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Orientation XYZ

Symmetry Operations

In the Oh Point Group, with orientation XYZ there are the following symmetry operations

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_3$	$\{1,1,1\}$ , $\{1,1,-1\}$ , $\{1,-1,1\}$ , $\{-1,1,1\}$ , $\{-1,-1,1\}$ , $\{-1,1,-1\}$ , $\{1,-1,-1\}$ , $\{-1,-1,-1\}$ ,
$C_2$	$\{1,1,0\}$ , $\{1,-1,0\}$ , $\{1,0,-1\}$ , $\{1,0,1\}$ , $\{0,1,1\}$ , $\{0,1,-1\}$ ,
$C_4$	$\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , $\{0,0,-1\}$ , $\{0,-1,0\}$ , $\{-1,0,0\}$ ,
$C_2$	$\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ ,
$\text{i}$	$\{0,0,0\}$ ,
$S_4$	$\{0,0,1\}$ , $\{0,1,0\}$ , $\{1,0,0\}$ , $\{0,0,-1\}$ , $\{0,-1,0\}$ , $\{-1,0,0\}$ ,
$S_6$	$\{1,1,1\}$ , $\{1,1,-1\}$ , $\{1,-1,1\}$ , $\{-1,1,1\}$ , $\{-1,-1,1\}$ , $\{-1,1,-1\}$ , $\{1,-1,-1\}$ , $\{-1,-1,-1\}$ ,
$\sigma _h$	$\{1,0,0\}$ , $\{0,1,0\}$ , $\{0,0,1\}$ ,
$\sigma _d$	$\{1,1,0\}$ , $\{1,-1,0\}$ , $\{1,0,-1\}$ , $\{1,0,1\}$ , $\{0,1,1\}$ , $\{0,1,-1\}$ ,

Character Table

	$\text{E} \,{\text{(1)}}$	$C_3 \,{\text{(8)}}$	$C_2 \,{\text{(6)}}$	$C_4 \,{\text{(6)}}$	$C_2 \,{\text{(3)}}$	$\text{i} \,{\text{(1)}}$	$S_4 \,{\text{(6)}}$	$S_6 \,{\text{(8)}}$	$\sigma_h \,{\text{(3)}}$	$\sigma_d \,{\text{(6)}}$
$A_{1g}$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
$A_{2g}$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$1$	$-1$
$E_g$	$2$	$-1$	$0$	$0$	$2$	$2$	$0$	$-1$	$2$	$0$
$T_{1g}$	$3$	$0$	$-1$	$1$	$-1$	$3$	$1$	$0$	$-1$	$-1$
$T_{2g}$	$3$	$0$	$1$	$-1$	$-1$	$3$	$-1$	$0$	$-1$	$1$
$A_{1u}$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$
$A_{2u}$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$1$
$E_u$	$2$	$-1$	$0$	$0$	$2$	$-2$	$0$	$1$	$-2$	$0$
$T_{1u}$	$3$	$0$	$-1$	$1$	$-1$	$-3$	$-1$	$0$	$1$	$1$
$T_{2u}$	$3$	$0$	$1$	$-1$	$-1$	$-3$	$1$	$0$	$1$	$-1$

Product Table

	$A_{1g}$	$A_{2g}$	$E_g$	$T_{1g}$	$T_{2g}$	$A_{1u}$	$A_{2u}$	$E_u$	$T_{1u}$	$T_{2u}$
$A_{1g}$	$A_{1g}$	$A_{2g}$	$E_g$	$T_{1g}$	$T_{2g}$	$A_{1u}$	$A_{2u}$	$E_u$	$T_{1u}$	$T_{2u}$
$A_{2g}$	$A_{2g}$	$A_{1g}$	$E_g$	$T_{2g}$	$T_{1g}$	$A_{2u}$	$A_{1u}$	$E_u$	$T_{2u}$	$T_{1u}$
$E_g$	$E_g$	$E_g$	$A_{1g}+A_{2g}+E_g$	$T_{1g}+T_{2g}$	$T_{1g}+T_{2g}$	$E_u$	$E_u$	$A_{1u}+A_{2u}+E_u$	$T_{1u}+T_{2u}$	$T_{1u}+T_{2u}$
$T_{1g}$	$T_{1g}$	$T_{2g}$	$T_{1g}+T_{2g}$	$A_{1g}+E_g+T_{1g}+T_{2g}$	$A_{2g}+E_g+T_{1g}+T_{2g}$	$T_{1u}$	$T_{2u}$	$T_{1u}+T_{2u}$	$A_{1u}+E_u+T_{1u}+T_{2u}$	$A_{2u}+E_u+T_{1u}+T_{2u}$
$T_{2g}$	$T_{2g}$	$T_{1g}$	$T_{1g}+T_{2g}$	$A_{2g}+E_g+T_{1g}+T_{2g}$	$A_{1g}+E_g+T_{1g}+T_{2g}$	$T_{2u}$	$T_{1u}$	$T_{1u}+T_{2u}$	$A_{2u}+E_u+T_{1u}+T_{2u}$	$A_{1u}+E_u+T_{1u}+T_{2u}$
$A_{1u}$	$A_{1u}$	$A_{2u}$	$E_u$	$T_{1u}$	$T_{2u}$	$A_{1g}$	$A_{2g}$	$E_g$	$T_{1g}$	$T_{2g}$
$A_{2u}$	$A_{2u}$	$A_{1u}$	$E_u$	$T_{2u}$	$T_{1u}$	$A_{2g}$	$A_{1g}$	$E_g$	$T_{2g}$	$T_{1g}$
$E_u$	$E_u$	$E_u$	$A_{1u}+A_{2u}+E_u$	$T_{1u}+T_{2u}$	$T_{1u}+T_{2u}$	$E_g$	$E_g$	$A_{1g}+A_{2g}+E_g$	$T_{1g}+T_{2g}$	$T_{1g}+T_{2g}$
$T_{1u}$	$T_{1u}$	$T_{2u}$	$T_{1u}+T_{2u}$	$A_{1u}+E_u+T_{1u}+T_{2u}$	$A_{2u}+E_u+T_{1u}+T_{2u}$	$T_{1g}$	$T_{2g}$	$T_{1g}+T_{2g}$	$A_{1g}+E_g+T_{1g}+T_{2g}$	$A_{2g}+E_g+T_{1g}+T_{2g}$
$T_{2u}$	$T_{2u}$	$T_{1u}$	$T_{1u}+T_{2u}$	$A_{2u}+E_u+T_{1u}+T_{2u}$	$A_{1u}+E_u+T_{1u}+T_{2u}$	$T_{2g}$	$T_{1g}$	$T_{1g}+T_{2g}$	$A_{2g}+E_g+T_{1g}+T_{2g}$	$A_{1g}+E_g+T_{1g}+T_{2g}$

Other Orientations

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