td>[6]</td><td>12</td></tr><tr><td>6m2</td><td>6m2</td><td> m ⋅ 3 : m   {\displaystyle m\cdot 3:m\ } </td><td>D3h</td><td>*322</td><td>[3,2]</td><td>12</td></tr><tr><td> 6 m 2 m 2 m {\displaystyle {\tfrac {6}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}} </td><td>6/mmm</td><td> m ⋅ 6 : m   {\displaystyle m\cdot 6:m\ } </td><td>D6h</td><td>*622</td><td>[6,2]</td><td>24</td></tr><tr><th rowspan="5" colspan="2"><a href="/facts/Cubic_crystal_system/92Co7BKM">Cubic</a></th><td>23</td><td>23</td><td> 3 / 2   {\displaystyle 3/2\ } </td><td>T</td><td>332</td><td>[3,3]+</td><td>12</td></tr><tr><td> 2 m {\displaystyle {\tfrac {2}{m}}} 3</td><td>m3</td><td> 6 ~ / 2 {\displaystyle {\tilde {6}}/2} </td><td>Th</td><td>3*2</td><td>[3+,4]</td><td>24</td></tr><tr><td>432</td><td>432</td><td> 3 / 4   {\displaystyle 3/4\ } </td><td>O</td><td>432</td><td>[4,3]+</td><td>24</td></tr><tr><td>43m</td><td>43m</td><td> 3 / 4 ~ {\displaystyle 3/{\tilde {4}}} </td><td>Td</td><td>*332</td><td>[3,3]</td><td>24</td></tr><tr><td> 4 m {\displaystyle {\tfrac {4}{m}}} 3 2 m {\displaystyle {\tfrac {2}{m}}} </td><td>m3m</td><td> 6 ~ / 4 {\displaystyle {\tilde {6}}/4} </td><td>Oh</td><td>*432</td><td>[4,3]</td><td>48</td></tr></tbody></table>
<h2 id="isomorphisms">Isomorphisms</h2>
See also: <a href="/facts/Crystal_structure/bJ4bCeHd">Crystal structure § Crystal systems</a>
Many of the crystallographic point groups share the same internal structure. For example, the point groups 1, 2, and m contain different geometric symmetry operations, (inversion, rotation, and reflection, respectively) but all share the structure of the <a href="/facts/Cyclic_group/JL5zfFuL">cyclic group</a> C2. All <a href="/facts/Group_isomorphism/LC4N1aDw">isomorphic</a> groups are of the same <a href="/facts/Order_(group_theory)/Cl3tKGFg">order</a>, but not all groups of the same order are isomorphic. The point groups which are isomorphic are shown in the following table:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<table><tbody><tr><th><a href="/facts/Hermann%25E2%2580%2593Mauguin_notation/NGHDFXKC">Hermann–Mauguin</a></th><th><a href="/facts/Schoenflies_notation/c4y0GtEt">Schoenflies</a></th><th><a href="/facts/Order_(group_theory)/Cl3tKGFg">Order</a></th><th colspan="2"><a href="/facts/List_of_small_groups/q1cKEj1G">Abstract group</a></th></tr><tr><td>1</td><td>C1</td><td>1</td><td><a href="/facts/Trivial_group/XPjQkQFm">C1</a></td><td> G 1 1 {\displaystyle G_{1}^{1}} </td></tr><tr><td>1</td><td>Ci = S2</td><td>2</td><td rowspan="3"><a href="/facts/Cyclic_group/JL5zfFuL">C2</a></td><td rowspan="3"> G 2 1 {\displaystyle G_{2}^{1}} </td></tr><tr><td>2</td><td>C2</td><td>2</td></tr><tr><td>m</td><td>Cs = C1h</td><td>2</td></tr><tr><td>3</td><td>C3</td><td>3</td><td><a href="/facts/Cyclic_group/JL5zfFuL">C3</a></td><td> G 3 1 {\displaystyle G_{3}^{1}} </td></tr><tr><td>4</td><td>C4</td><td>4</td><td rowspan="2"><a href="/facts/Cyclic_group/JL5zfFuL">C4</a></td><td rowspan="2"> G 4 1 {\displaystyle G_{4}^{1}} </td></tr><tr><td>4</td><td>S4</td><td>4</td></tr><tr><td>2/m</td><td> C2h</td><td>4</td><td rowspan="3"><a href="/facts/Klein_four-group/X2oQCYBy">D2</a> = C2 × C2</td><td rowspan="3"> G 4 2 {\displaystyle G_{4}^{2}} </td></tr><tr><td> 222</td><td>D2 = V</td><td>4</td></tr><tr><td>mm2</td><td>C2v</td><td> 4</td></tr><tr><td>3</td><td>C3i = S6</td><td>6</td><td rowspan="3"><a href="/facts/Cyclic_group/JL5zfFuL">C6</a></td><td rowspan="3"> G 6 1 {\displaystyle G_{6}^{1}} </td></tr><tr><td>6</td><td>C6</td><td>6</td></tr><tr><td>6</td><td>C3h</td><td>6</td></tr><tr><td>32</td><td>D3</td><td>6</td><td rowspan="2"><a href="/facts/Dihedral_group_of_order_6/AwrrQFhv">D3</a></td><td rowspan="2"> G 6 2 {\displaystyle G_{6}^{2}} </td></tr><tr><td>3m</td><td>C3v</td><td>6</td></tr><tr><td>mmm</td><td>D2h = Vh</td><td>8</td><td>D2 × C2</td><td> G 8 3 {\displaystyle G_{8}^{3}} </td></tr><tr><td> 4/m</td><td>C4h</td><td>8</td><td>C4 × C2</td><td> G 8 2 {\displaystyle G_{8}^{2}} </td></tr><tr><td>422</td><td>D4</td><td>8</td><td rowspan="3"><a href="/facts/Dihedral_group_of_order_8/S0waDM2z">D4</a></td><td rowspan="3"> G 8 4 {\displaystyle G_{8}^{4}} </td></tr><tr><td>4mm</td><td>C4v</td><td>8</td></tr><tr><td>42m</td><td>D2d = Vd</td><td>8</td></tr><tr><td>6/m</td><td>C6h</td><td>12</td><td>C6 × C2</td><td> G 12 2 {\displaystyle G_{12}^{2}} </td></tr><tr><td>23</td><td>T</td><td>12</td><td><a href="/facts/Alternating_group/yethKwyo">A4</a></td><td> G 12 5 {\displaystyle G_{12}^{5}} </td></tr><tr><td>3m</td><td>D3d</td><td>12</td><td rowspan="4"><a href="/facts/Dihedral_group/1PUlRh4M">D6</a></td><td rowspan="4"> G 12 3 {\displaystyle G_{12}^{3}} </td></tr><tr><td>622</td><td>D6</td><td>12</td></tr><tr><td>6mm</td><td>C6v</td><td>12</td></tr><tr><td>6m2</td><td>D3h</td><td>12</td></tr><tr><td>4/mmm</td><td>D4h</td><td>16</td><td>D4 × C2</td><td> G 16 9 {\displaystyle G_{16}^{9}} </td></tr><tr><td>6/mmm</td><td>D6h</td><td>24</td><td>D6 × C2</td><td> G 24 5 {\displaystyle G_{24}^{5}} </td></tr><tr><td>m3</td><td>Th</td><td>24</td><td>A4 × C2</td><td> G 24 10 {\displaystyle G_{24}^{10}} </td></tr><tr><td>432</td><td>O  </td><td>24</td><td rowspan="2"><a href="/facts/Symmetric_group/RUksK5l5">S4</a></td><td rowspan="2"> G 24 7 {\displaystyle G_{24}^{7}} </td></tr><tr><td>43m</td><td>Td</td><td>24</td></tr><tr><td>m3m</td><td>Oh</td><td>48</td><td>S4 × C2</td><td> G 48 7 {\displaystyle G_{48}^{7}} </td></tr></tbody></table>
This table makes use of <a href="/facts/Cyclic_group/JL5zfFuL">cyclic groups</a> (C1, C2, C3, C4, C6), <a href="/facts/Dihedral_group/1PUlRh4M">dihedral groups</a> (D2, D3, D4, D6), one of the <a href="/facts/Alternating_group/yethKwyo">alternating groups</a> (A4), and one of the <a href="/facts/Symmetric_group/RUksK5l5">symmetric groups</a> (S4). Here the symbol " × " indicates a <a href="/facts/Direct_product_of_groups/4MWCxY0I">direct product</a>.

<h2 id="deriving-the-crystallographic-point-group-crystal-class-from-the-space-group">Deriving the crystallographic point group (crystal class) from the space group</h2>
<ol><li>Leave out the <a href="/facts/Bravais_lattice/7gQKTsXk">Bravais lattice</a> type.</li>
<li>Convert all symmetry elements with translational components into their respective symmetry elements without translation symmetry. (Glide planes are converted into simple mirror planes; screw axes are converted into simple axes of rotation.)</li>
<li>Axes of rotation, <a href="/facts/Improper_rotation/kcLdCD9r">rotoinversion</a> axes, and mirror planes remain unchanged.</li></ol>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Molecular_symmetry/BrDxAkTK">Molecular symmetry</a></li>
<li><a href="/facts/Point_group/41NSqUN6">Point group</a></li>
<li><a href="/facts/Space_group/2XQx0J7M">Space group</a></li>
<li><a href="/facts/Point_groups_in_three_dimensions/Uc4MRIyl">Point groups in three dimensions</a></li>
<li><a href="/facts/Crystal_system/dMQLk1q4">Crystal system</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://archive.today/20130704033345/http://it.iucr.org/Ab/ch12o1v0001/">Point-group symbols in International Tables for Crystallography (2006). Vol. A, ch. 12.1, pp. 818-820</a></li>
<li><a href="https://archive.today/20130704032551/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</a></li>
<li><a href="https://web.archive.org/web/20120204104121/http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/Solids.html">Pictorial overview of the 32 groups</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Authier, André (2015). "12. The Birth and Rise of the Space-Lattice Concept". Early days of X-ray crystallography. Oxford: Oxford University Press. doi:10.1093/acprof:oso/9780199659845.003.0012. ISBN 9780198754053. Retrieved 24 December 2024. <a href="9780198754053" target="_blank">9780198754053</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">"(International Tables) Abstract". Archived from the original on 2013-07-04. Retrieved 2011-11-25. <a href="https://archive.today/20130704042455/http://it.iucr.org/Ab/ch12o1v0001/sec12o1o3/" target="_blank">https://archive.today/20130704042455/http://it.iucr.org/Ab/ch12o1v0001/sec12o1o3/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Novak, I (1995-07-18). "Molecular isomorphism". European Journal of Physics. 16 (4). IOP Publishing: 151–153. Bibcode:1995EJPh...16..151N. doi:10.1088/0143-0807/16/4/001. ISSN 0143-0807. S2CID 250887121. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Crystallographic point group open-in-new

Crystallographic point group