| Name: | $\mathrm{U}(1)\times\mathrm{U}(1)_2$ |
| $\mathbb{R}$-dimension: | $6$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&\alpha I_2&0\\0&0&\bar\alpha I_2\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),\ \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ |
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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}$ |
| Maximal subgroups: | $L_1(C_2)$, $L_1(C_3)$ |
| Minimal supergroups: | $L_1(D_6)$${}^{\times 2}$, $L_1(O)$, $L_1(D_{6,1})$, $L(J(D_3),D_3)$, $L(D_{6,1},D_3)$, $L(D_6,D_3)$, $L_1(J(D_3))$, $L_2(D_3)$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$48$ |
$0$ |
$960$ |
$0$ |
$24080$ |
$0$ |
$673344$ |
$0$ |
$19982424$ |
| $a_2$ |
$1$ |
$2$ |
$12$ |
$96$ |
$968$ |
$10992$ |
$133706$ |
$1696704$ |
$22139760$ |
$294568416$ |
$3975491342$ |
$54241833384$ |
$746516723926$ |
| $a_3$ |
$1$ |
$0$ |
$18$ |
$0$ |
$2622$ |
$0$ |
$660060$ |
$0$ |
$196185598$ |
$0$ |
$62823357828$ |
$0$ |
$20989718895684$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$4$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$12$ |
$6$ |
$22$ |
$48$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$18$ |
$96$ |
$56$ |
$200$ |
$114$ |
$432$ |
$960$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$148$ |
$968$ |
$552$ |
$322$ |
$2122$ |
$1212$ |
$4730$ |
$2680$ |
$10630$ |
$24080$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1580$ |
$10992$ |
$894$ |
$6192$ |
$3500$ |
$24760$ |
$13916$ |
$7860$ |
$56184$ |
$31500$ |
$128112$ |
$71610$ |
$293216$ |
$673344$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2622$ |
$18400$ |
$133706$ |
$10344$ |
$74552$ |
$41672$ |
$305752$ |
$23328$ |
$170148$ |
$94872$ |
$701892$ |
$389832$ |
$216974$ |
$1615984$ |
| $$ |
$895972$ |
$3729516$ |
$2064384$ |
$8624952$ |
$19982424$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&4&0&3&0&4&0&0&8\\0&4&0&2&0&12&0&0&16&0&10&0&18&36&0\\1&0&9&0&8&0&23&24&0&15&0&46&0&0&96\\0&2&0&10&0&20&0&0&26&0&10&0&54&64&0\\2&0&8&0&16&0&24&36&0&30&0&64&0&0&132\\0&12&0&20&0&78&0&0&116&0&58&0&182&278&0\\1&0&23&0&24&0&87&86&0&63&0&176&0&0&400\\4&0&24&0&36&0&86&118&0&96&0&208&0&0&484\\0&16&0&26&0&116&0&0&200&0&90&0&294&468&0\\3&0&15&0&30&0&63&96&0&93&0&166&0&0&408\\0&10&0&10&0&58&0&0&90&0&54&0&132&224&0\\4&0&46&0&64&0&176&208&0&166&0&420&0&0&952\\0&18&0&54&0&182&0&0&294&0&132&0&520&738&0\\0&36&0&64&0&278&0&0&468&0&224&0&738&1166&0\\8&0&96&0&132&0&400&484&0&408&0&952&0&0&2300\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&9&10&16&78&87&118&200&93&54&420&520&1166&2300&1160&1160&2802&2368&710\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
