====== Point Group Oh ======

===== Character Table =====

###

|  $  $  ^  $ \text{E} \,{\text{(1)}} $  ^  $ C_3 \,{\text{(8)}} $  ^  $ C_2 \,{\text{(6)}} $  ^  $ C_4 \,{\text{(6)}} $  ^  $ C_2 \,{\text{(3)}} $  ^  $ \text{i} \,{\text{(1)}} $  ^  $ S_4 \,{\text{(6)}} $  ^  $ S_6 \,{\text{(8)}} $  ^  $ \sigma_h \,{\text{(3)}} $  ^  $ \sigma_d \,{\text{(6)}} $  ^
^ $ A_{1g} $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |
^ $ A_{2g} $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |
^ $ E_g $ |  $ 2 $ |  $ -1 $ |  $ 0 $ |  $ 0 $ |  $ 2 $ |  $ 2 $ |  $ 0 $ |  $ -1 $ |  $ 2 $ |  $ 0 $ |
^ $ T_{1g} $ |  $ 3 $ |  $ 0 $ |  $ -1 $ |  $ 1 $ |  $ -1 $ |  $ 3 $ |  $ 1 $ |  $ 0 $ |  $ -1 $ |  $ -1 $ |
^ $ T_{2g} $ |  $ 3 $ |  $ 0 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ 3 $ |  $ -1 $ |  $ 0 $ |  $ -1 $ |  $ 1 $ |
^ $ A_{1u} $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |  $ -1 $ |
^ $ A_{2u} $ |  $ 1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ 1 $ |  $ -1 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ 1 $ |
^ $ E_u $ |  $ 2 $ |  $ -1 $ |  $ 0 $ |  $ 0 $ |  $ 2 $ |  $ -2 $ |  $ 0 $ |  $ 1 $ |  $ -2 $ |  $ 0 $ |
^ $ T_{1u} $ |  $ 3 $ |  $ 0 $ |  $ -1 $ |  $ 1 $ |  $ -1 $ |  $ -3 $ |  $ -1 $ |  $ 0 $ |  $ 1 $ |  $ 1 $ |
^ $ T_{2u} $ |  $ 3 $ |  $ 0 $ |  $ 1 $ |  $ -1 $ |  $ -1 $ |  $ -3 $ |  $ 1 $ |  $ 0 $ |  $ 1 $ |  $ -1 $ |

###

===== Product Table =====

###

|  $  $  ^  $ A_{1g} $  ^  $ A_{2g} $  ^  $ E_g $  ^  $ T_{1g} $  ^  $ T_{2g} $  ^  $ A_{1u} $  ^  $ A_{2u} $  ^  $ E_u $  ^  $ T_{1u} $  ^  $ T_{2u} $  ^
^ $ A_{1g} $  | $ A_{1g} $  | $ A_{2g} $  | $ E_g $  | $ T_{1g} $  | $ T_{2g} $  | $ A_{1u} $  | $ A_{2u} $  | $ E_u $  | $ T_{1u} $  | $ T_{2u} $  |
^ $ A_{2g} $  | $ A_{2g} $  | $ A_{1g} $  | $ E_g $  | $ T_{2g} $  | $ T_{1g} $  | $ A_{2u} $  | $ A_{1u} $  | $ E_u $  | $ T_{2u} $  | $ T_{1u} $  |
^ $ E_g $  | $ E_g $  | $ E_g $  | $ A_{1g}+A_{2g}+E_g $  | $ T_{1g}+T_{2g} $  | $ T_{1g}+T_{2g} $  | $ E_u $  | $ E_u $  | $ A_{1u}+A_{2u}+E_u $  | $ T_{1u}+T_{2u} $  | $ T_{1u}+T_{2u} $  |
^ $ T_{1g} $  | $ T_{1g} $  | $ T_{2g} $  | $ T_{1g}+T_{2g} $  | $ A_{1g}+E_g+T_{1g}+T_{2g} $  | $ A_{2g}+E_g+T_{1g}+T_{2g} $  | $ T_{1u} $  | $ T_{2u} $  | $ T_{1u}+T_{2u} $  | $ A_{1u}+E_u+T_{1u}+T_{2u} $  | $ A_{2u}+E_u+T_{1u}+T_{2u} $  |
^ $ T_{2g} $  | $ T_{2g} $  | $ T_{1g} $  | $ T_{1g}+T_{2g} $  | $ A_{2g}+E_g+T_{1g}+T_{2g} $  | $ A_{1g}+E_g+T_{1g}+T_{2g} $  | $ T_{2u} $  | $ T_{1u} $  | $ T_{1u}+T_{2u} $  | $ A_{2u}+E_u+T_{1u}+T_{2u} $  | $ A_{1u}+E_u+T_{1u}+T_{2u} $  |
^ $ A_{1u} $  | $ A_{1u} $  | $ A_{2u} $  | $ E_u $  | $ T_{1u} $  | $ T_{2u} $  | $ A_{1g} $  | $ A_{2g} $  | $ E_g $  | $ T_{1g} $  | $ T_{2g} $  |
^ $ A_{2u} $  | $ A_{2u} $  | $ A_{1u} $  | $ E_u $  | $ T_{2u} $  | $ T_{1u} $  | $ A_{2g} $  | $ A_{1g} $  | $ E_g $  | $ T_{2g} $  | $ T_{1g} $  |
^ $ E_u $  | $ E_u $  | $ E_u $  | $ A_{1u}+A_{2u}+E_u $  | $ T_{1u}+T_{2u} $  | $ T_{1u}+T_{2u} $  | $ E_g $  | $ E_g $  | $ A_{1g}+A_{2g}+E_g $  | $ T_{1g}+T_{2g} $  | $ T_{1g}+T_{2g} $  |
^ $ T_{1u} $  | $ T_{1u} $  | $ T_{2u} $  | $ T_{1u}+T_{2u} $  | $ A_{1u}+E_u+T_{1u}+T_{2u} $  | $ A_{2u}+E_u+T_{1u}+T_{2u} $  | $ T_{1g} $  | $ T_{2g} $  | $ T_{1g}+T_{2g} $  | $ A_{1g}+E_g+T_{1g}+T_{2g} $  | $ A_{2g}+E_g+T_{1g}+T_{2g} $  |
^ $ T_{2u} $  | $ T_{2u} $  | $ T_{1u} $  | $ T_{1u}+T_{2u} $  | $ A_{2u}+E_u+T_{1u}+T_{2u} $  | $ A_{1u}+E_u+T_{1u}+T_{2u} $  | $ T_{2g} $  | $ T_{1g} $  | $ T_{1g}+T_{2g} $  | $ A_{2g}+E_g+T_{1g}+T_{2g} $  | $ A_{1g}+E_g+T_{1g}+T_{2g} $  |

###

===== Implemented Settings =====

[[physics_chemistry:point_groups:oh:orientation_0sqrt2-1z| Oh_0sqrt2-1z ]] [[physics_chemistry:point_groups:oh:orientation_0sqrt21z| Oh_0sqrt21z ]] [[physics_chemistry:point_groups:oh:orientation_111z| Oh_111z ]] [[physics_chemistry:point_groups:oh:orientation_sqrt20-1z| Oh_sqrt20-1z ]] [[physics_chemistry:point_groups:oh:orientation_sqrt201z| Oh_sqrt201z ]] [[physics_chemistry:point_groups:oh:orientation_xyz| Oh_XYZ ]] 

==== Setting 0sqrt2-1z ====

[[physics_chemistry:point_groups:oh:orientation_0sqrt2-1z|Details of the Oh group in with setting 0sqrt2-1z]]

{{:physics_chemistry:pointgroup:oh_0sqrt2-1z.png }}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{-\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$ , $\left\{0,-\frac{2 \sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{-\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{0,\frac{2 \sqrt{2}}{3},\frac{1}{3}\right\}$ , $\left\{\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$ , |
^ $C_2$ | $\left\{-\sqrt{2},0,0\right\}$ , $\left\{-\sqrt{2},-\sqrt{6},0\right\}$ , $\left\{-\sqrt{2},\sqrt{6},0\right\}$ , $\left\{0,-\sqrt{2},-2\right\}$ , $\left\{-\sqrt{6},-\sqrt{2},4\right\}$ , $\left\{-\sqrt{6},\sqrt{2},-4\right\}$ , |
^ $C_4$ | $\left\{0,-\sqrt{2},1\right\}$ , $\left\{0,\sqrt{2},-1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$ , $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$ , $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$ , |
^ $C_2$ | $\left\{0,-\sqrt{2},1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_4$ | $\left\{0,-\sqrt{2},1\right\}$ , $\left\{0,\sqrt{2},-1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$ , $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$ , $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$ , |
^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{-\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$ , $\left\{0,-\frac{2 \sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{-\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{0,\frac{2 \sqrt{2}}{3},\frac{1}{3}\right\}$ , $\left\{\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$ , |
^ $\sigma _h$ | $\left\{0,-\sqrt{2},1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$ , |
^ $\sigma _d$ | $\left\{-\sqrt{2},0,0\right\}$ , $\left\{-\sqrt{2},-\sqrt{6},0\right\}$ , $\left\{-\sqrt{2},\sqrt{6},0\right\}$ , $\left\{0,-\sqrt{2},-2\right\}$ , $\left\{-\sqrt{6},-\sqrt{2},4\right\}$ , $\left\{-\sqrt{6},\sqrt{2},-4\right\}$ , |

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==== Setting 0sqrt21z ====

[[physics_chemistry:point_groups:oh:orientation_0sqrt21z|Details of the Oh group in with setting 0sqrt21z]]

{{:physics_chemistry:pointgroup:oh_0sqrt21z.png }}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$ , $\left\{0,\frac{2 \sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{-\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{0,-\frac{2 \sqrt{2}}{3},\frac{1}{3}\right\}$ , $\left\{-\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$ , |
^ $C_2$ | $\left\{\sqrt{2},0,0\right\}$ , $\left\{\sqrt{2},\sqrt{6},0\right\}$ , $\left\{\sqrt{2},-\sqrt{6},0\right\}$ , $\left\{0,\sqrt{2},-2\right\}$ , $\left\{\sqrt{6},\sqrt{2},4\right\}$ , $\left\{\sqrt{6},-\sqrt{2},-4\right\}$ , |
^ $C_4$ | $\left\{0,\sqrt{2},1\right\}$ , $\left\{0,-\sqrt{2},-1\right\}$ , $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$ , $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$ , |
^ $C_2$ | $\left\{0,\sqrt{2},1\right\}$ , $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$ , $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_4$ | $\left\{0,\sqrt{2},1\right\}$ , $\left\{0,-\sqrt{2},-1\right\}$ , $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$ , $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$ , |
^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$ , $\left\{0,\frac{2 \sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{-\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{0,-\frac{2 \sqrt{2}}{3},\frac{1}{3}\right\}$ , $\left\{-\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$ , |
^ $\sigma _h$ | $\left\{0,\sqrt{2},1\right\}$ , $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$ , $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$ , |
^ $\sigma _d$ | $\left\{\sqrt{2},0,0\right\}$ , $\left\{\sqrt{2},\sqrt{6},0\right\}$ , $\left\{\sqrt{2},-\sqrt{6},0\right\}$ , $\left\{0,\sqrt{2},-2\right\}$ , $\left\{\sqrt{6},\sqrt{2},4\right\}$ , $\left\{\sqrt{6},-\sqrt{2},-4\right\}$ , |

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==== Setting 111z ====

[[physics_chemistry:point_groups:oh:orientation_111z|Details of the Oh group in with setting 111z]]

{{:physics_chemistry:pointgroup:oh_111z.png }}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{\frac{1}{3} \left(1+\sqrt{3}\right),\frac{1}{3} \left(1-\sqrt{3}\right),\frac{1}{3}\right\}$ , $\left\{\frac{2}{3},\frac{2}{3},-\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(\sqrt{3}-1\right),\frac{1}{3} \left(-1-\sqrt{3}\right),-\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(-1-\sqrt{3}\right),\frac{1}{3} \left(\sqrt{3}-1\right),-\frac{1}{3}\right\}$ , $\left\{-\frac{2}{3},-\frac{2}{3},\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(1-\sqrt{3}\right),\frac{1}{3} \left(1+\sqrt{3}\right),\frac{1}{3}\right\}$ , |
^ $C_2$ | $\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , $\{1,1,-2\}$ , $\left\{1+\sqrt{3},1-\sqrt{3},4\right\}$ , $\left\{\sqrt{3}-1,-1-\sqrt{3},-4\right\}$ , |
^ $C_4$ | $\{1,1,1\}$ , $\{-1,-1,-1\}$ , $\left\{\frac{\sqrt{3}}{2}-\frac{1}{2},-\frac{1}{2}-\frac{\sqrt{3}}{2},1\right\}$ , $\left\{-\frac{1}{2}-\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2}-\frac{1}{2},1\right\}$ , $\left\{\frac{1}{2}-\frac{\sqrt{3}}{2},\frac{1}{2}+\frac{\sqrt{3}}{2},-1\right\}$ , $\left\{\frac{1}{2}+\frac{\sqrt{3}}{2},\frac{1}{2}-\frac{\sqrt{3}}{2},-1\right\}$ , |
^ $C_2$ | $\{1,1,1\}$ , $\left\{\frac{\sqrt{3}}{2}-\frac{1}{2},-\frac{1}{2}-\frac{\sqrt{3}}{2},1\right\}$ , $\left\{\frac{1}{2}+\frac{\sqrt{3}}{2},\frac{1}{2}-\frac{\sqrt{3}}{2},-1\right\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_4$ | $\{1,1,1\}$ , $\{-1,-1,-1\}$ , $\left\{\frac{\sqrt{3}}{2}-\frac{1}{2},-\frac{1}{2}-\frac{\sqrt{3}}{2},1\right\}$ , $\left\{-\frac{1}{2}-\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2}-\frac{1}{2},1\right\}$ , $\left\{\frac{1}{2}-\frac{\sqrt{3}}{2},\frac{1}{2}+\frac{\sqrt{3}}{2},-1\right\}$ , $\left\{\frac{1}{2}+\frac{\sqrt{3}}{2},\frac{1}{2}-\frac{\sqrt{3}}{2},-1\right\}$ , |
^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{\frac{1}{3} \left(1+\sqrt{3}\right),\frac{1}{3} \left(1-\sqrt{3}\right),\frac{1}{3}\right\}$ , $\left\{\frac{2}{3},\frac{2}{3},-\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(\sqrt{3}-1\right),\frac{1}{3} \left(-1-\sqrt{3}\right),-\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(-1-\sqrt{3}\right),\frac{1}{3} \left(\sqrt{3}-1\right),-\frac{1}{3}\right\}$ , $\left\{-\frac{2}{3},-\frac{2}{3},\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(1-\sqrt{3}\right),\frac{1}{3} \left(1+\sqrt{3}\right),\frac{1}{3}\right\}$ , |
^ $\sigma _h$ | $\{1,1,1\}$ , $\left\{\frac{\sqrt{3}}{2}-\frac{1}{2},-\frac{1}{2}-\frac{\sqrt{3}}{2},1\right\}$ , $\left\{\frac{1}{2}+\frac{\sqrt{3}}{2},\frac{1}{2}-\frac{\sqrt{3}}{2},-1\right\}$ , |
^ $\sigma _d$ | $\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , $\{1,1,-2\}$ , $\left\{1+\sqrt{3},1-\sqrt{3},4\right\}$ , $\left\{\sqrt{3}-1,-1-\sqrt{3},-4\right\}$ , |

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==== Setting sqrt20-1z ====

[[physics_chemistry:point_groups:oh:orientation_sqrt20-1z|Details of the Oh group in with setting sqrt20-1z]]

{{:physics_chemistry:pointgroup:oh_sqrt20-1z.png }}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{-\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$ , $\left\{-\frac{2 \sqrt{2}}{3},0,-\frac{1}{3}\right\}$ , $\left\{\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$ , $\left\{\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$ , $\left\{\frac{2 \sqrt{2}}{3},0,\frac{1}{3}\right\}$ , $\left\{-\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$ , |
^ $C_2$ | $\left\{0,\sqrt{2},0\right\}$ , $\left\{-\sqrt{6},\sqrt{2},0\right\}$ , $\left\{\sqrt{6},\sqrt{2},0\right\}$ , $\left\{-\sqrt{2},0,-2\right\}$ , $\left\{-\sqrt{2},\sqrt{6},4\right\}$ , $\left\{\sqrt{2},\sqrt{6},-4\right\}$ , |
^ $C_4$ | $\left\{-\sqrt{2},0,1\right\}$ , $\left\{\sqrt{2},0,-1\right\}$ , $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$ , $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$ , |
^ $C_2$ | $\left\{-\sqrt{2},0,1\right\}$ , $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_4$ | $\left\{-\sqrt{2},0,1\right\}$ , $\left\{\sqrt{2},0,-1\right\}$ , $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$ , $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$ , |
^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{-\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$ , $\left\{-\frac{2 \sqrt{2}}{3},0,-\frac{1}{3}\right\}$ , $\left\{\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$ , $\left\{\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$ , $\left\{\frac{2 \sqrt{2}}{3},0,\frac{1}{3}\right\}$ , $\left\{-\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$ , |
^ $\sigma _h$ | $\left\{-\sqrt{2},0,1\right\}$ , $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$ , |
^ $\sigma _d$ | $\left\{0,\sqrt{2},0\right\}$ , $\left\{-\sqrt{6},\sqrt{2},0\right\}$ , $\left\{\sqrt{6},\sqrt{2},0\right\}$ , $\left\{-\sqrt{2},0,-2\right\}$ , $\left\{-\sqrt{2},\sqrt{6},4\right\}$ , $\left\{\sqrt{2},\sqrt{6},-4\right\}$ , |

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==== Setting sqrt201z ====

[[physics_chemistry:point_groups:oh:orientation_sqrt201z|Details of the Oh group in with setting sqrt201z]]

{{:physics_chemistry:pointgroup:oh_sqrt201z.png }}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$ , $\left\{\frac{2 \sqrt{2}}{3},0,-\frac{1}{3}\right\}$ , $\left\{-\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$ , $\left\{-\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$ , $\left\{-\frac{2 \sqrt{2}}{3},0,\frac{1}{3}\right\}$ , $\left\{\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$ , |
^ $C_2$ | $\left\{0,-\sqrt{2},0\right\}$ , $\left\{\sqrt{6},-\sqrt{2},0\right\}$ , $\left\{-\sqrt{6},-\sqrt{2},0\right\}$ , $\left\{\sqrt{2},0,-2\right\}$ , $\left\{\sqrt{2},-\sqrt{6},4\right\}$ , $\left\{-\sqrt{2},-\sqrt{6},-4\right\}$ , |
^ $C_4$ | $\left\{\sqrt{2},0,1\right\}$ , $\left\{-\sqrt{2},0,-1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$ , $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$ , $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$ , |
^ $C_2$ | $\left\{\sqrt{2},0,1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$ , $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_4$ | $\left\{\sqrt{2},0,1\right\}$ , $\left\{-\sqrt{2},0,-1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$ , $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$ , $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$ , |
^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$ , $\left\{\frac{2 \sqrt{2}}{3},0,-\frac{1}{3}\right\}$ , $\left\{-\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$ , $\left\{-\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$ , $\left\{-\frac{2 \sqrt{2}}{3},0,\frac{1}{3}\right\}$ , $\left\{\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$ , |
^ $\sigma _h$ | $\left\{\sqrt{2},0,1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$ , $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$ , |
^ $\sigma _d$ | $\left\{0,-\sqrt{2},0\right\}$ , $\left\{\sqrt{6},-\sqrt{2},0\right\}$ , $\left\{-\sqrt{6},-\sqrt{2},0\right\}$ , $\left\{\sqrt{2},0,-2\right\}$ , $\left\{\sqrt{2},-\sqrt{6},4\right\}$ , $\left\{-\sqrt{2},-\sqrt{6},-4\right\}$ , |

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==== Setting XYZ ====

[[physics_chemistry:point_groups:oh:orientation_xyz|Details of the Oh group in with setting XYZ]]

{{:physics_chemistry:pointgroup:oh_xyz.png }}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{1,1,1\}$ , $\{1,1,-1\}$ , $\{1,-1,1\}$ , $\{-1,1,1\}$ , $\{-1,-1,1\}$ , $\{-1,1,-1\}$ , $\{1,-1,-1\}$ , $\{-1,-1,-1\}$ , |
^ $C_2$ | $\{1,1,0\}$ , $\{1,-1,0\}$ , $\{1,0,-1\}$ , $\{1,0,1\}$ , $\{0,1,1\}$ , $\{0,1,-1\}$ , |
^ $C_4$ | $\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , $\{0,0,-1\}$ , $\{0,-1,0\}$ , $\{-1,0,0\}$ , |
^ $C_2$ | $\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_4$ | $\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , $\{0,0,-1\}$ , $\{0,-1,0\}$ , $\{-1,0,0\}$ , |
^ $S_6$ | $\{1,1,1\}$ , $\{1,1,-1\}$ , $\{1,-1,1\}$ , $\{-1,1,1\}$ , $\{-1,-1,1\}$ , $\{-1,1,-1\}$ , $\{1,-1,-1\}$ , $\{-1,-1,-1\}$ , |
^ $\sigma _h$ | $\{1,0,0\}$ , $\{0,1,0\}$ , $\{0,0,1\}$ , |
^ $\sigma _d$ | $\{1,1,0\}$ , $\{1,-1,0\}$ , $\{1,0,-1\}$ , $\{1,0,1\}$ , $\{0,1,1\}$ , $\{0,1,-1\}$ , |

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===== Table of several point groups =====

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[[physics_chemistry:point_groups|Return to Main page on Point Groups]]

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^Nonaxial groups      | [[physics_chemistry:point_groups:c1|C]]<sub>[[physics_chemistry:point_groups:c1|1]]</sub> | [[physics_chemistry:point_groups:cs|C]]<sub>[[physics_chemistry:point_groups:cs|s]]</sub> | [[physics_chemistry:point_groups:ci|C]]<sub>[[physics_chemistry:point_groups:ci|i]]</sub> | | | | |
^C<sub>n</sub> groups | [[physics_chemistry:point_groups:c2|C]]<sub>[[physics_chemistry:point_groups:c2|2]]</sub> | [[physics_chemistry:point_groups:c3|C]]<sub>[[physics_chemistry:point_groups:c3|3]]</sub> | [[physics_chemistry:point_groups:c4|C]]<sub>[[physics_chemistry:point_groups:c4|4]]</sub> | [[physics_chemistry:point_groups:c5|C]]<sub>[[physics_chemistry:point_groups:c5|5]]</sub> | [[physics_chemistry:point_groups:c6|C]]<sub>[[physics_chemistry:point_groups:c6|6]]</sub> | [[physics_chemistry:point_groups:c7|C]]<sub>[[physics_chemistry:point_groups:c7|7]]</sub> | [[physics_chemistry:point_groups:c8|C]]<sub>[[physics_chemistry:point_groups:c8|8]]</sub> | 
^D<sub>n</sub> groups | [[physics_chemistry:point_groups:d2|D]]<sub>[[physics_chemistry:point_groups:d2|2]]</sub> | [[physics_chemistry:point_groups:d3|D]]<sub>[[physics_chemistry:point_groups:d3|3]]</sub> | [[physics_chemistry:point_groups:d4|D]]<sub>[[physics_chemistry:point_groups:d4|4]]</sub> | [[physics_chemistry:point_groups:d5|D]]<sub>[[physics_chemistry:point_groups:d5|5]]</sub> | [[physics_chemistry:point_groups:d6|D]]<sub>[[physics_chemistry:point_groups:d6|6]]</sub> | [[physics_chemistry:point_groups:d7|D]]<sub>[[physics_chemistry:point_groups:d7|7]]</sub> | [[physics_chemistry:point_groups:d8|D]]<sub>[[physics_chemistry:point_groups:d8|8]]</sub> | 
^C<sub>nv</sub> groups | [[physics_chemistry:point_groups:c2v|C]]<sub>[[physics_chemistry:point_groups:c2v|2v]]</sub> | [[physics_chemistry:point_groups:c3v|C]]<sub>[[physics_chemistry:point_groups:c3v|3v]]</sub> | [[physics_chemistry:point_groups:c4v|C]]<sub>[[physics_chemistry:point_groups:c4v|4v]]</sub> | [[physics_chemistry:point_groups:c5v|C]]<sub>[[physics_chemistry:point_groups:c5v|5v]]</sub> | [[physics_chemistry:point_groups:c6v|C]]<sub>[[physics_chemistry:point_groups:c6v|6v]]</sub> | [[physics_chemistry:point_groups:c7v|C]]<sub>[[physics_chemistry:point_groups:c7v|7v]]</sub> | [[physics_chemistry:point_groups:c8v|C]]<sub>[[physics_chemistry:point_groups:c8v|8v]]</sub> | 
^C<sub>nh</sub> groups | [[physics_chemistry:point_groups:c2h|C]]<sub>[[physics_chemistry:point_groups:c2h|2h]]</sub> | [[physics_chemistry:point_groups:c3h|C]]<sub>[[physics_chemistry:point_groups:c3h|3h]]</sub> | [[physics_chemistry:point_groups:c4h|C]]<sub>[[physics_chemistry:point_groups:c4h|4h]]</sub> | [[physics_chemistry:point_groups:c5h|C]]<sub>[[physics_chemistry:point_groups:c5h|5h]]</sub> | [[physics_chemistry:point_groups:c6h|C]]<sub>[[physics_chemistry:point_groups:c6h|6h]]</sub> | | | 
^D<sub>nh</sub> groups | [[physics_chemistry:point_groups:d2h|D]]<sub>[[physics_chemistry:point_groups:d2h|2h]]</sub> | [[physics_chemistry:point_groups:d3h|D]]<sub>[[physics_chemistry:point_groups:d3h|3h]]</sub> | [[physics_chemistry:point_groups:d4h|D]]<sub>[[physics_chemistry:point_groups:d4h|4h]]</sub> | [[physics_chemistry:point_groups:d5h|D]]<sub>[[physics_chemistry:point_groups:d5h|5h]]</sub> | [[physics_chemistry:point_groups:d6h|D]]<sub>[[physics_chemistry:point_groups:d6h|6h]]</sub> | [[physics_chemistry:point_groups:d7h|D]]<sub>[[physics_chemistry:point_groups:d7h|7h]]</sub> | [[physics_chemistry:point_groups:d8h|D]]<sub>[[physics_chemistry:point_groups:d8h|8h]]</sub> | 
^D<sub>nd</sub> groups | [[physics_chemistry:point_groups:d2d|D]]<sub>[[physics_chemistry:point_groups:d2d|2d]]</sub> | [[physics_chemistry:point_groups:d3d|D]]<sub>[[physics_chemistry:point_groups:d3d|3d]]</sub> | [[physics_chemistry:point_groups:d4d|D]]<sub>[[physics_chemistry:point_groups:d4d|4d]]</sub> | [[physics_chemistry:point_groups:d5d|D]]<sub>[[physics_chemistry:point_groups:d5d|5d]]</sub> | [[physics_chemistry:point_groups:d6d|D]]<sub>[[physics_chemistry:point_groups:d6d|6d]]</sub> | [[physics_chemistry:point_groups:d7d|D]]<sub>[[physics_chemistry:point_groups:d7d|7d]]</sub> | [[physics_chemistry:point_groups:d8d|D]]<sub>[[physics_chemistry:point_groups:d8d|8d]]</sub> | 
^S<sub>n</sub> groups | [[physics_chemistry:point_groups:S2|S]]<sub>[[physics_chemistry:point_groups:S2|2]]</sub> | [[physics_chemistry:point_groups:S4|S]]<sub>[[physics_chemistry:point_groups:S4|4]]</sub> | [[physics_chemistry:point_groups:S6|S]]<sub>[[physics_chemistry:point_groups:S6|6]]</sub> | [[physics_chemistry:point_groups:S8|S]]<sub>[[physics_chemistry:point_groups:S8|8]]</sub> | [[physics_chemistry:point_groups:S10|S]]<sub>[[physics_chemistry:point_groups:S10|10]]</sub> | [[physics_chemistry:point_groups:S12|S]]<sub>[[physics_chemistry:point_groups:S12|12]]</sub> |  | 
^Cubic groups | [[physics_chemistry:point_groups:T|T]] | [[physics_chemistry:point_groups:Th|T]]<sub>[[physics_chemistry:point_groups:Th|h]]</sub> | [[physics_chemistry:point_groups:Td|T]]<sub>[[physics_chemistry:point_groups:Td|d]]</sub> | [[physics_chemistry:point_groups:O|O]] | [[physics_chemistry:point_groups:Oh|O]]<sub>[[physics_chemistry:point_groups:Oh|h]]</sub> | [[physics_chemistry:point_groups:I|I]] | [[physics_chemistry:point_groups:Ih|I]]<sub>[[physics_chemistry:point_groups:Ih|h]]</sub> | 
^Linear groups      | [[physics_chemistry:point_groups:cinfv|C]]<sub>[[physics_chemistry:point_groups:cinfv|$\infty$v]]</sub> | [[physics_chemistry:point_groups:cinfv|D]]<sub>[[physics_chemistry:point_groups:dinfh|$\infty$h]]</sub> | | | | | |

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