| 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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $L(C_{2,1},C_1)$, $L_1(C_3)$ |
| Minimal supergroups: | $L(J(D_3),J(C_3))$, $L(J(D_3),D_3)$, $L(D_{6,1},D_3)$, $L(O_1,T)$, $L_2(D_{3,2})$, $L(D_{6,1},C_{6,1})$, $L(D_{6,2},D_{3,2})$, $L(D_{6,2},C_6)$${}^{\times 2}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$45$ |
$0$ |
$950$ |
$0$ |
$24045$ |
$0$ |
$673218$ |
$0$ |
$19981962$ |
| $a_2$ |
$1$ |
$3$ |
$15$ |
$106$ |
$999$ |
$11088$ |
$134000$ |
$1697601$ |
$22142487$ |
$294576688$ |
$3975516390$ |
$54241909131$ |
$746516952752$ |
| $a_3$ |
$1$ |
$0$ |
$16$ |
$0$ |
$2604$ |
$0$ |
$659860$ |
$0$ |
$196183148$ |
$0$ |
$62823326076$ |
$0$ |
$20989718468796$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$15$ |
$6$ |
$21$ |
$45$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$16$ |
$106$ |
$54$ |
$197$ |
$114$ |
$429$ |
$950$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$146$ |
$999$ |
$548$ |
$316$ |
$2115$ |
$1206$ |
$4721$ |
$2680$ |
$10620$ |
$24045$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1572$ |
$11088$ |
$894$ |
$6180$ |
$3494$ |
$24741$ |
$13904$ |
$7840$ |
$56163$ |
$31480$ |
$128082$ |
$71610$ |
$293181$ |
$673218$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2604$ |
$18380$ |
$134000$ |
$10326$ |
$74520$ |
$41648$ |
$305701$ |
$23328$ |
$170112$ |
$94852$ |
$701835$ |
$389792$ |
$216904$ |
$1615914$ |
| $$ |
$895902$ |
$3729411$ |
$2064384$ |
$8624826$ |
$19981962$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&6&0&0&0&3&0&0&8\\0&3&0&3&0&12&0&0&15&0&9&0&21&35&0\\2&0&10&0&6&0&20&31&0&13&0&43&0&0&96\\0&3&0&7&0&20&0&0&29&0&13&0&47&67&0\\0&0&6&0&18&0&30&25&0&31&0&69&0&0&132\\0&12&0&20&0&78&0&0&116&0&58&0&182&278&0\\1&0&20&0&30&0&83&85&0&75&0&178&0&0&400\\6&0&31&0&25&0&85&130&0&77&0&201&0&0&484\\0&15&0&29&0&116&0&0&195&0&87&0&303&465&0\\0&0&13&0&31&0&75&77&0&86&0&174&0&0&408\\0&9&0&13&0&58&0&0&87&0&50&0&140&221&0\\3&0&43&0&69&0&178&201&0&174&0&423&0&0&952\\0&21&0&47&0&182&0&0&303&0&140&0&498&747&0\\0&35&0&67&0&278&0&0&465&0&221&0&747&1161&0\\8&0&96&0&132&0&400&484&0&408&0&952&0&0&2300\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&10&7&18&78&83&130&195&86&50&423&498&1161&2300&1133&1191&2760&2362&657\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
