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--- physics_chemistry:point_groups:ih [2018/03/21 18:50] – created Stefano Agrestini
+++ physics_chemistry:point_groups:ih [2018/03/23 10:52] (current) – Stefano Agrestini
@@ Line 1: / Line 1: @@
-====== Ih ======
+====== Point Group Ih ======
+===== Character Table =====
 ###
-alligned paragraph text
+|    ^  $ \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)}} $  ^
+^  $A_g$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |
+^  $T_{1g}$  |   $3$  |   $\frac{1}{2} \left(1+\sqrt{5}\right)$  |   $\frac{1}{2} \left(1-\sqrt{5}\right)$  |   $0$  |   $-1$  |   $3$  |   $\frac{1}{2} \left(1-\sqrt{5}\right)$  |   $\frac{1}{2} \left(1+\sqrt{5}\right)$  |   $0$  |   $-1$  |
+^  $T_{2g}$  |   $3$  |   $\frac{1}{2} \left(1-\sqrt{5}\right)$  |   $\frac{1}{2} \left(1+\sqrt{5}\right)$  |   $0$  |   $-1$  |   $3$  |   $\frac{1}{2} \left(1+\sqrt{5}\right)$  |   $\frac{1}{2} \left(1-\sqrt{5}\right)$  |   $0$  |   $-1$  |
+^  $G_g$  |   $4$  |   $-1$  |   $-1$  |   $1$  |   $0$  |   $4$  |   $-1$  |   $-1$  |   $1$  |   $0$  |
+^  $H_g$  |   $5$  |   $0$  |   $0$  |   $-1$  |   $1$  |   $5$  |   $0$  |   $0$  |   $-1$  |   $1$  |
+^  $A_u$  |   $1$  |   $1$  |   $1$  |   $1$  |   $1$  |   $-1$  |   $-1$  |   $-1$  |   $-1$  |   $-1$  |
+^  $T_{1u}$  |   $3$  |   $\frac{1}{2} \left(1+\sqrt{5}\right)$  |   $\frac{1}{2} \left(1-\sqrt{5}\right)$  |   $0$  |   $-1$  |   $-3$  |   $\frac{1}{2} \left(-1+\sqrt{5}\right)$  |   $\frac{1}{2} \left(-1-\sqrt{5}\right)$  |   $0$  |   $1$  |
+^  $T_{2u}$  |   $3$  |   $\frac{1}{2} \left(1-\sqrt{5}\right)$  |   $\frac{1}{2} \left(1+\sqrt{5}\right)$  |   $0$  |   $-1$  |   $-3$  |   $\frac{1}{2} \left(-1-\sqrt{5}\right)$  |   $\frac{1}{2} \left(-1+\sqrt{5}\right)$  |   $0$  |   $1$  |
+^  $G_u$  |   $4$  |   $-1$  |   $-1$  |   $1$  |   $0$  |   $-4$  |   $1$  |   $1$  |   $-1$  |   $0$  |
+^  $H_u$  |   $5$  |   $0$  |   $0$  |   $-1$  |   $1$  |   $-5$  |   $0$  |   $0$  |   $1$  |   $-1$  |
 ###
-===== Example =====
+===== Product Table =====
 ###
-description text
+|    ^   $A_g$   ^   $T_{1g}$   ^   $T_{2g}$   ^   $G_g$   ^   $H_g$   ^   $A_u$   ^   $T_{1u}$   ^   $T_{2u}$   ^   $G_u$   ^   $H_u$   ^
+^  $A_g$   |  $A_g$   |  $T_{1g}$   |  $T_{2g}$   |  $G_g$   |  $H_g$   |  $A_u$   |  $T_{1u}$   |  $T_{2u}$   |  $G_u$   |  $H_u$   |
+^  $T_{1g}$   |  $T_{1g}$   |  $A_g+H_g+T_{1g}$   |  $G_g+H_g$   |  $G_g+H_g+T_{2g}$   |  $G_g+H_g+T_{1g}+T_{2g}$   |  $T_{1u}$   |  $A_u+H_u+T_{1u}$   |  $G_u+H_u$   |  $G_u+H_u+T_{2u}$   |  $G_u+H_u+T_{1u}+T_{2u}$   |
+^  $T_{2g}$   |  $T_{2g}$   |  $G_g+H_g$   |  $A_g+H_g+T_{2g}$   |  $G_g+H_g+T_{1g}$   |  $G_g+H_g+T_{1g}+T_{2g}$   |  $T_{2u}$   |  $G_u+H_u$   |  $A_u+H_u+T_{2u}$   |  $G_u+H_u+T_{1u}$   |  $G_u+H_u+T_{1u}+T_{2u}$   |
+^  $G_g$   |  $G_g$   |  $G_g+H_g+T_{2g}$   |  $G_g+H_g+T_{1g}$   |  $A_g+G_g+H_g+T_{1g}+T_{2g}$   |  $G_g+2 H_g+T_{1g}+T_{2g}$   |  $G_u$   |  $G_u+H_u+T_{2u}$   |  $G_u+H_u+T_{1u}$   |  $A_u+G_u+H_u+T_{1u}+T_{2u}$   |  $G_u+2 H_u+T_{1u}+T_{2u}$   |
+^  $H_g$   |  $H_g$   |  $G_g+H_g+T_{1g}+T_{2g}$   |  $G_g+H_g+T_{1g}+T_{2g}$   |  $G_g+2 H_g+T_{1g}+T_{2g}$   |  $A_g+2 G_g+2 H_g+T_{1g}+T_{2g}$   |  $H_u$   |  $G_u+H_u+T_{1u}+T_{2u}$   |  $G_u+H_u+T_{1u}+T_{2u}$   |  $G_u+2 H_u+T_{1u}+T_{2u}$   |  $A_u+2 G_u+2 H_u+T_{1u}+T_{2u}$   |
+^  $A_u$   |  $A_u$   |  $T_{1u}$   |  $T_{2u}$   |  $G_u$   |  $H_u$   |  $A_g$   |  $T_{1g}$   |  $T_{2g}$   |  $G_g$   |  $H_g$   |
+^  $T_{1u}$   |  $T_{1u}$   |  $A_u+H_u+T_{1u}$   |  $G_u+H_u$   |  $G_u+H_u+T_{2u}$   |  $G_u+H_u+T_{1u}+T_{2u}$   |  $T_{1g}$   |  $A_g+H_g+T_{1g}$   |  $G_g+H_g$   |  $G_g+H_g+T_{2g}$   |  $G_g+H_g+T_{1g}+T_{2g}$   |
+^  $T_{2u}$   |  $T_{2u}$   |  $G_u+H_u$   |  $A_u+H_u+T_{2u}$   |  $G_u+H_u+T_{1u}$   |  $G_u+H_u+T_{1u}+T_{2u}$   |  $T_{2g}$   |  $G_g+H_g$   |  $A_g+H_g+T_{2g}$   |  $G_g+H_g+T_{1g}$   |  $G_g+H_g+T_{1g}+T_{2g}$   |
+^  $G_u$   |  $G_u$   |  $G_u+H_u+T_{2u}$   |  $G_u+H_u+T_{1u}$   |  $A_u+G_u+H_u+T_{1u}+T_{2u}$   |  $G_u+2 H_u+T_{1u}+T_{2u}$   |  $G_g$   |  $G_g+H_g+T_{2g}$   |  $G_g+H_g+T_{1g}$   |  $A_g+G_g+H_g+T_{1g}+T_{2g}$   |  $G_g+2 H_g+T_{1g}+T_{2g}$   |
+^  $H_u$   |  $H_u$   |  $G_u+H_u+T_{1u}+T_{2u}$   |  $G_u+H_u+T_{1u}+T_{2u}$   |  $G_u+2 H_u+T_{1u}+T_{2u}$   |  $A_u+2 G_u+2 H_u+T_{1u}+T_{2u}$   |  $H_g$   |  $G_g+H_g+T_{1g}+T_{2g}$   |  $G_g+H_g+T_{1g}+T_{2g}$   |  $G_g+2 H_g+T_{1g}+T_{2g}$   |  $A_g+2 G_g+2 H_g+T_{1g}+T_{2g}$   |
 ###
-==== Input ====
+===== Implemented Settings =====
-<code Quanty Example.Quanty>
--- some example code
-</code>
-==== Result ====
+[[physics_chemistry:point_groups:ih:orientation_xyz| Ih_xyz ]]
-<WRAP center box 100%>
-text produced as output
-</WRAP>
-===== Table of contents =====
+==== Setting xyz ====
-{{indexmenu>.#1}}
+[[physics_chemistry:point_groups:ih:orientation_xyz|Details of the Ih group in with setting xyz]]
+{{:physics_chemistry:pointgroup:ih_xyz.png }}
+###
+^ Operator ^ Orientation ^
+^  $\text{E}$  |  $\{0,0,0\}$  , |
+^  $C_5$  |  $\left\{1+\sqrt{5},2,0\right\}$  ,  $\left\{-1-\sqrt{5},-2,0\right\}$  ,  $\left\{1+\sqrt{5},-2,0\right\}$  ,  $\left\{-1-\sqrt{5},2,0\right\}$  ,  $\left\{0,1+\sqrt{5},2\right\}$  ,  $\left\{0,-1-\sqrt{5},-2\right\}$  ,  $\left\{0,1+\sqrt{5},-2\right\}$  ,  $\left\{0,-1-\sqrt{5},2\right\}$  ,  $\left\{2,0,1+\sqrt{5}\right\}$  ,  $\left\{-2,0,-1-\sqrt{5}\right\}$  ,  $\left\{-2,0,1+\sqrt{5}\right\}$  ,  $\left\{2,0,-1-\sqrt{5}\right\}$  , |
+^  $C_5^2$  |  $\left\{1+\sqrt{5},2,0\right\}$  ,  $\left\{-1-\sqrt{5},-2,0\right\}$  ,  $\left\{1+\sqrt{5},-2,0\right\}$  ,  $\left\{-1-\sqrt{5},2,0\right\}$  ,  $\left\{0,1+\sqrt{5},2\right\}$  ,  $\left\{0,-1-\sqrt{5},-2\right\}$  ,  $\left\{0,1+\sqrt{5},-2\right\}$  ,  $\left\{0,-1-\sqrt{5},2\right\}$  ,  $\left\{2,0,1+\sqrt{5}\right\}$  ,  $\left\{-2,0,-1-\sqrt{5}\right\}$  ,  $\left\{-2,0,1+\sqrt{5}\right\}$  ,  $\left\{2,0,-1-\sqrt{5}\right\}$  , |
+^  $C_3$  |  $\{-1,-1,-1\}$  ,  $\left\{0,\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$  ,  $\left\{0,\frac{1}{2} \left(\sqrt{5}-1\right),\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$  ,  $\{1,1,1\}$  ,  $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),0,\frac{1}{2} \left(1-\sqrt{5}\right)\right\}$  ,  $\left\{\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1-\sqrt{5}\right),0\right\}$  ,  $\left\{\frac{1}{2} \left(\sqrt{5}-1\right),\frac{1}{2} \left(1+\sqrt{5}\right),0\right\}$  ,  $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),0,\frac{1}{2} \left(\sqrt{5}-1\right)\right\}$  ,  $\left\{\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$  ,  $\left\{\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),-\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$  ,  $\left\{\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},-\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$  ,  $\left\{\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),-\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},-\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$  ,  $\left\{-\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),-\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$  ,  $\left\{\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$  ,  $\left\{-\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$  ,  $\left\{\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),-\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$  ,  $\left\{-\frac{10 \left(-4 \sqrt{25-5 \sqrt{5}}-10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},-\frac{10 \left(7 \sqrt{25-5 \sqrt{5}}+15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right)\right\}$  ,  $\left\{\frac{10 \left(-4 \sqrt{25-5 \sqrt{5}}-10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{10 \left(7 \sqrt{25-5 \sqrt{5}}+15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right)\right\}$  ,  $\left\{-\frac{10 \left(4 \sqrt{25-5 \sqrt{5}}+10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},-\frac{10 \left(-7 \sqrt{25-5 \sqrt{5}}-15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right)\right\}$  ,  $\left\{\frac{10 \left(4 \sqrt{25-5 \sqrt{5}}+10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{10 \left(-7 \sqrt{25-5 \sqrt{5}}-15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right)\right\}$  , |
+^  $C_2$  |  $\{0,0,1\}$  ,  $\{0,1,0\}$  ,  $\{1,0,0\}$  ,  $\left\{10 \left(2+\sqrt{5}\right),-5 \left(1+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$  ,  $\left\{10 \left(2+\sqrt{5}\right),-5 \left(1+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$  ,  $\left\{10 \left(2+\sqrt{5}\right),5 \left(1+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$  ,  $\left\{10 \left(2+\sqrt{5}\right),5 \left(1+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$  ,  $\left\{-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$  ,  $\left\{-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$  ,  $\left\{\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$  ,  $\left\{\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$  ,  $\left\{-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$  ,  $\left\{-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$  ,  $\left\{\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$  ,  $\left\{\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$  , |
+^  $\text{i}$  |  $\{0,0,0\}$  , |
+^  $S_{10}$  |  $\left\{1+\sqrt{5},2,0\right\}$  ,  $\left\{-1-\sqrt{5},-2,0\right\}$  ,  $\left\{1+\sqrt{5},-2,0\right\}$  ,  $\left\{-1-\sqrt{5},2,0\right\}$  ,  $\left\{0,1+\sqrt{5},2\right\}$  ,  $\left\{0,-1-\sqrt{5},-2\right\}$  ,  $\left\{0,1+\sqrt{5},-2\right\}$  ,  $\left\{0,-1-\sqrt{5},2\right\}$  ,  $\left\{2,0,1+\sqrt{5}\right\}$  ,  $\left\{-2,0,-1-\sqrt{5}\right\}$  ,  $\left\{-2,0,1+\sqrt{5}\right\}$  ,  $\left\{2,0,-1-\sqrt{5}\right\}$  , |
+^  $S_{10}^3$  |  $\left\{1+\sqrt{5},2,0\right\}$  ,  $\left\{-1-\sqrt{5},-2,0\right\}$  ,  $\left\{1+\sqrt{5},-2,0\right\}$  ,  $\left\{-1-\sqrt{5},2,0\right\}$  ,  $\left\{0,1+\sqrt{5},2\right\}$  ,  $\left\{0,-1-\sqrt{5},-2\right\}$  ,  $\left\{0,1+\sqrt{5},-2\right\}$  ,  $\left\{0,-1-\sqrt{5},2\right\}$  ,  $\left\{2,0,1+\sqrt{5}\right\}$  ,  $\left\{-2,0,-1-\sqrt{5}\right\}$  ,  $\left\{-2,0,1+\sqrt{5}\right\}$  ,  $\left\{2,0,-1-\sqrt{5}\right\}$  , |
+^  $S_6$  |  $\{-1,-1,-1\}$  ,  $\left\{0,\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1-\sqrt{5}\right)\right\}$  ,  $\left\{0,\frac{1}{2} \left(\sqrt{5}-1\right),\frac{1}{2} \left(1+\sqrt{5}\right)\right\}$  ,  $\{1,1,1\}$  ,  $\left\{\frac{1}{2} \left(-1-\sqrt{5}\right),0,\frac{1}{2} \left(1-\sqrt{5}\right)\right\}$  ,  $\left\{\frac{1}{2} \left(1-\sqrt{5}\right),\frac{1}{2} \left(-1-\sqrt{5}\right),0\right\}$  ,  $\left\{\frac{1}{2} \left(\sqrt{5}-1\right),\frac{1}{2} \left(1+\sqrt{5}\right),0\right\}$  ,  $\left\{\frac{1}{2} \left(1+\sqrt{5}\right),0,\frac{1}{2} \left(\sqrt{5}-1\right)\right\}$  ,  $\left\{\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$  ,  $\left\{\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),-\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$  ,  $\left\{\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},-\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$  ,  $\left\{\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),-\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)},-\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)}\right\}$  ,  $\left\{-\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),-\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$  ,  $\left\{\frac{-3 \sqrt{30-10 \sqrt{5}}-5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),\frac{20+8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$  ,  $\left\{-\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right),\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$  ,  $\left\{\frac{3 \sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}+2 \left(5+\sqrt{5}\right)}{4 \left(5+\sqrt{5}\right)},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right),-\frac{-20-8 \sqrt{5}+\sqrt{30-10 \sqrt{5}}+5 \sqrt{6-2 \sqrt{5}}}{4 \left(5+\sqrt{5}\right)}\right\}$  ,  $\left\{-\frac{10 \left(-4 \sqrt{25-5 \sqrt{5}}-10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},-\frac{10 \left(7 \sqrt{25-5 \sqrt{5}}+15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right)\right\}$  ,  $\left\{\frac{10 \left(-4 \sqrt{25-5 \sqrt{5}}-10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{10 \left(7 \sqrt{25-5 \sqrt{5}}+15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right)\right\}$  ,  $\left\{-\frac{10 \left(4 \sqrt{25-5 \sqrt{5}}+10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},-\frac{10 \left(-7 \sqrt{25-5 \sqrt{5}}-15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(-\sqrt{30-10 \sqrt{5}}+\sqrt{6-2 \sqrt{5}}+2 \left(1+\sqrt{5}\right)\right)\right\}$  ,  $\left\{\frac{10 \left(4 \sqrt{25-5 \sqrt{5}}+10 \sqrt{5-\sqrt{5}}+\sqrt{5+\sqrt{5}} \left(25+11 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{10 \left(-7 \sqrt{25-5 \sqrt{5}}-15 \sqrt{5-\sqrt{5}}+2 \sqrt{5+\sqrt{5}} \left(5+2 \sqrt{5}\right)\right)}{\left(5+\sqrt{5}\right)^{7/2}},\frac{1}{8} \left(\sqrt{30-10 \sqrt{5}}-\sqrt{6-2 \sqrt{5}}-2 \left(1+\sqrt{5}\right)\right)\right\}$  , |
+^  $\sigma _h$  |  $\{0,0,1\}$  ,  $\{0,1,0\}$  ,  $\{1,0,0\}$  ,  $\left\{10 \left(2+\sqrt{5}\right),-5 \left(1+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$  ,  $\left\{10 \left(2+\sqrt{5}\right),-5 \left(1+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$  ,  $\left\{10 \left(2+\sqrt{5}\right),5 \left(1+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$  ,  $\left\{10 \left(2+\sqrt{5}\right),5 \left(1+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2\right\}$  ,  $\left\{-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$  ,  $\left\{-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$  ,  $\left\{\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$  ,  $\left\{\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right),\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right)\right\}$  ,  $\left\{-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$  ,  $\left\{-\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$  ,  $\left\{\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),-\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$  ,  $\left\{\frac{1}{2} \left(5+\sqrt{5}\right)^2,\frac{1}{2} \left(5+\sqrt{5}\right) \left(5+3 \sqrt{5}\right),\frac{1}{4} \left(\sqrt{5}-5\right) \left(1+\sqrt{5}\right) \left(5+\sqrt{5}\right)\right\}$  , |
+###
+===== Table of several point groups =====
+###
+[[physics_chemistry:point_groups|Return to Main page on Point Groups]]
+###
+###
+^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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