====== Point Group D3d ======

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

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

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

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

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

[[physics_chemistry:point_groups:d3d:orientation_111| D3d_111 ]] [[physics_chemistry:point_groups:d3d:orientation_zx| D3d_Zx ]] [[physics_chemistry:point_groups:d3d:orientation_zx_a| D3d_Zx_A ]] [[physics_chemistry:point_groups:d3d:orientation_zx_b| D3d_Zx_B ]] [[physics_chemistry:point_groups:d3d:orientation_z(x-y)| D3d_Z(x-y) ]] [[physics_chemistry:point_groups:d3d:orientation_z(x-y)_a| D3d_Z(x-y)_A ]] [[physics_chemistry:point_groups:d3d:orientation_z(x-y)_b| D3d_Z(x-y)_B ]] [[physics_chemistry:point_groups:d3d:orientation_zy| D3d_Zy ]] [[physics_chemistry:point_groups:d3d:orientation_zy_a| D3d_Zy_A ]] [[physics_chemistry:point_groups:d3d:orientation_zy_b| D3d_Zy_B ]] 

==== Setting 111 ====

[[physics_chemistry:point_groups:d3d:orientation_111|Details of the D3d group in with setting 111]]

{{:physics_chemistry:pointgroup:d3d_111.png }}

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

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==== Setting Zx ====

[[physics_chemistry:point_groups:d3d:orientation_zx|Details of the D3d group in with setting Zx]]

{{:physics_chemistry:pointgroup:d3d_zx.png }}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $C_2$ | $\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{-1,\sqrt{3},0\right\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $\sigma _d$ | $\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{-1,\sqrt{3},0\right\}$ , |

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==== Setting Zx_A ====

[[physics_chemistry:point_groups:d3d:orientation_zx_a|Details of the D3d group in with setting Zx_A]]

{{:physics_chemistry:pointgroup:d3d_zx_a.png }}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $C_2$ | $\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{-1,\sqrt{3},0\right\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $\sigma _d$ | $\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{-1,\sqrt{3},0\right\}$ , |

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==== Setting Zx_B ====

[[physics_chemistry:point_groups:d3d:orientation_zx_b|Details of the D3d group in with setting Zx_B]]

{{:physics_chemistry:pointgroup:d3d_zx_b.png }}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $C_2$ | $\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{-1,\sqrt{3},0\right\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $\sigma _d$ | $\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{-1,\sqrt{3},0\right\}$ , |

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==== Setting Z(x-y) ====

[[physics_chemistry:point_groups:d3d:orientation_z(x-y)|Details of the D3d group in with setting Z(x-y)]]

{{:physics_chemistry:pointgroup:d3d_z(x-y).png }}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $C_2$ | $\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $\sigma _d$ | $\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , |

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==== Setting Z(x-y)_A ====

[[physics_chemistry:point_groups:d3d:orientation_z(x-y)_a|Details of the D3d group in with setting Z(x-y)_A]]

{{:physics_chemistry:pointgroup:d3d_z(x-y)_a.png }}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $C_2$ | $\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $\sigma _d$ | $\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , |

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==== Setting Z(x-y)_B ====

[[physics_chemistry:point_groups:d3d:orientation_z(x-y)_b|Details of the D3d group in with setting Z(x-y)_B]]

{{:physics_chemistry:pointgroup:d3d_z(x-y)_b.png }}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $C_2$ | $\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $\sigma _d$ | $\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , |

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==== Setting Zy ====

[[physics_chemistry:point_groups:d3d:orientation_zy|Details of the D3d group in with setting Zy]]

{{:physics_chemistry:pointgroup:d3d_zy.png }}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $C_2$ | $\{0,1,0\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},-1,0\right\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $\sigma _d$ | $\{0,1,0\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},-1,0\right\}$ , |

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==== Setting Zy_A ====

[[physics_chemistry:point_groups:d3d:orientation_zy_a|Details of the D3d group in with setting Zy_A]]

{{:physics_chemistry:pointgroup:d3d_zy_a.png }}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $C_2$ | $\{0,1,0\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},-1,0\right\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $\sigma _d$ | $\{0,1,0\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},-1,0\right\}$ , |

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==== Setting Zy_B ====

[[physics_chemistry:point_groups:d3d:orientation_zy_b|Details of the D3d group in with setting Zy_B]]

{{:physics_chemistry:pointgroup:d3d_zy_b.png }}

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^ Operator ^ Orientation ^
^ $\text{E}$ | $\{0,0,0\}$ , |
^ $C_3$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $C_2$ | $\{0,1,0\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},-1,0\right\}$ , |
^ $\text{i}$ | $\{0,0,0\}$ , |
^ $S_6$ | $\{0,0,1\}$ , $\{0,0,-1\}$ , |
^ $\sigma _d$ | $\{0,1,0\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},-1,0\right\}$ , |

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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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