| 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: | $C_2\times S_4$ |
| Order: | $48$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 \\0 & 0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0& 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0& -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$24$ |
$0$ |
$340$ |
$0$ |
$6475$ |
$0$ |
$145278$ |
$0$ |
$3643332$ |
| $a_2$ |
$1$ |
$2$ |
$8$ |
$44$ |
$329$ |
$2962$ |
$29980$ |
$328757$ |
$3827507$ |
$46676330$ |
$590165678$ |
$7674277327$ |
$101983083350$ |
| $a_3$ |
$1$ |
$0$ |
$10$ |
$0$ |
$780$ |
$0$ |
$131720$ |
$0$ |
$31046400$ |
$0$ |
$8800210440$ |
$0$ |
$2768901971484$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$8$ |
$4$ |
$12$ |
$24$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$10$ |
$44$ |
$26$ |
$80$ |
$48$ |
$161$ |
$340$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$60$ |
$329$ |
$194$ |
$120$ |
$662$ |
$398$ |
$1395$ |
$830$ |
$2985$ |
$6475$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$496$ |
$2962$ |
$294$ |
$1736$ |
$1028$ |
$6294$ |
$3696$ |
$2190$ |
$13632$ |
$7980$ |
$29800$ |
$17360$ |
$65590$ |
$145278$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$780$ |
$4684$ |
$29980$ |
$2752$ |
$17338$ |
$10088$ |
$65637$ |
$5886$ |
$37890$ |
$21958$ |
$145055$ |
$83450$ |
$48220$ |
$322490$ |
| $$ |
$184940$ |
$720566$ |
$411894$ |
$1617042$ |
$3643332$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&0&2&0&1&0&1&0&0&2\\0&3&0&1&0&5&0&0&5&0&4&0&4&10&0\\1&0&5&0&3&0&8&8&0&3&0&13&0&0&22\\0&1&0&5&0&7&0&0&7&0&2&0&16&14&0\\1&0&3&0&8&0&7&10&0&9&0&19&0&0&28\\0&5&0&7&0&25&0&0&29&0&17&0&43&62&0\\0&0&8&0&7&0&27&20&0&12&0&42&0&0&80\\2&0&8&0&10&0&20&33&0&21&0&45&0&0&92\\0&5&0&7&0&29&0&0&47&0&20&0&56&90&0\\1&0&3&0&9&0&12&21&0&26&0&30&0&0&72\\0&4&0&2&0&17&0&0&20&0&18&0&26&47&0\\1&0&13&0&19&0&42&45&0&30&0&94&0&0&170\\0&4&0&16&0&43&0&0&56&0&26&0&110&131&0\\0&10&0&14&0&62&0&0&90&0&47&0&131&210&0\\2&0&22&0&28&0&80&92&0&72&0&170&0&0&380\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&5&5&8&25&27&33&47&26&18&94&110&210&380&194&187&432&353&128\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/48$ | $0$ | $1/8$ | $1/6$ | $3/16$ |
|---|
| $a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $1/8$ | $1/8$ | $0$ | $0$ | $1/8$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
