| 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: | $C_4$ |
| Order: | $4$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 &0 & \zeta_{8}^{1} \\0 & 0 & 0 & 0 & \zeta_{8}^{3} & 0 \\0 & 0 & 0 & \zeta_{8}^{3} & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $L_1(C_2)$ |
| Minimal supergroups: | $L_2(C_{4,1})$, $L_1(D_{4,1})$, $L(J(C_4),C_{4,1})$, $L(D_{4,1},C_{4,1})$, $L_1(J(C_4))$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$54$ |
$0$ |
$1240$ |
$0$ |
$33670$ |
$0$ |
$978264$ |
$0$ |
$29546748$ |
| $a_2$ |
$1$ |
$2$ |
$12$ |
$113$ |
$1270$ |
$15337$ |
$192897$ |
$2492380$ |
$32839506$ |
$439210589$ |
$5944149907$ |
$81223529290$ |
$1118751586795$ |
| $a_3$ |
$1$ |
$0$ |
$18$ |
$0$ |
$3570$ |
$0$ |
$966600$ |
$0$ |
$292513298$ |
$0$ |
$94093174728$ |
$0$ |
$31472712201480$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$4$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$12$ |
$6$ |
$24$ |
$54$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$18$ |
$113$ |
$64$ |
$246$ |
$138$ |
$548$ |
$1240$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$182$ |
$1270$ |
$714$ |
$406$ |
$2852$ |
$1604$ |
$6464$ |
$3620$ |
$14720$ |
$33670$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$2120$ |
$15337$ |
$1188$ |
$8562$ |
$4790$ |
$34986$ |
$19502$ |
$10900$ |
$80142$ |
$44600$ |
$184112$ |
$102270$ |
$423948$ |
$978264$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$3570$ |
$25976$ |
$192897$ |
$14482$ |
$106994$ |
$59440$ |
$444074$ |
$33060$ |
$245982$ |
$136436$ |
$1024756$ |
$566872$ |
$314014$ |
$2369232$ |
| $$ |
$1309000$ |
$5486600$ |
$3027780$ |
$12723984$ |
$29546748$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&4&0&3&0&6&0&0&12\\0&4&0&2&0&14&0&0&20&0&12&0&28&50&0\\1&0&9&0&10&0&29&33&0&24&0&63&0&0&140\\0&2&0&10&0&26&0&0&40&0&16&0&70&94&0\\2&0&10&0&18&0&34&47&0&39&0&89&0&0&196\\0&14&0&26&0&104&0&0&166&0&80&0&264&404&0\\1&0&29&0&34&0&117&127&0&100&0&253&0&0&588\\4&0&33&0&47&0&127&162&0&133&0&305&0&0&716\\0&20&0&40&0&166&0&0&284&0&130&0&442&688&0\\3&0&24&0&39&0&100&133&0&119&0&250&0&0&604\\0&12&0&16&0&80&0&0&130&0&70&0&202&326&0\\6&0&63&0&89&0&253&305&0&250&0&610&0&0&1420\\0&28&0&70&0&264&0&0&442&0&202&0&744&1106&0\\0&50&0&94&0&404&0&0&688&0&326&0&1106&1728&0\\12&0&140&0&196&0&588&716&0&604&0&1420&0&0&3424\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&9&10&18&104&117&162&284&119&70&610&744&1728&3424&1698&1721&4148&3515&1004\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $1/2$ | $0$ | $0$ |
|---|
| $a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $1/2$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
