| 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: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $L(D_{4,1},D_2)$${}^{\times 2}$, $L(J(D_2),D_2)$${}^{\times 2}$, $L(D_{4,2},C_4)$, $L_1(D_4)$, $L(J(C_4),C_4)$ |
| Minimal supergroups: | $L_2(J(D_4))$, $L(J(O),O)$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$24$ |
$0$ |
$470$ |
$0$ |
$11235$ |
$0$ |
$297612$ |
$0$ |
$8425032$ |
| $a_2$ |
$1$ |
$2$ |
$9$ |
$56$ |
$492$ |
$5172$ |
$59691$ |
$726945$ |
$9178434$ |
$119000576$ |
$1574504409$ |
$21165153495$ |
$288100598979$ |
| $a_3$ |
$1$ |
$0$ |
$9$ |
$0$ |
$1242$ |
$0$ |
$284820$ |
$0$ |
$79207730$ |
$0$ |
$24435080964$ |
$0$ |
$8010398049192$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$3$ |
$11$ |
$24$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$9$ |
$56$ |
$28$ |
$99$ |
$57$ |
$213$ |
$470$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$73$ |
$492$ |
$269$ |
$159$ |
$1017$ |
$588$ |
$2250$ |
$1290$ |
$5010$ |
$11235$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$755$ |
$5172$ |
$432$ |
$2904$ |
$1661$ |
$11380$ |
$6471$ |
$3700$ |
$25581$ |
$14510$ |
$57760$ |
$32655$ |
$130886$ |
$297612$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1242$ |
$8439$ |
$59691$ |
$4800$ |
$33509$ |
$18939$ |
$134721$ |
$10722$ |
$75756$ |
$42700$ |
$306406$ |
$171910$ |
$96705$ |
$699014$ |
| $$ |
$391384$ |
$1598968$ |
$893466$ |
$3666306$ |
$8425032$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&0&0&1&0&0&4\\0&2&0&1&0&6&0&0&8&0&5&0&9&17&0\\1&0&6&0&2&0&10&15&0&5&0&21&0&0&44\\0&1&0&5&0&10&0&0&12&0&5&0&26&29&0\\0&0&2&0&11&0&14&12&0&18&0&33&0&0&60\\0&6&0&10&0&38&0&0&54&0&28&0&84&125&0\\0&0&10&0&14&0&43&36&0&29&0&83&0&0&174\\4&0&15&0&12&0&36&66&0&35&0&88&0&0&208\\0&8&0&12&0&54&0&0&89&0&42&0&127&202&0\\0&0&5&0&18&0&29&35&0&48&0&76&0&0&170\\0&5&0&5&0&28&0&0&42&0&25&0&59&99&0\\1&0&21&0&33&0&83&88&0&76&0&191&0&0&404\\0&9&0&26&0&84&0&0&127&0&59&0&227&313&0\\0&17&0&29&0&125&0&0&202&0&99&0&313&490&0\\4&0&44&0&60&0&174&208&0&170&0&404&0&0&942\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&5&11&38&43&66&89&48&25&191&227&490&942&469&490&1113&940&285\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/16$ | $0$ | $1/8$ | $0$ | $5/16$ |
|---|
| $a_1=0$ | $1/2$ | $1/2$ | $1/16$ | $0$ | $1/8$ | $0$ | $5/16$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $1/16$ | $0$ | $1/8$ | $0$ | $5/16$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $1/16$ | $0$ | $1/8$ | $0$ | $5/16$ |
|---|
