| 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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 &0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $L_1(J(C_3))$, $L(D_3,C_3)$, $L(J(C_2),J(C_1))$, $L(D_{3,2},C_3)$ |
| Minimal supergroups: | $L_2(J(D_3))$, $L(J(D_6),J(D_3))$, $L(J(O),J(T))$, $L(J(D_6),J(C_6))$${}^{\times 2}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$24$ |
$0$ |
$480$ |
$0$ |
$12040$ |
$0$ |
$336672$ |
$0$ |
$9991212$ |
| $a_2$ |
$1$ |
$2$ |
$9$ |
$56$ |
$507$ |
$5562$ |
$67046$ |
$848920$ |
$11071563$ |
$147289214$ |
$1987760604$ |
$27120961320$ |
$373258495506$ |
| $a_3$ |
$1$ |
$0$ |
$9$ |
$0$ |
$1311$ |
$0$ |
$330040$ |
$0$ |
$98093079$ |
$0$ |
$31411684584$ |
$0$ |
$10494859549944$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$3$ |
$11$ |
$24$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$9$ |
$56$ |
$28$ |
$100$ |
$57$ |
$216$ |
$480$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$73$ |
$507$ |
$275$ |
$161$ |
$1060$ |
$606$ |
$2365$ |
$1340$ |
$5315$ |
$12040$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$788$ |
$5562$ |
$444$ |
$3093$ |
$1747$ |
$12376$ |
$6955$ |
$3930$ |
$28089$ |
$15750$ |
$64056$ |
$35805$ |
$146608$ |
$336672$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1311$ |
$9192$ |
$67046$ |
$5169$ |
$37265$ |
$20830$ |
$152861$ |
$11654$ |
$85065$ |
$47426$ |
$350934$ |
$194906$ |
$108487$ |
$807982$ |
| $$ |
$447986$ |
$1864758$ |
$1032192$ |
$4312476$ |
$9991212$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&0&0&1&0&0&4\\0&2&0&1&0&6&0&0&8&0&5&0&9&18&0\\1&0&6&0&2&0&10&15&0&5&0&22&0&0&48\\0&1&0&5&0&10&0&0&12&0&5&0&28&32&0\\0&0&2&0&11&0&14&13&0&19&0&34&0&0&66\\0&6&0&10&0&39&0&0&58&0&29&0&91&139&0\\0&0&10&0&14&0&45&39&0&32&0&90&0&0&200\\4&0&15&0&13&0&39&71&0&43&0&98&0&0&242\\0&8&0&12&0&58&0&0&102&0&46&0&144&235&0\\0&0&5&0&19&0&32&43&0&56&0&86&0&0&204\\0&5&0&5&0&29&0&0&46&0&27&0&65&112&0\\1&0&22&0&34&0&90&98&0&86&0&213&0&0&476\\0&9&0&28&0&91&0&0&144&0&65&0&264&368&0\\0&18&0&32&0&139&0&0&235&0&112&0&368&583&0\\4&0&48&0&66&0&200&242&0&204&0&476&0&0&1150\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&5&11&39&45&71&102&56&27&213&264&583&1150&586&607&1422&1209&383\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/12$ | $0$ | $0$ | $1/6$ | $1/4$ |
|---|
| $a_1=0$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_3=0$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
