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−Table of Contents

Point Group D5h

Point Group D5h

Character Table

	$\text{E} \,{\text{(1)}}$	$C_5 \,{\text{(2)}}$	$C_5^2{} \,{\text{(2)}}$	$C_2 \,{\text{(5)}}$	$\sigma_h \,{\text{(1)}}$	$S_5 \,{\text{(2)}}$	$S_5^3{} \,{\text{(2)}}$	$\sigma_v \,{\text{(5)}}$
$A'_1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$A''_1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
$A'_2$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$
$A''_2$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$
$E'_1$	$2$	$\frac{1}{2} \left(-1+\sqrt{5}\right)$	$\frac{1}{2} \left(-1-\sqrt{5}\right)$	$0$	$2$	$\frac{1}{2} \left(-1+\sqrt{5}\right)$	$\frac{1}{2} \left(-1-\sqrt{5}\right)$	$0$
$E''_1$	$2$	$\frac{1}{2} \left(-1+\sqrt{5}\right)$	$\frac{1}{2} \left(-1-\sqrt{5}\right)$	$0$	$-2$	$\frac{1}{2} \left(1-\sqrt{5}\right)$	$\frac{1}{2} \left(1+\sqrt{5}\right)$	$0$
$E'_2$	$2$	$\frac{1}{2} \left(-1-\sqrt{5}\right)$	$\frac{1}{2} \left(-1+\sqrt{5}\right)$	$0$	$2$	$\frac{1}{2} \left(-1-\sqrt{5}\right)$	$\frac{1}{2} \left(-1+\sqrt{5}\right)$	$0$
$E''_2$	$2$	$\frac{1}{2} \left(-1-\sqrt{5}\right)$	$\frac{1}{2} \left(-1+\sqrt{5}\right)$	$0$	$-2$	$\frac{1}{2} \left(1+\sqrt{5}\right)$	$\frac{1}{2} \left(1-\sqrt{5}\right)$	$0$

Product Table

	$A'_1$	$A''_1$	$A'_2$	$A''_2$	$E'_1$	$E''_1$	$E'_2$	$E''_2$
$A'_1$	$A'_1$	$A''_1$	$A'_2$	$A''_2$	$E'_1$	$E''_1$	$E'_2$	$E''_2$
$A''_1$	$A''_1$	$A'_1$	$A''_2$	$A'_2$	$E''_1$	$E'_1$	$E''_2$	$E'_2$
$A'_2$	$A'_2$	$A''_2$	$A'_1$	$A''_1$	$E'_1$	$E''_1$	$E'_2$	$E''_2$
$A''_2$	$A''_2$	$A'_2$	$A''_1$	$A'_1$	$E''_1$	$E'_1$	$E''_2$	$E'_2$
$E'_1$	$E'_1$	$E''_1$	$E'_1$	$E''_1$	$A'_1+A'_2+E'_2$	$A''_1+A''_2+E''_2$	$E'_1+E'_2$	$E''_1+E''_2$
$E''_1$	$E''_1$	$E'_1$	$E''_1$	$E'_1$	$A''_1+A''_2+E''_2$	$A'_1+A'_2+E'_2$	$E''_1+E''_2$	$E'_1+E'_2$
$E'_2$	$E'_2$	$E''_2$	$E'_2$	$E''_2$	$E'_1+E'_2$	$E''_1+E''_2$	$A'_1+A'_2+E'_1$	$A''_1+A''_2+E''_1$
$E''_2$	$E''_2$	$E'_2$	$E''_2$	$E'_2$	$E''_1+E''_2$	$E'_1+E'_2$	$A''_1+A''_2+E''_1$	$A'_1+A'_2+E'_1$

Implemented Settings

Setting Zx

Details of the D5h group in with setting Zx

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_5$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$C_5^2$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$C_2$	$\{1,0,0\}$ , $\left\{1,\sqrt{5+2 \sqrt{5}},0\right\}$ , $\left\{-\sqrt{1+\frac{2}{\sqrt{5}}},1,0\right\}$ , $\left\{\sqrt{1+\frac{2}{\sqrt{5}}},1,0\right\}$ , $\left\{1,-\sqrt{5+2 \sqrt{5}},0\right\}$ ,
$\sigma _h$	$\{0,0,1\}$ ,
$S_5$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$S_5^3$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$\sigma _v$	$\{0,1,0\}$ , $\left\{-\sqrt{5+2 \sqrt{5}},1,0\right\}$ , $\left\{1,\sqrt{1+\frac{2}{\sqrt{5}}},0\right\}$ , $\left\{1,-\sqrt{1+\frac{2}{\sqrt{5}}},0\right\}$ , $\left\{\sqrt{5+2 \sqrt{5}},1,0\right\}$ ,

Setting Zy

Details of the D5h group in with setting Zy

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_5$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$C_5^2$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$C_2$	$\{0,1,0\}$ , $\left\{-\sqrt{5+2 \sqrt{5}},1,0\right\}$ , $\left\{1,\sqrt{1+\frac{2}{\sqrt{5}}},0\right\}$ , $\left\{1,-\sqrt{1+\frac{2}{\sqrt{5}}},0\right\}$ , $\left\{\sqrt{5+2 \sqrt{5}},1,0\right\}$ ,
$\sigma _h$	$\{0,0,1\}$ ,
$S_5$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$S_5^3$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$\sigma _v$	$\{1,0,0\}$ , $\left\{1,\sqrt{5+2 \sqrt{5}},0\right\}$ , $\left\{-\sqrt{1+\frac{2}{\sqrt{5}}},1,0\right\}$ , $\left\{\sqrt{1+\frac{2}{\sqrt{5}}},1,0\right\}$ , $\left\{1,-\sqrt{5+2 \sqrt{5}},0\right\}$ ,

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}