| 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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $L_1(C_{2,1})$, $L_1(C_3)$ |
| Minimal supergroups: | $L_1(J(D_3))$, $L(D_{6,2},D_{3,2})$, $L(J(D_3),D_{3,2})$, $L(D_{6,1},D_{3,2})$, $L_1(D_{6,2})$${}^{\times 2}$, $L_2(D_{3,2})$, $L_1(O_1)$, $L_1(D_{6,1})$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$48$ |
$0$ |
$960$ |
$0$ |
$24080$ |
$0$ |
$673344$ |
$0$ |
$19982424$ |
| $a_2$ |
$1$ |
$3$ |
$15$ |
$106$ |
$999$ |
$11088$ |
$134000$ |
$1697601$ |
$22142487$ |
$294576688$ |
$3975516390$ |
$54241909131$ |
$746516952752$ |
| $a_3$ |
$1$ |
$0$ |
$20$ |
$0$ |
$2652$ |
$0$ |
$660500$ |
$0$ |
$196192108$ |
$0$ |
$62823455100$ |
$0$ |
$20989720361148$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$4$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$15$ |
$8$ |
$24$ |
$48$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$20$ |
$106$ |
$60$ |
$206$ |
$120$ |
$438$ |
$960$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$158$ |
$999$ |
$566$ |
$328$ |
$2142$ |
$1224$ |
$4748$ |
$2700$ |
$10650$ |
$24080$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1608$ |
$11088$ |
$918$ |
$6234$ |
$3530$ |
$24822$ |
$13958$ |
$7880$ |
$56244$ |
$31540$ |
$128172$ |
$71680$ |
$293286$ |
$673344$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2652$ |
$18488$ |
$134000$ |
$10398$ |
$74682$ |
$41756$ |
$305944$ |
$23408$ |
$170274$ |
$94972$ |
$702078$ |
$389972$ |
$217044$ |
$1616184$ |
| $$ |
$896112$ |
$3729726$ |
$2064636$ |
$8625204$ |
$19982424$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&1&0&2&4&0&2&0&3&0&0&8\\0&4&0&4&0&12&0&0&16&0&8&0&20&34&0\\2&0&10&0&8&0&22&27&0&17&0&43&0&0&96\\0&4&0&8&0&20&0&0&30&0&12&0&46&66&0\\1&0&8&0&15&0&28&31&0&27&0&66&0&0&132\\0&12&0&20&0&78&0&0&116&0&58&0&182&278&0\\2&0&22&0&28&0&82&89&0&73&0&175&0&0&400\\4&0&27&0&31&0&89&118&0&85&0&207&0&0&484\\0&16&0&30&0&116&0&0&196&0&86&0&302&464&0\\2&0&17&0&27&0&73&85&0&82&0&168&0&0&408\\0&8&0&12&0&58&0&0&86&0&52&0&142&220&0\\3&0&43&0&66&0&175&207&0&168&0&423&0&0&952\\0&20&0&46&0&182&0&0&302&0&142&0&500&746&0\\0&34&0&66&0&278&0&0&464&0&220&0&746&1166&0\\8&0&96&0&132&0&400&484&0&408&0&952&0&0&2300\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&10&8&15&78&82&118&196&82&52&423&500&1166&2300&1138&1164&2762&2353&662\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
