| 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: | $C_2^2\times C_6$ |
| Order: | $24$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{12}^{11} & 0 & 0 \\0 & 0 & 0 & 0& \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{1} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$ |
| Maximal subgroups: | $L_2(J(C_2))$, $L_1(J(C_6))$, $L(J(C_6),C_6)$, $L_2(C_6)$, $L_2(C_{6,1})$, $L_2(J(C_3))$, $L(J(C_6),J(C_3))$, $L(J(C_6),C_{6,1})$ |
| Minimal supergroups: | $L_2(J(D_6))$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$33$ |
$0$ |
$570$ |
$0$ |
$12425$ |
$0$ |
$309078$ |
$0$ |
$8338638$ |
| $a_2$ |
$1$ |
$2$ |
$10$ |
$65$ |
$554$ |
$5547$ |
$61317$ |
$720652$ |
$8817606$ |
$110969399$ |
$1426114865$ |
$18631476492$ |
$246708662691$ |
| $a_3$ |
$1$ |
$0$ |
$13$ |
$0$ |
$1383$ |
$0$ |
$283800$ |
$0$ |
$73713255$ |
$0$ |
$21334378968$ |
$0$ |
$6593232660168$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$10$ |
$5$ |
$16$ |
$33$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$13$ |
$65$ |
$38$ |
$126$ |
$75$ |
$264$ |
$570$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$93$ |
$554$ |
$323$ |
$195$ |
$1169$ |
$690$ |
$2546$ |
$1490$ |
$5600$ |
$12425$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$862$ |
$5547$ |
$504$ |
$3201$ |
$1863$ |
$12206$ |
$7045$ |
$4094$ |
$27183$ |
$15640$ |
$60880$ |
$34895$ |
$136920$ |
$309078$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1383$ |
$8992$ |
$61317$ |
$5187$ |
$34921$ |
$19982$ |
$137785$ |
$11468$ |
$78339$ |
$44686$ |
$311268$ |
$176530$ |
$100455$ |
$705510$ |
| $$ |
$399210$ |
$1603448$ |
$905310$ |
$3652740$ |
$8338638$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&3&0&1&0&2&0&0&4\\0&3&0&2&0&8&0&0&9&0&6&0&10&18&0\\1&0&7&0&5&0&13&14&0&7&0&25&0&0&44\\0&2&0&6&0&12&0&0&13&0&6&0&27&30&0\\1&0&5&0&11&0&15&18&0&15&0&35&0&0&60\\0&8&0&12&0&44&0&0&56&0&32&0&88&128&0\\1&0&13&0&15&0&45&41&0&27&0&86&0&0&172\\3&0&14&0&18&0&41&59&0&40&0&96&0&0&204\\0&9&0&13&0&56&0&0&89&0&43&0&127&199&0\\1&0&7&0&15&0&27&40&0&43&0&71&0&0&164\\0&6&0&6&0&32&0&0&43&0&29&0&63&101&0\\2&0&25&0&35&0&86&96&0&71&0&198&0&0&396\\0&10&0&27&0&88&0&0&127&0&63&0&228&309&0\\0&18&0&30&0&128&0&0&199&0&101&0&309&479&0\\4&0&44&0&60&0&172&204&0&164&0&396&0&0&900\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&7&6&11&44&45&59&89&43&29&198&228&479&900&440&437&1016&834&251\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/12$ | $1/6$ | $0$ | $1/6$ | $1/12$ |
|---|
| $a_1=0$ | $7/24$ | $1/4$ | $1/24$ | $1/12$ | $0$ | $1/12$ | $1/24$ |
|---|
| $a_3=0$ | $7/24$ | $1/4$ | $1/24$ | $1/12$ | $0$ | $1/12$ | $1/24$ |
|---|
| $a_1=a_3=0$ | $7/24$ | $1/4$ | $1/24$ | $1/12$ | $0$ | $1/12$ | $1/24$ |
|---|
