| 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: | $D_6$ |
| Order: | $12$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}0 & 0 & \zeta_{12}^{5} & 0 \\0 & 0 & 0 & \zeta_{12}^{7} \\\zeta_{12}^{1} & 0 & 0 & 0 \\0 & \zeta_{12}^{11} & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 \\0 & \zeta_{6}^{5} &0 & 0 \\0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{1} \\0 & 0 & 0 & 0 & \zeta_{12}^{5} & 0 \\0 & 0 & 0 & \zeta_{12}^{5} & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $L(D_3,C_3)$, $L(D_{2,1},C_{2,1})$, $L_1(D_{3,2})$, $L(C_{6,1},C_3)$ |
| Minimal supergroups: | $L_2(D_{6,1})$, $L(J(D_6),D_{6,2})$${}^{\times 2}$, $L(J(D_6),J(D_3))$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$24$ |
$0$ |
$480$ |
$0$ |
$12040$ |
$0$ |
$336672$ |
$0$ |
$9991212$ |
| $a_2$ |
$1$ |
$2$ |
$9$ |
$57$ |
$511$ |
$5577$ |
$67096$ |
$849081$ |
$11072067$ |
$147290769$ |
$1987765354$ |
$27120975741$ |
$373258539110$ |
| $a_3$ |
$1$ |
$0$ |
$10$ |
$0$ |
$1326$ |
$0$ |
$330250$ |
$0$ |
$98096054$ |
$0$ |
$31411727550$ |
$0$ |
$10494860180574$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$4$ |
$12$ |
$24$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$10$ |
$57$ |
$30$ |
$103$ |
$60$ |
$219$ |
$480$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$79$ |
$511$ |
$283$ |
$164$ |
$1071$ |
$612$ |
$2374$ |
$1350$ |
$5325$ |
$12040$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$804$ |
$5577$ |
$459$ |
$3117$ |
$1765$ |
$12411$ |
$6979$ |
$3940$ |
$28122$ |
$15770$ |
$64086$ |
$35840$ |
$146643$ |
$336672$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1326$ |
$9244$ |
$67096$ |
$5199$ |
$37341$ |
$20878$ |
$152972$ |
$11704$ |
$85137$ |
$47486$ |
$351039$ |
$194986$ |
$108522$ |
$808092$ |
| $$ |
$448056$ |
$1864863$ |
$1032318$ |
$4312602$ |
$9991212$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&1&3&0&0&0&1&0&0&4\\0&2&0&2&0&6&0&0&8&0&4&0&10&17&0\\1&0&6&0&3&0&10&15&0&8&0&21&0&0&48\\0&2&0&4&0&10&0&0&15&0&6&0&23&33&0\\0&0&3&0&9&0&15&13&0&15&0&34&0&0&66\\0&6&0&10&0&39&0&0&58&0&29&0&91&139&0\\1&0&10&0&15&0&42&43&0&37&0&88&0&0&200\\3&0&15&0&13&0&43&64&0&39&0&101&0&0&242\\0&8&0&15&0&58&0&0&98&0&43&0&151&232&0\\0&0&8&0&15&0&37&39&0&44&0&86&0&0&204\\0&4&0&6&0&29&0&0&43&0&26&0&71&110&0\\1&0&21&0&34&0&88&101&0&86&0&213&0&0&476\\0&10&0&23&0&91&0&0&151&0&71&0&250&373&0\\0&17&0&33&0&139&0&0&232&0&110&0&373&583&0\\4&0&48&0&66&0&200&242&0&204&0&476&0&0&1150\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&4&9&39&42&64&98&44&26&213&250&583&1150&569&592&1381&1182&331\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $1/6$ | $0$ | $0$ | $1/3$ |
|---|
| $a_1=0$ | $1/2$ | $1/4$ | $0$ | $1/6$ | $0$ | $0$ | $1/12$ |
|---|
| $a_3=0$ | $1/2$ | $1/4$ | $0$ | $1/6$ | $0$ | $0$ | $1/12$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/4$ | $0$ | $1/6$ | $0$ | $0$ | $1/12$ |
|---|
