| 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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 \\0 & 0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$15$ |
$0$ |
$195$ |
$0$ |
$3780$ |
$0$ |
$93177$ |
$0$ |
$2612148$ |
| $a_2$ |
$1$ |
$2$ |
$7$ |
$32$ |
$209$ |
$1822$ |
$19275$ |
$227999$ |
$2872091$ |
$37536350$ |
$501961547$ |
$6816222679$ |
$93576095750$ |
| $a_3$ |
$1$ |
$0$ |
$7$ |
$0$ |
$471$ |
$0$ |
$89720$ |
$0$ |
$25001207$ |
$0$ |
$7889431032$ |
$0$ |
$2626711883928$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$7$ |
$3$ |
$8$ |
$15$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$7$ |
$32$ |
$17$ |
$49$ |
$30$ |
$95$ |
$195$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$38$ |
$209$ |
$117$ |
$73$ |
$390$ |
$234$ |
$813$ |
$480$ |
$1735$ |
$3780$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$298$ |
$1822$ |
$177$ |
$1037$ |
$609$ |
$3814$ |
$2206$ |
$1290$ |
$8334$ |
$4790$ |
$18460$ |
$10535$ |
$41300$ |
$93177$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$471$ |
$2867$ |
$19275$ |
$1663$ |
$10884$ |
$6220$ |
$42665$ |
$3569$ |
$24126$ |
$13702$ |
$96009$ |
$54047$ |
$30561$ |
$217639$ |
| $$ |
$122059$ |
$496174$ |
$277326$ |
$1136310$ |
$2612148$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&0&0&0&0&0&1\\0&2&0&1&0&3&0&0&3&0&2&0&2&5&0\\1&0&4&0&1&0&4&6&0&1&0&6&0&0&13\\0&1&0&3&0&4&0&0&4&0&1&0&9&8&0\\0&0&1&0&6&0&5&3&0&7&0&11&0&0&17\\0&3&0&4&0&14&0&0&17&0&9&0&25&38&0\\0&0&4&0&5&0&16&10&0&9&0&25&0&0&53\\2&0&6&0&3&0&10&25&0&10&0&25&0&0&63\\0&3&0&4&0&17&0&0&30&0&12&0&37&61&0\\0&0&1&0&7&0&9&10&0&20&0&22&0&0&53\\0&2&0&1&0&9&0&0&12&0&10&0&16&30&0\\0&0&6&0&11&0&25&25&0&22&0&60&0&0&121\\0&2&0&9&0&25&0&0&37&0&16&0&73&93&0\\0&5&0&8&0&38&0&0&61&0&30&0&93&153&0\\1&0&13&0&17&0&53&63&0&53&0&121&0&0&294\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&3&6&14&16&25&30&20&10&60&73&153&294&156&164&365&315&107\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/48$ | $0$ | $1/8$ | $1/6$ | $3/16$ |
|---|
| $a_1=0$ | $3/8$ | $1/4$ | $0$ | $0$ | $1/8$ | $0$ | $1/8$ |
|---|
| $a_3=0$ | $3/8$ | $1/4$ | $0$ | $0$ | $1/8$ | $0$ | $1/8$ |
|---|
| $a_1=a_3=0$ | $3/8$ | $1/4$ | $0$ | $0$ | $1/8$ | $0$ | $1/8$ |
|---|
