Table of Contents

Point Group Ih

Character Table

$ $ $ \text{E} \,{\text{(1)}} $ $ C_5 \,{\text{(12)}} $ $ C_5^2{} \,{\text{(12)}} $ $ C_3 \,{\text{(20)}} $ $ C_2 \,{\text{(15)}} $ $ \text{i} \,{\text{(1)}} $ $ S_{10} \,{\text{(12)}} $ $ S_{10}^3{} \,{\text{(12)}} $ $ S_6 \,{\text{(20)}} $ $ \sigma_h \,{\text{(15)}} $
$ A_g $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $
$ T_{1g} $ $ 3 $ $ \frac{1}{2} \left(1+\sqrt{5}\right) $ $ \frac{1}{2} \left(1-\sqrt{5}\right) $ $ 0 $ $ -1 $ $ 3 $ $ \frac{1}{2} \left(1-\sqrt{5}\right) $ $ \frac{1}{2} \left(1+\sqrt{5}\right) $ $ 0 $ $ -1 $
$ T_{2g} $ $ 3 $ $ \frac{1}{2} \left(1-\sqrt{5}\right) $ $ \frac{1}{2} \left(1+\sqrt{5}\right) $ $ 0 $ $ -1 $ $ 3 $ $ \frac{1}{2} \left(1+\sqrt{5}\right) $ $ \frac{1}{2} \left(1-\sqrt{5}\right) $ $ 0 $ $ -1 $
$ G_g $ $ 4 $ $ -1 $ $ -1 $ $ 1 $ $ 0 $ $ 4 $ $ -1 $ $ -1 $ $ 1 $ $ 0 $
$ H_g $ $ 5 $ $ 0 $ $ 0 $ $ -1 $ $ 1 $ $ 5 $ $ 0 $ $ 0 $ $ -1 $ $ 1 $
$ A_u $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ 1 $ $ -1 $ $ -1 $ $ -1 $ $ -1 $ $ -1 $
$ T_{1u} $ $ 3 $ $ \frac{1}{2} \left(1+\sqrt{5}\right) $ $ \frac{1}{2} \left(1-\sqrt{5}\right) $ $ 0 $ $ -1 $ $ -3 $ $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ $ 0 $ $ 1 $
$ T_{2u} $ $ 3 $ $ \frac{1}{2} \left(1-\sqrt{5}\right) $ $ \frac{1}{2} \left(1+\sqrt{5}\right) $ $ 0 $ $ -1 $ $ -3 $ $ \frac{1}{2} \left(-1-\sqrt{5}\right) $ $ \frac{1}{2} \left(-1+\sqrt{5}\right) $ $ 0 $ $ 1 $
$ G_u $ $ 4 $ $ -1 $ $ -1 $ $ 1 $ $ 0 $ $ -4 $ $ 1 $ $ 1 $ $ -1 $ $ 0 $
$ H_u $ $ 5 $ $ 0 $ $ 0 $ $ -1 $ $ 1 $ $ -5 $ $ 0 $ $ 0 $ $ 1 $ $ -1 $

Product Table

$ $ $ A_g $ $ T_{1g} $ $ T_{2g} $ $ G_g $ $ H_g $ $ A_u $ $ T_{1u} $ $ T_{2u} $ $ G_u $ $ H_u $
$ A_g $ $ A_g $ $ T_{1g} $ $ T_{2g} $ $ G_g $ $ H_g $ $ A_u $ $ T_{1u} $ $ T_{2u} $ $ G_u $ $ H_u $
$ T_{1g} $ $ T_{1g} $ $ A_g+H_g+T_{1g} $ $ G_g+H_g $ $ G_g+H_g+T_{2g} $ $ G_g+H_g+T_{1g}+T_{2g} $ $ T_{1u} $ $ A_u+H_u+T_{1u} $ $ G_u+H_u $ $ G_u+H_u+T_{2u} $ $ G_u+H_u+T_{1u}+T_{2u} $
$ T_{2g} $ $ T_{2g} $ $ G_g+H_g $ $ A_g+H_g+T_{2g} $ $ G_g+H_g+T_{1g} $ $ G_g+H_g+T_{1g}+T_{2g} $ $ T_{2u} $ $ G_u+H_u $ $ A_u+H_u+T_{2u} $ $ G_u+H_u+T_{1u} $ $ G_u+H_u+T_{1u}+T_{2u} $
$ G_g $ $ G_g $ $ G_g+H_g+T_{2g} $ $ G_g+H_g+T_{1g} $ $ A_g+G_g+H_g+T_{1g}+T_{2g} $ $ G_g+2 H_g+T_{1g}+T_{2g} $ $ G_u $ $ G_u+H_u+T_{2u} $ $ G_u+H_u+T_{1u} $ $ A_u+G_u+H_u+T_{1u}+T_{2u} $ $ G_u+2 H_u+T_{1u}+T_{2u} $
$ H_g $ $ H_g $ $ G_g+H_g+T_{1g}+T_{2g} $ $ G_g+H_g+T_{1g}+T_{2g} $ $ G_g+2 H_g+T_{1g}+T_{2g} $ $ A_g+2 G_g+2 H_g+T_{1g}+T_{2g} $ $ H_u $ $ G_u+H_u+T_{1u}+T_{2u} $ $ G_u+H_u+T_{1u}+T_{2u} $ $ G_u+2 H_u+T_{1u}+T_{2u} $ $ A_u+2 G_u+2 H_u+T_{1u}+T_{2u} $
$ A_u $ $ A_u $ $ T_{1u} $ $ T_{2u} $ $ G_u $ $ H_u $ $ A_g $ $ T_{1g} $ $ T_{2g} $ $ G_g $ $ H_g $
$ T_{1u} $ $ T_{1u} $ $ A_u+H_u+T_{1u} $ $ G_u+H_u $ $ G_u+H_u+T_{2u} $ $ G_u+H_u+T_{1u}+T_{2u} $ $ T_{1g} $ $ A_g+H_g+T_{1g} $ $ G_g+H_g $ $ G_g+H_g+T_{2g} $ $ G_g+H_g+T_{1g}+T_{2g} $
$ T_{2u} $ $ T_{2u} $ $ G_u+H_u $ $ A_u+H_u+T_{2u} $ $ G_u+H_u+T_{1u} $ $ G_u+H_u+T_{1u}+T_{2u} $ $ T_{2g} $ $ G_g+H_g $ $ A_g+H_g+T_{2g} $ $ G_g+H_g+T_{1g} $ $ G_g+H_g+T_{1g}+T_{2g} $
$ G_u $ $ G_u $ $ G_u+H_u+T_{2u} $ $ G_u+H_u+T_{1u} $ $ A_u+G_u+H_u+T_{1u}+T_{2u} $ $ G_u+2 H_u+T_{1u}+T_{2u} $ $ G_g $ $ G_g+H_g+T_{2g} $ $ G_g+H_g+T_{1g} $ $ A_g+G_g+H_g+T_{1g}+T_{2g} $ $ G_g+2 H_g+T_{1g}+T_{2g} $
$ H_u $ $ H_u $ $ G_u+H_u+T_{1u}+T_{2u} $ $ G_u+H_u+T_{1u}+T_{2u} $ $ G_u+2 H_u+T_{1u}+T_{2u} $ $ A_u+2 G_u+2 H_u+T_{1u}+T_{2u} $ $ H_g $ $ G_g+H_g+T_{1g}+T_{2g} $ $ G_g+H_g+T_{1g}+T_{2g} $ $ G_g+2 H_g+T_{1g}+T_{2g} $ $ A_g+2 G_g+2 H_g+T_{1g}+T_{2g} $

Implemented Settings

Ih_xyz

Setting xyz

Details of the Ih group in with setting xyz

Operator Orientation
$\text{E}$ $\{0,0,0\}$ ,
$C_5$ $\left\{1+\sqrt{5},2,0\right\}$ , $\left\{-1-\sqrt{5},-2,0\right\}$ , $\left\{1+\sqrt{5},-2,0\right\}$ , $\left\{-1-\sqrt{5},2,0\right\}$ , $\left\{0,1+\sqrt{5},2\right\}$ , $\left\{0,-1-\sqrt{5},-2\right\}$ , $\left\{0,1+\sqrt{5},-2\right\}$ , $\left\{0,-1-\sqrt{5},2\right\}$ , $\left\{2,0,1+\sqrt{5}\right\}$ , $\left\{-2,0,-1-\sqrt{5}\right\}$ , $\left\{-2,0,1+\sqrt{5}\right\}$ , $\left\{2,0,-1-\sqrt{5}\right\}$ ,
$C_5^2$ $\left\{1+\sqrt{5},2,0\right\}$ , $\left\{-1-\sqrt{5},-2,0\right\}$ , $\left\{1+\sqrt{5},-2,0\right\}$ , $\left\{-1-\sqrt{5},2,0\right\}$ , $\left\{0,1+\sqrt{5},2\right\}$ , $\left\{0,-1-\sqrt{5},-2\right\}$ , $\left\{0,1+\sqrt{5},-2\right\}$ , $\left\{0,-1-\sqrt{5},2\right\}$ , $\left\{2,0,1+\sqrt{5}\right\}$ , $\left\{-2,0,-1-\sqrt{5}\right\}$ , $\left\{-2,0,1+\sqrt{5}\right\}$ , $\left\{2,0,-1-\sqrt{5}\right\}$ ,
$C_3$ $\{-1,-1,-1\}$ , $\left\{0,\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$ , $\left\{0,\frac{1}{2} \left(\sqrt{5}-1\right),\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$ , $\{1,1,1\}$ , $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),0,\frac{1}{2} \left(1-\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1-\sqrt{5}\right),0\right\}$ , $\left\{\frac{1}{2} \left(\sqrt{5}-1\right),\frac{1}{2} \left(1+\sqrt{5}\right),0\right\}$ , $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),0,\frac{1}{2} \left(\sqrt{5}-1\right)\right\}$ , $\left\{\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),-\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},-\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),-\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},-\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{-\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),-\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{-\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),-\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{-\frac{10 \left(-4 \sqrt{25-5 \sqrt{5}}-10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},-\frac{10 \left(7 \sqrt{25-5 \sqrt{5}}+15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right)\right\}$ , $\left\{\frac{10 \left(-4 \sqrt{25-5 \sqrt{5}}-10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{10 \left(7 \sqrt{25-5 \sqrt{5}}+15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right)\right\}$ , $\left\{-\frac{10 \left(4 \sqrt{25-5 \sqrt{5}}+10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},-\frac{10 \left(-7 \sqrt{25-5 \sqrt{5}}-15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right)\right\}$ , $\left\{\frac{10 \left(4 \sqrt{25-5 \sqrt{5}}+10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{10 \left(-7 \sqrt{25-5 \sqrt{5}}-15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right)\right\}$ ,
$C_2$ $\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , $\left\{10 \left(2+\sqrt{5}\right),-5 \left(1+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$ , $\left\{10 \left(2+\sqrt{5}\right),-5 \left(1+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$ , $\left\{10 \left(2+\sqrt{5}\right),5 \left(1+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$ , $\left\{10 \left(2+\sqrt{5}\right),5 \left(1+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$ , $\left\{-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$ , $\left\{-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$ , $\left\{\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$ , $\left\{\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$ , $\left\{-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$ , $\left\{-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$ ,
$\text{i}$ $\{0,0,0\}$ ,
$S_{10}$ $\left\{1+\sqrt{5},2,0\right\}$ , $\left\{-1-\sqrt{5},-2,0\right\}$ , $\left\{1+\sqrt{5},-2,0\right\}$ , $\left\{-1-\sqrt{5},2,0\right\}$ , $\left\{0,1+\sqrt{5},2\right\}$ , $\left\{0,-1-\sqrt{5},-2\right\}$ , $\left\{0,1+\sqrt{5},-2\right\}$ , $\left\{0,-1-\sqrt{5},2\right\}$ , $\left\{2,0,1+\sqrt{5}\right\}$ , $\left\{-2,0,-1-\sqrt{5}\right\}$ , $\left\{-2,0,1+\sqrt{5}\right\}$ , $\left\{2,0,-1-\sqrt{5}\right\}$ ,
$S_{10}^3$ $\left\{1+\sqrt{5},2,0\right\}$ , $\left\{-1-\sqrt{5},-2,0\right\}$ , $\left\{1+\sqrt{5},-2,0\right\}$ , $\left\{-1-\sqrt{5},2,0\right\}$ , $\left\{0,1+\sqrt{5},2\right\}$ , $\left\{0,-1-\sqrt{5},-2\right\}$ , $\left\{0,1+\sqrt{5},-2\right\}$ , $\left\{0,-1-\sqrt{5},2\right\}$ , $\left\{2,0,1+\sqrt{5}\right\}$ , $\left\{-2,0,-1-\sqrt{5}\right\}$ , $\left\{-2,0,1+\sqrt{5}\right\}$ , $\left\{2,0,-1-\sqrt{5}\right\}$ ,
$S_6$ $\{-1,-1,-1\}$ , $\left\{0,\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$ , $\left\{0,\frac{1}{2} \left(\sqrt{5}-1\right),\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$ , $\{1,1,1\}$ , $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),0,\frac{1}{2} \left(1-\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1-\sqrt{5}\right),0\right\}$ , $\left\{\frac{1}{2} \left(\sqrt{5}-1\right),\frac{1}{2} \left(1+\sqrt{5}\right),0\right\}$ , $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),0,\frac{1}{2} \left(\sqrt{5}-1\right)\right\}$ , $\left\{\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),-\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},-\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),-\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},-\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{-\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),-\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{-\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),-\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$ , $\left\{-\frac{10 \left(-4 \sqrt{25-5 \sqrt{5}}-10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},-\frac{10 \left(7 \sqrt{25-5 \sqrt{5}}+15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right)\right\}$ , $\left\{\frac{10 \left(-4 \sqrt{25-5 \sqrt{5}}-10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{10 \left(7 \sqrt{25-5 \sqrt{5}}+15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right)\right\}$ , $\left\{-\frac{10 \left(4 \sqrt{25-5 \sqrt{5}}+10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},-\frac{10 \left(-7 \sqrt{25-5 \sqrt{5}}-15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right)\right\}$ , $\left\{\frac{10 \left(4 \sqrt{25-5 \sqrt{5}}+10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{10 \left(-7 \sqrt{25-5 \sqrt{5}}-15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right)\right\}$ ,
$\sigma _h$ $\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , $\left\{10 \left(2+\sqrt{5}\right),-5 \left(1+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$ , $\left\{10 \left(2+\sqrt{5}\right),-5 \left(1+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$ , $\left\{10 \left(2+\sqrt{5}\right),5 \left(1+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$ , $\left\{10 \left(2+\sqrt{5}\right),5 \left(1+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$ , $\left\{-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$ , $\left\{-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$ , $\left\{\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$ , $\left\{\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$ , $\left\{-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$ , $\left\{-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$ , $\left\{\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$ ,

Table of several point groups

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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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