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
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
| 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
| 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
| 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
| 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 ?? group in with setting ??
| 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 |
