| 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: | $C_2\times A_4$ |
| Order: | $24$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 \\0 & 0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$14$ |
$0$ |
$185$ |
$0$ |
$3444$ |
$0$ |
$77658$ |
$0$ |
$1957164$ |
| $a_2$ |
$1$ |
$2$ |
$7$ |
$32$ |
$206$ |
$1702$ |
$16560$ |
$178194$ |
$2043424$ |
$24438104$ |
$300910448$ |
$3784827422$ |
$48386311451$ |
| $a_3$ |
$1$ |
$0$ |
$7$ |
$0$ |
$442$ |
$0$ |
$70740$ |
$0$ |
$16119222$ |
$0$ |
$4269443640$ |
$0$ |
$1223582836080$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$7$ |
$3$ |
$8$ |
$14$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$7$ |
$32$ |
$17$ |
$49$ |
$29$ |
$92$ |
$185$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$38$ |
$206$ |
$116$ |
$70$ |
$379$ |
$225$ |
$769$ |
$455$ |
$1610$ |
$3444$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$290$ |
$1702$ |
$170$ |
$980$ |
$575$ |
$3475$ |
$2027$ |
$1191$ |
$7396$ |
$4305$ |
$16020$ |
$9289$ |
$35112$ |
$77658$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$442$ |
$2614$ |
$16560$ |
$1524$ |
$9512$ |
$5506$ |
$35670$ |
$3197$ |
$20519$ |
$11842$ |
$78264$ |
$44896$ |
$25838$ |
$173416$ |
| $$ |
$99211$ |
$387002$ |
$220836$ |
$868392$ |
$1957164$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&0&0&0&0&0&1\\0&2&0&1&0&3&0&0&3&0&1&0&2&6&0\\1&0&4&0&1&0&4&6&0&1&0&6&0&0&12\\0&1&0&3&0&4&0&0&4&0&1&0&7&9&0\\0&0&1&0&5&0&5&4&0&5&0&10&0&0&15\\0&3&0&4&0&14&0&0&16&0&8&0&22&34&0\\0&0&4&0&5&0&14&11&0&8&0&22&0&0&43\\2&0&6&0&4&0&11&21&0&9&0&22&0&0&50\\0&3&0&4&0&16&0&0&26&0&10&0&30&49&0\\0&0&1&0&5&0&8&9&0&13&0&17&0&0&38\\0&1&0&1&0&8&0&0&10&0&10&0&16&23&0\\0&0&6&0&10&0&22&22&0&17&0&51&0&0&92\\0&2&0&7&0&22&0&0&30&0&16&0&56&72&0\\0&6&0&9&0&34&0&0&49&0&23&0&72&115&0\\1&0&12&0&15&0&43&50&0&38&0&92&0&0&204\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&3&5&14&14&21&26&13&10&51&56&115&204&95&107&219&170&52\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/24$ | $0$ | $0$ | $1/3$ | $1/8$ |
|---|
| $a_1=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
