| 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: | $C_6$ |
| Order: | $6$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{12}^{11} & 0 & 0 \\0 & 0 & 0 & 0& \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{1} \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$5$ |
$0$ |
$62$ |
$0$ |
$1065$ |
$0$ |
$21714$ |
$0$ |
$492366$ |
$0$ |
$12006588$ |
| $a_2$ |
$1$ |
$3$ |
$17$ |
$123$ |
$1053$ |
$10023$ |
$102551$ |
$1105317$ |
$12393133$ |
$143358471$ |
$1701053811$ |
$20618447193$ |
$254490862879$ |
| $a_3$ |
$1$ |
$0$ |
$24$ |
$0$ |
$2600$ |
$0$ |
$446730$ |
$0$ |
$95350584$ |
$0$ |
$23185477812$ |
$0$ |
$6164009039496$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$5$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$17$ |
$10$ |
$31$ |
$62$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$24$ |
$123$ |
$74$ |
$245$ |
$148$ |
$506$ |
$1065$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$184$ |
$1053$ |
$626$ |
$376$ |
$2199$ |
$1308$ |
$4674$ |
$2770$ |
$10035$ |
$21714$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1644$ |
$10023$ |
$976$ |
$5886$ |
$3472$ |
$21533$ |
$12620$ |
$7424$ |
$46694$ |
$27290$ |
$101901$ |
$59388$ |
$223510$ |
$492366$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2600$ |
$16044$ |
$102551$ |
$9400$ |
$59546$ |
$34680$ |
$224379$ |
$20256$ |
$129988$ |
$75504$ |
$493682$ |
$285314$ |
$165312$ |
$1090879$ |
| $$ |
$629020$ |
$2419214$ |
$1391964$ |
$5381922$ |
$12006588$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&2&0&3&4&0&2&0&5&0&0&8\\0&5&0&5&0&16&0&0&17&0&10&0&22&33&0\\2&0&12&0&12&0&26&28&0&16&0&46&0&0&80\\0&5&0&9&0&24&0&0&29&0&14&0&42&57&0\\2&0&12&0&17&0&31&34&0&22&0&62&0&0&104\\0&16&0&24&0&84&0&0&104&0&56&0&152&216&0\\3&0&26&0&31&0&78&82&0&56&0&147&0&0&280\\4&0&28&0&34&0&82&96&0&64&0&164&0&0&320\\0&17&0&29&0&104&0&0&149&0&70&0&214&309&0\\2&0&16&0&22&0&56&64&0&52&0&114&0&0&240\\0&10&0&14&0&56&0&0&70&0&45&0&109&156&0\\5&0&46&0&62&0&147&164&0&114&0&313&0&0&592\\0&22&0&42&0&152&0&0&214&0&109&0&333&468&0\\0&33&0&57&0&216&0&0&309&0&156&0&468&689&0\\8&0&80&0&104&0&280&320&0&240&0&592&0&0&1240\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&5&12&9&17&84&78&96&149&52&45&313&333&689&1240&537&539&1179&896&223\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
