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