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: | $C_2^2\times S_4$ |
Order: | $96$ |
Abelian: | no |
Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 \\0 & 0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0& 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0& -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0& 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$ |
Maximal subgroups: | $L_1(J(O))$, $L_2(O)$, $L(J(O),J(T))$, $L_2(J(D_3))$, $L(J(O),O_1)$, $L_2(J(T))$, $L(J(O),O)$, $L_2(O_1)$, $L_2(J(D_4))$ |
Minimal supergroups: | |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$15$ |
$0$ |
$195$ |
$0$ |
$3500$ |
$0$ |
$75852$ |
$0$ |
$1864863$ |
$a_2$ |
$1$ |
$2$ |
$7$ |
$32$ |
$203$ |
$1647$ |
$15785$ |
$168534$ |
$1937038$ |
$23475674$ |
$295926932$ |
$3842467884$ |
$51025882748$ |
$a_3$ |
$1$ |
$0$ |
$7$ |
$0$ |
$438$ |
$0$ |
$67460$ |
$0$ |
$15590400$ |
$0$ |
$4403395332$ |
$0$ |
$1384627920654$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$7$ |
$3$ |
$8$ |
$15$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$7$ |
$32$ |
$17$ |
$49$ |
$30$ |
$95$ |
$195$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$38$ |
$203$ |
$115$ |
$72$ |
$375$ |
$228$ |
$773$ |
$465$ |
$1630$ |
$3500$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$284$ |
$1647$ |
$171$ |
$956$ |
$572$ |
$3377$ |
$1999$ |
$1195$ |
$7233$ |
$4265$ |
$15695$ |
$9205$ |
$34370$ |
$75852$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$438$ |
$2518$ |
$15785$ |
$1492$ |
$9129$ |
$5346$ |
$34090$ |
$3143$ |
$19779$ |
$11529$ |
$74943$ |
$43315$ |
$25160$ |
$166025$ |
$$ |
$95620$ |
$370034$ |
$212373$ |
$828870$ |
$1864863$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&2&0&0&0&0&0&0&1\\0&2&0&1&0&3&0&0&3&0&2&0&2&5&0\\1&0&4&0&1&0&4&6&0&1&0&6&0&0&11\\0&1&0&3&0&4&0&0&4&0&1&0&8&7&0\\0&0&1&0&6&0&5&3&0&6&0&11&0&0&14\\0&3&0&4&0&14&0&0&15&0&9&0&22&32&0\\0&0&4&0&5&0&15&9&0&7&0&22&0&0&40\\2&0&6&0&3&0&9&22&0&7&0&21&0&0&46\\0&3&0&4&0&15&0&0&24&0&10&0&28&45&0\\0&0&1&0&6&0&7&7&0&16&0&16&0&0&36\\0&2&0&1&0&9&0&0&10&0&10&0&14&24&0\\0&0&6&0&11&0&22&21&0&16&0&51&0&0&85\\0&2&0&8&0&22&0&0&28&0&14&0&56&66&0\\0&5&0&7&0&32&0&0&45&0&24&0&66&107&0\\1&0&11&0&14&0&40&46&0&36&0&85&0&0&190\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&3&6&14&15&22&24&16&10&51&56&107&190&99&104&217&185&66\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $1/2$ | $1/48$ | $0$ | $1/8$ | $1/6$ | $3/16$ |
---|
$a_1=0$ | $11/32$ | $1/4$ | $1/96$ | $0$ | $1/16$ | $1/12$ | $3/32$ |
---|
$a_3=0$ | $13/32$ | $5/16$ | $1/96$ | $0$ | $1/8$ | $1/12$ | $3/32$ |
---|
$a_1=a_3=0$ | $11/32$ | $1/4$ | $1/96$ | $0$ | $1/16$ | $1/12$ | $3/32$ |
---|