Table of Contents

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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

Oh_0sqrt2-1z Oh_0sqrt21z Oh_111z Oh_sqrt20-1z Oh_sqrt201z Oh_XYZ

Setting 0sqrt2-1z

Details of the Oh group in with setting 0sqrt2-1z

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

Setting 0sqrt21z

Details of the Oh group in with setting 0sqrt21z

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

Setting 111z

Details of the Oh group in with setting 111z

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

Setting sqrt20-1z

Details of the Oh group in with setting sqrt20-1z

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

Setting sqrt201z

Details of the Oh group in with setting sqrt201z

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

Setting XYZ

Details of the Oh group in with setting XYZ

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

Table of several point groups

Return to Main page on Point Groups

Nonaxial groups C1 Cs Ci
Cn groups C2 C3 C4 C5 C6 C7 C8
Dn groups D2 D3 D4 D5 D6 D7 D8
Cnv groups C2v C3v C4v C5v C6v C7v C8v
Cnh groups C2h C3h C4h C5h C6h
Dnh groups D2h D3h D4h D5h D6h D7h D8h
Dnd groups D2d D3d D4d D5d D6d D7d D8d
Sn groups S2 S4 S6 S8 S10 S12
Cubic groups T Th Td O Oh I Ih
Linear groups C$\infty$v D$\infty$h

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