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−Table of Contents

Point Group D6h

Point Group D6h

Character Table

	$\text{E} \,{\text{(1)}}$	$C_6 \,{\text{(2)}}$	$C_3 \,{\text{(2)}}$	$C_2 \,{\text{(1)}}$	$C_2' \,{\text{(3)}}$	$C_2\text{''} \,{\text{(3)}}$	$\text{i} \,{\text{(1)}}$	$S_6 \,{\text{(2)}}$	$S_3 \,{\text{(2)}}$	$\sigma_h \,{\text{(1)}}$	$\sigma_v \,{\text{(3)}}$	$\sigma_d \,{\text{(3)}}$
$A_{1g}$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$A_{2g}$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$
$B_{1g}$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$1$
$B_{2g}$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$
$E_{1g}$	$2$	$1$	$-1$	$-2$	$0$	$0$	$2$	$-1$	$1$	$-2$	$0$	$0$
$E_{2g}$	$2$	$-1$	$-1$	$2$	$0$	$0$	$2$	$-1$	$-1$	$2$	$0$	$0$
$A_{1u}$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
$A_{2u}$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
$B_{1u}$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$
$B_{2u}$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$
$E_{1u}$	$2$	$1$	$-1$	$-2$	$0$	$0$	$-2$	$1$	$-1$	$2$	$0$	$0$
$E_{2u}$	$2$	$-1$	$-1$	$2$	$0$	$0$	$-2$	$1$	$1$	$-2$	$0$	$0$

Product Table

	$A_{1g}$	$A_{2g}$	$B_{1g}$	$B_{2g}$	$E_{1g}$	$E_{2g}$	$A_{1u}$	$A_{2u}$	$B_{1u}$	$B_{2u}$	$E_{1u}$	$E_{2u}$
$A_{1g}$	$A_{1g}$	$A_{2g}$	$B_{1g}$	$B_{2g}$	$E_{1g}$	$E_{2g}$	$A_{1u}$	$A_{2u}$	$B_{1u}$	$B_{2u}$	$E_{1u}$	$E_{2u}$
$A_{2g}$	$A_{2g}$	$A_{1g}$	$B_{2g}$	$B_{1g}$	$E_{1g}$	$E_{2g}$	$A_{2u}$	$A_{1u}$	$B_{2u}$	$B_{1u}$	$E_{1u}$	$E_{2u}$
$B_{1g}$	$B_{1g}$	$B_{2g}$	$A_{1g}$	$A_{2g}$	$E_{2g}$	$E_{1g}$	$B_{1u}$	$B_{2u}$	$A_{1u}$	$A_{2u}$	$E_{2u}$	$E_{1u}$
$B_{2g}$	$B_{2g}$	$B_{1g}$	$A_{2g}$	$A_{1g}$	$E_{2g}$	$E_{1g}$	$B_{2u}$	$B_{1u}$	$A_{2u}$	$A_{1u}$	$E_{2u}$	$E_{1u}$
$E_{1g}$	$E_{1g}$	$E_{1g}$	$E_{2g}$	$E_{2g}$	$A_{1g}+A_{2g}+E_{2g}$	$B_{1g}+B_{2g}+E_{1g}$	$E_{1u}$	$E_{1u}$	$E_{2u}$	$E_{2u}$	$A_{1u}+A_{2u}+E_{2u}$	$B_{1u}+B_{2u}+E_{1u}$
$E_{2g}$	$E_{2g}$	$E_{2g}$	$E_{1g}$	$E_{1g}$	$B_{1g}+B_{2g}+E_{1g}$	$A_{1g}+A_{2g}+E_{2g}$	$E_{2u}$	$E_{2u}$	$E_{1u}$	$E_{1u}$	$B_{1u}+B_{2u}+E_{1u}$	$A_{1u}+A_{2u}+E_{2u}$
$A_{1u}$	$A_{1u}$	$A_{2u}$	$B_{1u}$	$B_{2u}$	$E_{1u}$	$E_{2u}$	$A_{1g}$	$A_{2g}$	$B_{1g}$	$B_{2g}$	$E_{1g}$	$E_{2g}$
$A_{2u}$	$A_{2u}$	$A_{1u}$	$B_{2u}$	$B_{1u}$	$E_{1u}$	$E_{2u}$	$A_{2g}$	$A_{1g}$	$B_{2g}$	$B_{1g}$	$E_{1g}$	$E_{2g}$
$B_{1u}$	$B_{1u}$	$B_{2u}$	$A_{1u}$	$A_{2u}$	$E_{2u}$	$E_{1u}$	$B_{1g}$	$B_{2g}$	$A_{1g}$	$A_{2g}$	$E_{2g}$	$E_{1g}$
$B_{2u}$	$B_{2u}$	$B_{1u}$	$A_{2u}$	$A_{1u}$	$E_{2u}$	$E_{1u}$	$B_{2g}$	$B_{1g}$	$A_{2g}$	$A_{1g}$	$E_{2g}$	$E_{1g}$
$E_{1u}$	$E_{1u}$	$E_{1u}$	$E_{2u}$	$E_{2u}$	$A_{1u}+A_{2u}+E_{2u}$	$B_{1u}+B_{2u}+E_{1u}$	$E_{1g}$	$E_{1g}$	$E_{2g}$	$E_{2g}$	$A_{1g}+A_{2g}+E_{2g}$	$B_{1g}+B_{2g}+E_{1g}$
$E_{2u}$	$E_{2u}$	$E_{2u}$	$E_{1u}$	$E_{1u}$	$B_{1u}+B_{2u}+E_{1u}$	$A_{1u}+A_{2u}+E_{2u}$	$E_{2g}$	$E_{2g}$	$E_{1g}$	$E_{1g}$	$B_{1g}+B_{2g}+E_{1g}$	$A_{1g}+A_{2g}+E_{2g}$

Implemented Settings

Setting Zx

Details of the D6h group in with setting Zx

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_6$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$C_3$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$C_2$	$\{0,0,1\}$ ,
$C_2'$	$\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{1,-\sqrt{3},0\right\}$ ,
$C_2\text{''}$	$\{0,1,0\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},-1,0\right\}$ ,
$\text{i}$	$\{0,0,0\}$ ,
$S_6$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$S_3$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$\sigma _h$	$\{0,0,1\}$ ,
$\sigma _v$	$\{0,1,0\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},-1,0\right\}$ ,
$\sigma _d$	$\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{1,-\sqrt{3},0\right\}$ ,

Setting Zy

Details of the D6h group in with setting Zy

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_6$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$C_3$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$C_2$	$\{0,0,1\}$ ,
$C_2'$	$\{0,1,0\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},-1,0\right\}$ ,
$C_2\text{''}$	$\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{1,-\sqrt{3},0\right\}$ ,
$\text{i}$	$\{0,0,0\}$ ,
$S_6$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$S_3$	$\{0,0,1\}$ , $\{0,0,-1\}$ ,
$\sigma _h$	$\{0,0,1\}$ ,
$\sigma _v$	$\{1,0,0\}$ , $\left\{1,\sqrt{3},0\right\}$ , $\left\{1,-\sqrt{3},0\right\}$ ,
$\sigma _d$	$\{0,1,0\}$ , $\left\{\sqrt{3},1,0\right\}$ , $\left\{\sqrt{3},-1,0\right\}$ ,

Table of several point groups

Return to Main page on Point Groups

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}