| Name: | $\mathrm{SU}(2)\times\mathrm{U}(1)_2$ |
| $\mathbb{R}$-dimension: | $4$ |
| Description: | $\left\{\begin{bmatrix}A&0&0\\0&\alpha I_2&0\\0&0&\bar\alpha I_2\end{bmatrix}: A\in\mathrm{SU}(2),\ \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ |
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: | $A_4$ |
| Order: | $12$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 \\0 & 0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$26$ |
$0$ |
$365$ |
$0$ |
$6874$ |
$0$ |
$155274$ |
$0$ |
$3914196$ |
| $a_2$ |
$1$ |
$2$ |
$9$ |
$52$ |
$381$ |
$3322$ |
$32895$ |
$355756$ |
$4085037$ |
$48870946$ |
$601805451$ |
$7569609192$ |
$96772487311$ |
| $a_3$ |
$1$ |
$0$ |
$12$ |
$0$ |
$872$ |
$0$ |
$141370$ |
$0$ |
$32237240$ |
$0$ |
$8538872916$ |
$0$ |
$2447165491848$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$5$ |
$14$ |
$26$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$12$ |
$52$ |
$31$ |
$93$ |
$56$ |
$180$ |
$365$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$72$ |
$381$ |
$225$ |
$136$ |
$746$ |
$444$ |
$1528$ |
$905$ |
$3210$ |
$6874$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$568$ |
$3322$ |
$336$ |
$1941$ |
$1142$ |
$6919$ |
$4040$ |
$2372$ |
$14768$ |
$8595$ |
$32015$ |
$18564$ |
$70196$ |
$155274$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$872$ |
$5196$ |
$32895$ |
$3032$ |
$18973$ |
$10988$ |
$71258$ |
$6384$ |
$41000$ |
$23664$ |
$156466$ |
$89757$ |
$51648$ |
$346772$ |
| $$ |
$198380$ |
$773934$ |
$441630$ |
$1736700$ |
$3914196$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&2&0&1&0&1&0&0&2\\0&3&0&2&0&6&0&0&6&0&3&0&5&11&0\\1&0&6&0&4&0&9&10&0&4&0&13&0&0&24\\0&2&0&5&0&8&0&0&9&0&3&0&13&18&0\\1&0&4&0&7&0&9&11&0&7&0&19&0&0&30\\0&6&0&8&0&28&0&0&32&0&16&0&44&68&0\\1&0&9&0&9&0&26&25&0&16&0&43&0&0&86\\2&0&10&0&11&0&25&34&0&20&0&48&0&0&100\\0&6&0&9&0&32&0&0&49&0&19&0&63&98&0\\1&0&4&0&7&0&16&20&0&20&0&33&0&0&76\\0&3&0&3&0&16&0&0&19&0&17&0&32&48&0\\1&0&13&0&19&0&43&48&0&33&0&99&0&0&184\\0&5&0&13&0&44&0&0&63&0&32&0&107&146&0\\0&11&0&18&0&68&0&0&98&0&48&0&146&227&0\\2&0&24&0&30&0&86&100&0&76&0&184&0&0&408\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&6&5&7&28&26&34&49&20&17&99&107&227&408&185&199&425&328&91\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.
