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Schoenflies notation
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==Point groups== {{Main|Point groups in three dimensions}} {{See also|Molecular symmetry}} In three dimensions, there are an infinite number of point groups, but all of them can be classified by several families. * ''C''<sub>''n''</sub> (for [[cyclic group|cyclic]]) has an ''n''-fold rotation axis. ** ''C''<sub>''n''h</sub> is ''C''<sub>''n''</sub> with the addition of a mirror (reflection) plane perpendicular to the axis of rotation (''horizontal plane''). ** ''C''<sub>''n''v</sub> is ''C''<sub>''n''</sub> with the addition of ''n'' mirror planes containing the axis of rotation (''vertical planes''). * ''C''<sub>s</sub> denotes a group with only mirror plane (for ''Spiegel'', German for mirror) and no other symmetry elements. * ''S''<sub>''n''</sub> (for ''Spiegel'', German for [[mirror]]) contains only a ''n''-fold [[rotation-reflection axis]]. The index, ''n'', should be even because when it is odd an ''n''-fold rotation-reflection axis is equivalent to a combination of an ''n''-fold rotation axis and a perpendicular plane, hence ''S''<sub>''n''</sub> = ''C''<sub>''n''h</sub> for odd ''n''. * ''C''<sub>''n''i</sub> has only a [[Improper rotation|rotoinversion axis]]. This notation is rarely used because any rotoinversion axis can be expressed instead as rotation-reflection axis: For odd ''n'', ''C''<sub>''n''i</sub> = ''S''<sub>2''n''</sub> and ''C''<sub>2''n''i</sub> = ''S''<sub>''n''</sub> = ''C''<sub>''n''h</sub>, and for even ''n'', ''C''<sub>2''n''i</sub> = ''S''<sub>2''n''</sub>. Only the notation ''C''<sub>i</sub> (meaning ''C''<sub>1i</sub>) is commonly used, and some sources write ''C''<sub>3i</sub>, ''C''<sub>5i</sub> etc. * ''D''<sub>''n''</sub> (for [[dihedral group|dihedral]], or two-sided) has an ''n''-fold rotation axis plus ''n'' twofold axes perpendicular to that axis. ** ''D''<sub>''n''h</sub> has, in addition, a horizontal mirror plane and, as a consequence, also ''n'' vertical mirror planes each containing the ''n''-fold axis and one of the twofold axes. ** ''D''<sub>''n''d</sub> has, in addition to the elements of ''D''<sub>''n''</sub>, ''n'' vertical mirror planes which pass between twofold axes (''diagonal planes''). * ''T'' (the chiral [[tetrahedron|tetrahedral]] group) has the rotation axes of a tetrahedron (three 2-fold axes and four 3-fold axes). ** ''T''<sub>d</sub> includes diagonal mirror planes (each diagonal plane contains only one twofold axis and passes between two other twofold axes, as in ''D''<sub>2d</sub>). This addition of diagonal planes results in three improper rotation operations '''S<sub>4</sub>'''. ** ''T''<sub>h</sub> includes three horizontal mirror planes. Each plane contains two twofold axes and is perpendicular to the third twofold axis, which results in inversion center '''i'''. * ''O'' (the chiral [[octahedron|octahedral]] group) has the rotation axes of an octahedron or [[cube]] (three 4-fold axes, four 3-fold axes, and six diagonal 2-fold axes). ** ''O''<sub>h</sub> includes horizontal mirror planes and, as a consequence, vertical mirror planes. It contains also inversion center and improper rotation operations. * ''I'' (the chiral [[icosahedron|icosahedral]] group) indicates that the group has the rotation axes of an icosahedron or [[dodecahedron]] (six 5-fold axes, ten 3-fold axes, and 15 2-fold axes). ** ''I''<sub>h</sub> includes horizontal mirror planes and contains also inversion center and improper rotation operations. All groups that do not contain more than one higher-order axis (order 3 or more) can be arranged as shown in a table below; symbols in red are rarely used. {| class="wikitable" border="1" |- ! ||''n'' = 1||2||3||4||5||6||7||8||...||β |- ! ''C''<sub>''n''</sub> | ''C''<sub>1</sub> | ''C''<sub>2</sub> | ''C''<sub>3</sub> | ''C''<sub>4</sub> |style="background:silver"| ''C''<sub>5</sub> | ''C''<sub>6</sub> |style="background:silver"| ''C''<sub>7</sub> |style="background:silver"| ''C''<sub>8</sub> |style="background:silver"| {{center|...}} |style="background:silver"| ''C''<sub>β</sub> |- ! ''C''<sub>''n''v</sub> | <span style="color:red;">''C''<sub>1v</sub></span> = ''C''<sub>1h</sub> | ''C''<sub>2v</sub> | ''C''<sub>3v</sub> | ''C''<sub>4v</sub> |style="background:silver"| ''C''<sub>5v</sub> | ''C''<sub>6v</sub> |style="background:silver"| ''C''<sub>7v</sub> |style="background:silver"| ''C''<sub>8v</sub> |style="background:silver"| {{center|...}} |style="background:silver"| ''C''<sub>βv</sub> |- ! ''C''<sub>''n''h</sub> | <span style="color:red;">''C''<sub>1h</sub></span> = ''C''<sub>s</sub> | ''C''<sub>2h</sub> | ''C''<sub>3h</sub> | ''C''<sub>4h</sub> |style="background:silver"| ''C''<sub>5h</sub> | ''C''<sub>6h</sub> |style="background:silver"| ''C''<sub>7h</sub> |style="background:silver"| ''C''<sub>8h</sub> |style="background:silver"| {{center|...}} |style="background:silver"| ''C''<sub>βh</sub> |- ! ''S''<sub>''n''</sub> | <span style="color:red;">''S''<sub>1</sub></span> = ''C''<sub>s</sub> | ''S''<sub>2</sub> = ''C''<sub>i</sub> | <span style="color:red;">''S''<sub>3</sub></span> = ''C''<sub>3h</sub> | ''S''<sub>4</sub> |style="background:silver"| <span style="color:red;">''S''<sub>5</sub></span> = ''C''<sub>5h</sub> | ''S''<sub>6</sub> |style="background:silver"| <span style="color:red;">''S''<sub>7</sub></span> = ''C''<sub>7h</sub> |style="background:silver"| ''S''<sub>8</sub> |style="background:silver"| {{center|...}} |style="background:silver"| <span style="color:red;">''S''<sub>β</sub></span> = ''C''<sub>βh</sub> |- ! ''C''<sub>''n''i</sub> (redundant) | <span style="color:red;">''C''<sub>1i</sub></span> = ''C''<sub>i</sub> | ''C''<sub>2i</sub> = ''C''<sub>s</sub> | ''C''<sub>3i</sub> = ''S''<sub>6</sub> | ''C''<sub>4i</sub> = ''S''<sub>4</sub> |style="background:silver"| ''C''<sub>5i</sub> = ''S''<sub>10</sub> | ''C''<sub>6i</sub> = ''C''<sub>3h</sub> |style="background:silver"| ''C''<sub>7i</sub> = ''S''<sub>14</sub> |style="background:silver"| ''C''<sub>8i</sub> = ''S''<sub>8</sub> |style="background:silver"| {{center|...}} |style="background:silver"| <span style="color:red;">''C''<sub>βi</sub></span> = ''C''<sub>βh</sub> |- ! ''D''<sub>''n''</sub> | <span style="color:red;">''D''<sub>1</sub></span> = ''C''<sub>2</sub> | ''D''<sub>2</sub> | ''D''<sub>3</sub> | ''D''<sub>4</sub> |style="background:silver"| ''D''<sub>5</sub> | ''D''<sub>6</sub> |style="background:silver"| ''D''<sub>7</sub> |style="background:silver"| ''D''<sub>8</sub> |style="background:silver"| {{center|...}} |style="background:silver"| ''D''<sub>β</sub> |- ! ''D''<sub>''n''h</sub> | <span style="color:red;">''D''<sub>1h</sub></span> = ''C''<sub>2v</sub> | ''D''<sub>2h</sub> | ''D''<sub>3h</sub> | ''D''<sub>4h</sub> |style="background:silver"| ''D''<sub>5h</sub> | ''D''<sub>6h</sub> |style="background:silver"| ''D''<sub>7h</sub> |style="background:silver"| ''D''<sub>8h</sub> |style="background:silver"| {{center|...}} |style="background:silver"| ''D''<sub>βh</sub> |- ! ''D''<sub>''n''d</sub> | <span style="color:red;">''D''<sub>1d</sub></span> = ''C''<sub>2h</sub> | ''D''<sub>2d</sub> | ''D''<sub>3d</sub> |style="background:silver"| ''D''<sub>4d</sub> |style="background:silver"| ''D''<sub>5d</sub> |style="background:silver"| ''D''<sub>6d</sub> |style="background:silver"| ''D''<sub>7d</sub> |style="background:silver"| ''D''<sub>8d</sub> |style="background:silver"| {{center|...}} |style="background:silver"| <span style="color:red;">''D''<sub>βd</sub></span> = ''D''<sub>βh</sub> |} In crystallography, due to the [[crystallographic restriction theorem]], ''n'' is restricted to the values of 1, 2, 3, 4, or 6. The noncrystallographic groups are shown with grayed backgrounds. ''D''<sub>4d</sub> and ''D''<sub>6d</sub> are also 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 group|crystallographic point groups]]. Groups with ''n = β'' are called limit groups or [[Curie group]]s. There are two more limit groups, not listed in the table: ''K'' (for ''Kugel'', German for ball, sphere), the group of all rotations in 3-dimensional space; and ''K''<sub>h</sub>, the group of all rotations and reflections. In mathematics and theoretical physics they are known respectively as the ''special orthogonal group'' and the ''[[orthogonal group]]'' in three-dimensional space, with the symbols SO(3) and O(3).
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