====== 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 =====

[[physics_chemistry:point_groups:ih:orientation_xyz| Ih_xyz ]] 

==== Setting xyz ====

[[physics_chemistry:point_groups:ih:orientation_xyz|Details of the Ih group in with setting xyz]]

{{:physics_chemistry:pointgroup:ih_xyz.png }}

###

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

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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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