# Point Group D6h

## Character Table

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

## Product Table

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

## Implemented Settings

### Setting Zx

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

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

Nonaxial groups Cn groups Dn groups Cnv groups Cnh groups Dnh groups Dnd groups C1 Cs Ci C2 C3 C4 C5 C6 C7 C8 D2 D3 D4 D5 D6 D7 D8 C2v C3v C4v C5v C6v C7v C8v C2h C3h C4h C5h C6h D2h D3h D4h D5h D6h D7h D8h D2d D3d D4d D5d D6d D7d D8d S2 S4 S6 S8 S10 S12 T Th Td O Oh I Ih C$\infty$v D$\infty$h