Processing math: 100%

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Point Group Oh

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Point Group Oh

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

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

Implemented Settings

Setting 0sqrt2-1z

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_3$	$\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{-\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$ , $\left\{0,-\frac{2 \sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{-\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{0,\frac{2 \sqrt{2}}{3},\frac{1}{3}\right\}$ , $\left\{\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$ ,
$C_2$	$\left\{-\sqrt{2},0,0\right\}$ , $\left\{-\sqrt{2},-\sqrt{6},0\right\}$ , $\left\{-\sqrt{2},\sqrt{6},0\right\}$ , $\left\{0,-\sqrt{2},-2\right\}$ , $\left\{-\sqrt{6},-\sqrt{2},4\right\}$ , $\left\{-\sqrt{6},\sqrt{2},-4\right\}$ ,
$C_4$	$\left\{0,-\sqrt{2},1\right\}$ , $\left\{0,\sqrt{2},-1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$ , $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$ , $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$ ,
$C_2$	$\left\{0,-\sqrt{2},1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$ ,
$\text{i}$	$\{0,0,0\}$ ,
$S_4$	$\left\{0,-\sqrt{2},1\right\}$ , $\left\{0,\sqrt{2},-1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$ , $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$ , $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$ ,
$S_6$	$\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{-\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$ , $\left\{0,-\frac{2 \sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{-\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{0,\frac{2 \sqrt{2}}{3},\frac{1}{3}\right\}$ , $\left\{\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$ ,
$\sigma _h$	$\left\{0,-\sqrt{2},1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},-1\right\}$ ,
$\sigma _d$	$\left\{-\sqrt{2},0,0\right\}$ , $\left\{-\sqrt{2},-\sqrt{6},0\right\}$ , $\left\{-\sqrt{2},\sqrt{6},0\right\}$ , $\left\{0,-\sqrt{2},-2\right\}$ , $\left\{-\sqrt{6},-\sqrt{2},4\right\}$ , $\left\{-\sqrt{6},\sqrt{2},-4\right\}$ ,

Setting 0sqrt21z

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_3$	$\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$ , $\left\{0,\frac{2 \sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{-\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{0,-\frac{2 \sqrt{2}}{3},\frac{1}{3}\right\}$ , $\left\{-\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$ ,
$C_2$	$\left\{\sqrt{2},0,0\right\}$ , $\left\{\sqrt{2},\sqrt{6},0\right\}$ , $\left\{\sqrt{2},-\sqrt{6},0\right\}$ , $\left\{0,\sqrt{2},-2\right\}$ , $\left\{\sqrt{6},\sqrt{2},4\right\}$ , $\left\{\sqrt{6},-\sqrt{2},-4\right\}$ ,
$C_4$	$\left\{0,\sqrt{2},1\right\}$ , $\left\{0,-\sqrt{2},-1\right\}$ , $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$ , $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$ ,
$C_2$	$\left\{0,\sqrt{2},1\right\}$ , $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$ , $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$ ,
$\text{i}$	$\{0,0,0\}$ ,
$S_4$	$\left\{0,\sqrt{2},1\right\}$ , $\left\{0,-\sqrt{2},-1\right\}$ , $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$ , $\left\{-\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$ , $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$ ,
$S_6$	$\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$ , $\left\{0,\frac{2 \sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{-\sqrt{\frac{2}{3}},-\frac{\sqrt{2}}{3},-\frac{1}{3}\right\}$ , $\left\{0,-\frac{2 \sqrt{2}}{3},\frac{1}{3}\right\}$ , $\left\{-\sqrt{\frac{2}{3}},\frac{\sqrt{2}}{3},\frac{1}{3}\right\}$ ,
$\sigma _h$	$\left\{0,\sqrt{2},1\right\}$ , $\left\{\sqrt{\frac{3}{2}},-\frac{1}{\sqrt{2}},1\right\}$ , $\left\{\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}},-1\right\}$ ,
$\sigma _d$	$\left\{\sqrt{2},0,0\right\}$ , $\left\{\sqrt{2},\sqrt{6},0\right\}$ , $\left\{\sqrt{2},-\sqrt{6},0\right\}$ , $\left\{0,\sqrt{2},-2\right\}$ , $\left\{\sqrt{6},\sqrt{2},4\right\}$ , $\left\{\sqrt{6},-\sqrt{2},-4\right\}$ ,

Setting 111z

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_3$	$\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{\frac{1}{3} \left(1+\sqrt{3}\right),\frac{1}{3} \left(1-\sqrt{3}\right),\frac{1}{3}\right\}$ , $\left\{\frac{2}{3},\frac{2}{3},-\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(\sqrt{3}-1\right),\frac{1}{3} \left(-1-\sqrt{3}\right),-\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(-1-\sqrt{3}\right),\frac{1}{3} \left(\sqrt{3}-1\right),-\frac{1}{3}\right\}$ , $\left\{-\frac{2}{3},-\frac{2}{3},\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(1-\sqrt{3}\right),\frac{1}{3} \left(1+\sqrt{3}\right),\frac{1}{3}\right\}$ ,
$C_2$	$\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , $\{1,1,-2\}$ , $\left\{1+\sqrt{3},1-\sqrt{3},4\right\}$ , $\left\{\sqrt{3}-1,-1-\sqrt{3},-4\right\}$ ,
$C_4$	$\{1,1,1\}$ , $\{-1,-1,-1\}$ , $\left\{\frac{\sqrt{3}}{2}-\frac{1}{2},-\frac{1}{2}-\frac{\sqrt{3}}{2},1\right\}$ , $\left\{-\frac{1}{2}-\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2}-\frac{1}{2},1\right\}$ , $\left\{\frac{1}{2}-\frac{\sqrt{3}}{2},\frac{1}{2}+\frac{\sqrt{3}}{2},-1\right\}$ , $\left\{\frac{1}{2}+\frac{\sqrt{3}}{2},\frac{1}{2}-\frac{\sqrt{3}}{2},-1\right\}$ ,
$C_2$	$\{1,1,1\}$ , $\left\{\frac{\sqrt{3}}{2}-\frac{1}{2},-\frac{1}{2}-\frac{\sqrt{3}}{2},1\right\}$ , $\left\{\frac{1}{2}+\frac{\sqrt{3}}{2},\frac{1}{2}-\frac{\sqrt{3}}{2},-1\right\}$ ,
$\text{i}$	$\{0,0,0\}$ ,
$S_4$	$\{1,1,1\}$ , $\{-1,-1,-1\}$ , $\left\{\frac{\sqrt{3}}{2}-\frac{1}{2},-\frac{1}{2}-\frac{\sqrt{3}}{2},1\right\}$ , $\left\{-\frac{1}{2}-\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2}-\frac{1}{2},1\right\}$ , $\left\{\frac{1}{2}-\frac{\sqrt{3}}{2},\frac{1}{2}+\frac{\sqrt{3}}{2},-1\right\}$ , $\left\{\frac{1}{2}+\frac{\sqrt{3}}{2},\frac{1}{2}-\frac{\sqrt{3}}{2},-1\right\}$ ,
$S_6$	$\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{\frac{1}{3} \left(1+\sqrt{3}\right),\frac{1}{3} \left(1-\sqrt{3}\right),\frac{1}{3}\right\}$ , $\left\{\frac{2}{3},\frac{2}{3},-\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(\sqrt{3}-1\right),\frac{1}{3} \left(-1-\sqrt{3}\right),-\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(-1-\sqrt{3}\right),\frac{1}{3} \left(\sqrt{3}-1\right),-\frac{1}{3}\right\}$ , $\left\{-\frac{2}{3},-\frac{2}{3},\frac{1}{3}\right\}$ , $\left\{\frac{1}{3} \left(1-\sqrt{3}\right),\frac{1}{3} \left(1+\sqrt{3}\right),\frac{1}{3}\right\}$ ,
$\sigma _h$	$\{1,1,1\}$ , $\left\{\frac{\sqrt{3}}{2}-\frac{1}{2},-\frac{1}{2}-\frac{\sqrt{3}}{2},1\right\}$ , $\left\{\frac{1}{2}+\frac{\sqrt{3}}{2},\frac{1}{2}-\frac{\sqrt{3}}{2},-1\right\}$ ,
$\sigma _d$	$\{1,-1,0\}$ , $\left\{1+\sqrt{3},\sqrt{3}-1,0\right\}$ , $\left\{1-\sqrt{3},-1-\sqrt{3},0\right\}$ , $\{1,1,-2\}$ , $\left\{1+\sqrt{3},1-\sqrt{3},4\right\}$ , $\left\{\sqrt{3}-1,-1-\sqrt{3},-4\right\}$ ,

Setting sqrt20-1z

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_3$	$\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{-\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$ , $\left\{-\frac{2 \sqrt{2}}{3},0,-\frac{1}{3}\right\}$ , $\left\{\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$ , $\left\{\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$ , $\left\{\frac{2 \sqrt{2}}{3},0,\frac{1}{3}\right\}$ , $\left\{-\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$ ,
$C_2$	$\left\{0,\sqrt{2},0\right\}$ , $\left\{-\sqrt{6},\sqrt{2},0\right\}$ , $\left\{\sqrt{6},\sqrt{2},0\right\}$ , $\left\{-\sqrt{2},0,-2\right\}$ , $\left\{-\sqrt{2},\sqrt{6},4\right\}$ , $\left\{\sqrt{2},\sqrt{6},-4\right\}$ ,
$C_4$	$\left\{-\sqrt{2},0,1\right\}$ , $\left\{\sqrt{2},0,-1\right\}$ , $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$ , $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$ ,
$C_2$	$\left\{-\sqrt{2},0,1\right\}$ , $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$ ,
$\text{i}$	$\{0,0,0\}$ ,
$S_4$	$\left\{-\sqrt{2},0,1\right\}$ , $\left\{\sqrt{2},0,-1\right\}$ , $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$ , $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$ ,
$S_6$	$\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{-\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$ , $\left\{-\frac{2 \sqrt{2}}{3},0,-\frac{1}{3}\right\}$ , $\left\{\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$ , $\left\{\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$ , $\left\{\frac{2 \sqrt{2}}{3},0,\frac{1}{3}\right\}$ , $\left\{-\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$ ,
$\sigma _h$	$\left\{-\sqrt{2},0,1\right\}$ , $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$ ,
$\sigma _d$	$\left\{0,\sqrt{2},0\right\}$ , $\left\{-\sqrt{6},\sqrt{2},0\right\}$ , $\left\{\sqrt{6},\sqrt{2},0\right\}$ , $\left\{-\sqrt{2},0,-2\right\}$ , $\left\{-\sqrt{2},\sqrt{6},4\right\}$ , $\left\{\sqrt{2},\sqrt{6},-4\right\}$ ,

Setting sqrt201z

Operator	Orientation
$\text{E}$	$\{0,0,0\}$ ,
$C_3$	$\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$ , $\left\{\frac{2 \sqrt{2}}{3},0,-\frac{1}{3}\right\}$ , $\left\{-\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$ , $\left\{-\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$ , $\left\{-\frac{2 \sqrt{2}}{3},0,\frac{1}{3}\right\}$ , $\left\{\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$ ,
$C_2$	$\left\{0,-\sqrt{2},0\right\}$ , $\left\{\sqrt{6},-\sqrt{2},0\right\}$ , $\left\{-\sqrt{6},-\sqrt{2},0\right\}$ , $\left\{\sqrt{2},0,-2\right\}$ , $\left\{\sqrt{2},-\sqrt{6},4\right\}$ , $\left\{-\sqrt{2},-\sqrt{6},-4\right\}$ ,
$C_4$	$\left\{\sqrt{2},0,1\right\}$ , $\left\{-\sqrt{2},0,-1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$ , $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$ , $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$ ,
$C_2$	$\left\{\sqrt{2},0,1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$ , $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$ ,
$\text{i}$	$\{0,0,0\}$ ,
$S_4$	$\left\{\sqrt{2},0,1\right\}$ , $\left\{-\sqrt{2},0,-1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},1\right\}$ , $\left\{\frac{1}{\sqrt{2}},\sqrt{\frac{3}{2}},-1\right\}$ , $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$ ,
$S_6$	$\{0,0,1\}$ , $\{0,0,-1\}$ , $\left\{\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$ , $\left\{\frac{2 \sqrt{2}}{3},0,-\frac{1}{3}\right\}$ , $\left\{-\frac{\sqrt{2}}{3},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$ , $\left\{-\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},-\frac{1}{3}\right\}$ , $\left\{-\frac{2 \sqrt{2}}{3},0,\frac{1}{3}\right\}$ , $\left\{\frac{\sqrt{2}}{3},\sqrt{\frac{2}{3}},\frac{1}{3}\right\}$ ,
$\sigma _h$	$\left\{\sqrt{2},0,1\right\}$ , $\left\{-\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},1\right\}$ , $\left\{\frac{1}{\sqrt{2}},-\sqrt{\frac{3}{2}},-1\right\}$ ,
$\sigma _d$	$\left\{0,-\sqrt{2},0\right\}$ , $\left\{\sqrt{6},-\sqrt{2},0\right\}$ , $\left\{-\sqrt{6},-\sqrt{2},0\right\}$ , $\left\{\sqrt{2},0,-2\right\}$ , $\left\{\sqrt{2},-\sqrt{6},4\right\}$ , $\left\{-\sqrt{2},-\sqrt{6},-4\right\}$ ,

Setting XYZ

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

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