| 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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 \\0 & 0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\0 & 0 & 0 & 0 & \zeta_{8}^{3} & 0 \\0 & 0 & 0 & \zeta_{8}^{3} & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0\\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0& 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$15$ |
$0$ |
$205$ |
$0$ |
$4025$ |
$0$ |
$97587$ |
$0$ |
$2683758$ |
| $a_2$ |
$1$ |
$2$ |
$7$ |
$33$ |
$217$ |
$1882$ |
$19720$ |
$231275$ |
$2895849$ |
$37705926$ |
$503155552$ |
$6824542375$ |
$93633631338$ |
| $a_3$ |
$1$ |
$0$ |
$7$ |
$0$ |
$474$ |
$0$ |
$90410$ |
$0$ |
$25064662$ |
$0$ |
$7893957582$ |
$0$ |
$2627005356030$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$7$ |
$3$ |
$8$ |
$15$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$7$ |
$33$ |
$17$ |
$50$ |
$30$ |
$98$ |
$205$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$39$ |
$217$ |
$120$ |
$73$ |
$404$ |
$240$ |
$848$ |
$500$ |
$1830$ |
$4025$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$305$ |
$1882$ |
$180$ |
$1067$ |
$624$ |
$3940$ |
$2279$ |
$1330$ |
$8643$ |
$4980$ |
$19226$ |
$11025$ |
$43162$ |
$97587$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$474$ |
$2930$ |
$19720$ |
$1693$ |
$11141$ |
$6369$ |
$43670$ |
$3659$ |
$24750$ |
$14092$ |
$98417$ |
$55589$ |
$31541$ |
$223389$ |
| $$ |
$125783$ |
$509684$ |
$286146$ |
$1167600$ |
$2683758$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&0&0&0&0&0&1\\0&2&0&1&0&3&0&0&3&0&2&0&2&6&0\\1&0&4&0&1&0&4&7&0&1&0&6&0&0&13\\0&1&0&3&0&4&0&0&5&0&1&0&8&9&0\\0&0&1&0&6&0&5&3&0&6&0&13&0&0&17\\0&3&0&4&0&15&0&0&17&0&10&0&26&40&0\\0&0&4&0&5&0&16&11&0&10&0&26&0&0&53\\2&0&7&0&3&0&11&25&0&8&0&26&0&0&63\\0&3&0&5&0&17&0&0&29&0&11&0&39&61&0\\0&0&1&0&6&0&10&8&0&17&0&22&0&0&53\\0&2&0&1&0&10&0&0&11&0&12&0&19&31&0\\0&0&6&0&13&0&26&26&0&22&0&65&0&0&121\\0&2&0&8&0&26&0&0&39&0&19&0&72&95&0\\0&6&0&9&0&40&0&0&61&0&31&0&95&154&0\\1&0&13&0&17&0&53&63&0&53&0&121&0&0&294\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&3&6&15&16&25&29&17&12&65&72&154&294&153&165&357&314&95\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $1/4$ | $0$ | $1/4$ |
|---|
| $a_1=0$ | $5/16$ | $1/4$ | $0$ | $0$ | $1/8$ | $0$ | $1/8$ |
|---|
| $a_3=0$ | $7/16$ | $3/8$ | $0$ | $0$ | $1/4$ | $0$ | $1/8$ |
|---|
| $a_1=a_3=0$ | $5/16$ | $1/4$ | $0$ | $0$ | $1/8$ | $0$ | $1/8$ |
|---|
