| 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: | $S_4$ |
| Order: | $24$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 \\0 & 0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\0 &0 & 0 & 0 & \zeta_{8}^{3} & 0 \\0 & 0 & 0 & \zeta_{8}^{3} & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $L(D_{4,1},D_2)$, $L_1(T)$, $L(D_{3,2},C_3)$ |
| Minimal supergroups: | $L_2(O_1)$, $L(J(O),J(T))$, $L(J(O),O)$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$21$ |
$0$ |
$340$ |
$0$ |
$7245$ |
$0$ |
$184212$ |
$0$ |
$5209050$ |
| $a_2$ |
$1$ |
$2$ |
$8$ |
$45$ |
$349$ |
$3372$ |
$37405$ |
$450963$ |
$5721371$ |
$74967258$ |
$1003428913$ |
$13630106613$ |
$187141044942$ |
| $a_3$ |
$1$ |
$0$ |
$8$ |
$0$ |
$828$ |
$0$ |
$176660$ |
$0$ |
$49927724$ |
$0$ |
$15776754588$ |
$0$ |
$5253362581500$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$8$ |
$3$ |
$10$ |
$21$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$8$ |
$45$ |
$23$ |
$76$ |
$45$ |
$158$ |
$340$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$56$ |
$349$ |
$193$ |
$116$ |
$693$ |
$407$ |
$1495$ |
$870$ |
$3270$ |
$7245$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$516$ |
$3372$ |
$300$ |
$1905$ |
$1102$ |
$7258$ |
$4159$ |
$2400$ |
$16104$ |
$9190$ |
$36046$ |
$20475$ |
$81242$ |
$184212$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$828$ |
$5402$ |
$37405$ |
$3094$ |
$21039$ |
$11940$ |
$83690$ |
$6798$ |
$47139$ |
$26644$ |
$189487$ |
$106376$ |
$59932$ |
$431338$ |
| $$ |
$241437$ |
$986181$ |
$550494$ |
$2262960$ |
$5209050$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&3&0&0&0&1&0&0&2\\0&2&0&1&0&5&0&0&5&0&4&0&5&11&0\\1&0&5&0&2&0&7&11&0&2&0&13&0&0&26\\0&1&0&4&0&7&0&0&8&0&3&0&16&18&0\\0&0&2&0&9&0&9&7&0&11&0&22&0&0&34\\0&5&0&7&0&26&0&0&33&0&18&0&50&76&0\\0&0&7&0&9&0&28&22&0&19&0&50&0&0&106\\3&0&11&0&7&0&22&43&0&20&0&52&0&0&126\\0&5&0&8&0&33&0&0&56&0&23&0&78&122&0\\0&0&2&0&11&0&19&20&0&30&0&45&0&0&106\\0&4&0&3&0&18&0&0&23&0&18&0&34&60&0\\1&0&13&0&22&0&50&52&0&45&0&117&0&0&242\\0&5&0&16&0&50&0&0&78&0&34&0&138&189&0\\0&11&0&18&0&76&0&0&122&0&60&0&189&300&0\\2&0&26&0&34&0&106&126&0&106&0&242&0&0&588\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&4&9&26&28&43&56&30&18&117&138&300&588&298&316&708&609&182\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $1/4$ | $0$ | $1/4$ |
|---|
| $a_1=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $1/4$ | $0$ | $1/4$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $1/4$ | $0$ | $1/4$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $1/4$ | $0$ | $1/4$ |
|---|
