| 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: | $A_4$ |
| Order: | $12$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 \\0 & 0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}$ |
| Maximal subgroups: | $L_1(D_2)$, $L_1(C_3)$ |
| Minimal supergroups: | $L_1(O)$, $L_1(J(T))$, $L_2(T)$, $L(O_1,T)$, $L(O,T)$, $L_1(O_1)$, $L(J(T),T)$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$42$ |
$0$ |
$680$ |
$0$ |
$14490$ |
$0$ |
$368424$ |
$0$ |
$10418100$ |
| $a_2$ |
$1$ |
$2$ |
$11$ |
$76$ |
$657$ |
$6622$ |
$74445$ |
$900832$ |
$11439461$ |
$149924674$ |
$2006828301$ |
$27260124652$ |
$374281824163$ |
| $a_3$ |
$1$ |
$0$ |
$16$ |
$0$ |
$1656$ |
$0$ |
$353320$ |
$0$ |
$99855448$ |
$0$ |
$31553509176$ |
$0$ |
$10506725163000$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$4$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$11$ |
$6$ |
$20$ |
$42$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$16$ |
$76$ |
$46$ |
$152$ |
$90$ |
$316$ |
$680$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$112$ |
$657$ |
$386$ |
$232$ |
$1386$ |
$814$ |
$2990$ |
$1740$ |
$6540$ |
$14490$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1032$ |
$6622$ |
$600$ |
$3810$ |
$2204$ |
$14516$ |
$8318$ |
$4800$ |
$32208$ |
$18380$ |
$72092$ |
$40950$ |
$162484$ |
$368424$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1656$ |
$10804$ |
$74445$ |
$6188$ |
$42078$ |
$23880$ |
$167380$ |
$13596$ |
$94278$ |
$53288$ |
$378974$ |
$212752$ |
$119864$ |
$862676$ |
| $$ |
$482874$ |
$1972362$ |
$1100988$ |
$4525920$ |
$10418100$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&3&0&2&0&3&0&0&4\\0&4&0&2&0&10&0&0&10&0&8&0&10&22&0\\1&0&8&0&7&0&16&16&0&8&0&29&0&0&52\\0&2&0&8&0&14&0&0&16&0&6&0&32&36&0\\2&0&7&0&13&0&15&23&0&16&0&40&0&0&68\\0&10&0&14&0&52&0&0&66&0&36&0&100&152&0\\1&0&16&0&15&0&54&50&0&34&0&97&0&0&212\\3&0&16&0&23&0&50&68&0&52&0&113&0&0&252\\0&10&0&16&0&66&0&0&112&0&46&0&156&244&0\\2&0&8&0&16&0&34&52&0&52&0&84&0&0&212\\0&8&0&6&0&36&0&0&46&0&36&0&68&120&0\\3&0&29&0&40&0&97&113&0&84&0&229&0&0&484\\0&10&0&32&0&100&0&0&156&0&68&0&276&378&0\\0&22&0&36&0&152&0&0&244&0&120&0&378&600&0\\4&0&52&0&68&0&212&252&0&212&0&484&0&0&1176\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&8&8&13&52&54&68&112&52&36&229&276&600&1176&596&591&1416&1200&364\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
