| Name: | $\mathrm{U}(1)\times\mathrm{SU}(2)_2$ |
| $\mathbb{R}$-dimension: | $4$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&\overline{B}\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),B\in\mathrm{SU}(2)\right\}$ |
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: | $\mathrm{diag}(u,\bar u, u,\bar u,\bar u,u)$ |
| Name: | $D_4$ |
| Order: | $8$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$24$ |
$0$ |
$340$ |
$0$ |
$6454$ |
$0$ |
$142674$ |
$0$ |
$3470016$ |
| $a_2$ |
$1$ |
$2$ |
$8$ |
$44$ |
$329$ |
$2962$ |
$29790$ |
$321400$ |
$3642343$ |
$42827054$ |
$518216132$ |
$6415400884$ |
$80904420195$ |
| $a_3$ |
$1$ |
$0$ |
$10$ |
$0$ |
$780$ |
$0$ |
$129960$ |
$0$ |
$28651476$ |
$0$ |
$7309899048$ |
$0$ |
$2038904089200$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$8$ |
$4$ |
$12$ |
$24$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$10$ |
$44$ |
$26$ |
$80$ |
$48$ |
$161$ |
$340$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$60$ |
$329$ |
$194$ |
$120$ |
$662$ |
$398$ |
$1394$ |
$830$ |
$2980$ |
$6454$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$496$ |
$2962$ |
$294$ |
$1736$ |
$1028$ |
$6279$ |
$3690$ |
$2188$ |
$13559$ |
$7950$ |
$29536$ |
$17248$ |
$64736$ |
$142674$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$780$ |
$4684$ |
$29790$ |
$2752$ |
$17260$ |
$10056$ |
$64868$ |
$5874$ |
$37546$ |
$21808$ |
$142488$ |
$82264$ |
$47682$ |
$314668$ |
| $$ |
$181244$ |
$697920$ |
$401016$ |
$1553640$ |
$3470016$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&0&2&0&1&0&1&0&0&2\\0&3&0&1&0&5&0&0&5&0&4&0&4&10&0\\1&0&5&0&3&0&8&8&0&3&0&13&0&0&22\\0&1&0&5&0&7&0&0&7&0&2&0&16&14&0\\1&0&3&0&8&0&7&10&0&9&0&19&0&0&28\\0&5&0&7&0&25&0&0&29&0&17&0&43&62&0\\0&0&8&0&7&0&27&20&0&12&0&42&0&0&80\\2&0&8&0&10&0&20&33&0&21&0&45&0&0&92\\0&5&0&7&0&29&0&0&47&0&20&0&56&90&0\\1&0&3&0&9&0&12&21&0&26&0&30&0&0&72\\0&4&0&2&0&17&0&0&20&0&18&0&26&46&0\\1&0&13&0&19&0&42&45&0&30&0&93&0&0&166\\0&4&0&16&0&43&0&0&56&0&26&0&108&128&0\\0&10&0&14&0&62&0&0&90&0&46&0&128&205&0\\2&0&22&0&28&0&80&92&0&72&0&166&0&0&368\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&5&5&8&25&27&33&47&26&18&93&108&205&368&187&174&396&315&109\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
