| 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: | $C_2\times D_4$ |
| Order: | $16$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 &0 & \zeta_{8}^{1} \\0 & 0 & 0 & 0 & \zeta_{8}^{3} & 0 \\0 & 0 & 0 & \zeta_{8}^{3} & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0& 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0& 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $L(D_4,D_2)$, $L(D_{4,2},D_{2,1})$, $L(D_{4,1},C_{4,1})$, $L(J(D_2),D_{2,1})$, $L(J(D_2),D_2)$, $L_1(D_{4,1})$, $L(J(C_4),C_{4,1})$ |
| Minimal supergroups: | $L_2(J(D_4))$${}^{\times 2}$, $L(J(O),O_1)$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$18$ |
$0$ |
$335$ |
$0$ |
$8575$ |
$0$ |
$245637$ |
$0$ |
$7394310$ |
| $a_2$ |
$1$ |
$2$ |
$8$ |
$44$ |
$372$ |
$4032$ |
$48986$ |
$626187$ |
$8223018$ |
$109860596$ |
$1486300278$ |
$20307098847$ |
$279693611379$ |
| $a_3$ |
$1$ |
$0$ |
$8$ |
$0$ |
$957$ |
$0$ |
$243140$ |
$0$ |
$73167017$ |
$0$ |
$23524366068$ |
$0$ |
$7868208907812$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$8$ |
$3$ |
$9$ |
$18$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$8$ |
$44$ |
$22$ |
$73$ |
$42$ |
$153$ |
$335$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$57$ |
$372$ |
$201$ |
$118$ |
$759$ |
$433$ |
$1683$ |
$950$ |
$3780$ |
$8575$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$575$ |
$4032$ |
$327$ |
$2232$ |
$1260$ |
$8941$ |
$5008$ |
$2820$ |
$20325$ |
$11350$ |
$46470$ |
$25865$ |
$106666$ |
$245637$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$957$ |
$6676$ |
$48986$ |
$3747$ |
$27136$ |
$15125$ |
$111871$ |
$8445$ |
$62073$ |
$34504$ |
$257483$ |
$142597$ |
$79116$ |
$594303$ |
| $$ |
$328608$ |
$1374751$ |
$759024$ |
$3185826$ |
$7394310$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&3&0&0&0&0&0&0&3\\0&2&0&1&0&4&0&0&6&0&3&0&6&12&0\\1&0&5&0&1&0&7&11&0&5&0&14&0&0&35\\0&1&0&4&0&7&0&0&10&0&3&0&19&22&0\\0&0&1&0&8&0&10&8&0&14&0&24&0&0&49\\0&4&0&7&0&27&0&0&42&0&20&0&66&101&0\\0&0&7&0&10&0&33&28&0&25&0&64&0&0&147\\3&0&11&0&8&0&28&52&0&28&0&71&0&0&179\\0&6&0&10&0&42&0&0&73&0&33&0&108&172&0\\0&0&5&0&14&0&25&28&0&40&0&65&0&0&151\\0&3&0&3&0&20&0&0&33&0&19&0&49&82&0\\0&0&14&0&24&0&64&71&0&65&0&157&0&0&355\\0&6&0&19&0&66&0&0&108&0&49&0&192&274&0\\0&12&0&22&0&101&0&0&172&0&82&0&274&436&0\\3&0&35&0&49&0&147&179&0&151&0&355&0&0&856\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&4&8&27&33&52&73&40&19&157&192&436&856&434&454&1048&899&267\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/16$ | $0$ | $1/8$ | $0$ | $5/16$ |
|---|
| $a_1=0$ | $3/8$ | $1/4$ | $1/16$ | $0$ | $0$ | $0$ | $3/16$ |
|---|
| $a_3=0$ | $1/2$ | $3/8$ | $1/16$ | $0$ | $1/8$ | $0$ | $3/16$ |
|---|
| $a_1=a_3=0$ | $3/8$ | $1/4$ | $1/16$ | $0$ | $0$ | $0$ | $3/16$ |
|---|
