| Name: | $\mathrm{SU}(2)\times\mathrm{U}(1)_2$ |
| $\mathbb{R}$-dimension: | $4$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&\alpha I_2&0\\0&0&\bar\alpha I_2\end{bmatrix}: A\in\mathrm{SU}(2),\ \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ |
Symplectic form: | $\begin{bmatrix}J_2&0&0\\0&0&I_2\\0&-I_2&0\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ |
| Hodge circle: | $u\mapsto\mathrm{diag}(u,\bar u, u, u,\bar u, \bar u)$ |
| Name: | $S_3$ |
| Order: | $6$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$32$ |
$0$ |
$555$ |
$0$ |
$11984$ |
$0$ |
$291144$ |
$0$ |
$7601880$ |
| $a_2$ |
$1$ |
$2$ |
$10$ |
$66$ |
$560$ |
$5492$ |
$58886$ |
$668586$ |
$7895488$ |
$95945256$ |
$1191636542$ |
$15058177664$ |
$192989354542$ |
| $a_3$ |
$1$ |
$0$ |
$13$ |
$0$ |
$1370$ |
$0$ |
$263715$ |
$0$ |
$63318598$ |
$0$ |
$17000038398$ |
$0$ |
$4888747822332$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$10$ |
$5$ |
$16$ |
$32$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$13$ |
$66$ |
$38$ |
$126$ |
$74$ |
$260$ |
$555$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$94$ |
$560$ |
$324$ |
$192$ |
$1162$ |
$680$ |
$2496$ |
$1455$ |
$5440$ |
$11984$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$864$ |
$5492$ |
$502$ |
$3166$ |
$1836$ |
$11906$ |
$6870$ |
$3981$ |
$26204$ |
$15090$ |
$58150$ |
$33404$ |
$129808$ |
$291144$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1370$ |
$8812$ |
$58886$ |
$5080$ |
$33662$ |
$19306$ |
$130700$ |
$11097$ |
$74624$ |
$42706$ |
$292268$ |
$166550$ |
$95132$ |
$656752$ |
| $$ |
$373576$ |
$1481340$ |
$841176$ |
$3351432$ |
$7601880$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&3&0&1&0&2&0&0&4\\0&3&0&2&0&8&0&0&9&0&5&0&10&19&0\\1&0&7&0&5&0&13&15&0&7&0&24&0&0&44\\0&2&0&6&0&12&0&0&14&0&6&0&24&32&0\\1&0&5&0&10&0&15&18&0&13&0&34&0&0&58\\0&8&0&12&0&44&0&0&56&0&30&0&84&126&0\\1&0&13&0&15&0&43&43&0&29&0&82&0&0&164\\3&0&15&0&18&0&43&58&0&37&0&92&0&0&192\\0&9&0&14&0&56&0&0&87&0&39&0&122&187&0\\1&0&7&0&13&0&29&37&0&35&0&66&0&0&148\\0&5&0&6&0&30&0&0&39&0&28&0&62&94&0\\2&0&24&0&34&0&82&92&0&66&0&188&0&0&364\\0&10&0&24&0&84&0&0&122&0&62&0&206&288&0\\0&19&0&32&0&126&0&0&187&0&94&0&288&440&0\\4&0&44&0&58&0&164&192&0&148&0&364&0&0&800\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&7&6&10&44&43&58&87&35&28&188&206&440&800&362&380&834&648&174\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
