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−Table of Contents

Point Group Ih

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

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	C_s	C_i
C_n groups	C₂	C₃	C₄	C₅	C₆	C₇	C₈
D_n groups	D₂	D₃	D₄	D₅	D₆	D₇	D₈
C_nv groups	C_2v	C_3v	C_4v	C_5v	C_6v	C_7v	C_8v
C_nh groups	C_2h	C_3h	C_4h	C_5h	C_6h
D_nh groups	D_2h	D_3h	D_4h	D_5h	D_6h	D_7h	D_8h
D_nd groups	D_2d	D_3d	D_4d	D_5d	D_6d	D_7d	D_8d
S_n groups	S₂	S₄	S₆	S₈	S₁₀	S₁₂
Cubic groups	T	T_h	T_d	O	O_h	I	I_h
Linear groups	C_{$\infty$ v}	D_{$\infty$ h}

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