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Properties

Label 1.6.N.32.11b
  
Name \(B(4,4)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $32$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_4\wr C_2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$32$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)_3$
$\mathbb{R}$-dimension:$1$
Description:$\left\{\begin{bmatrix}\alpha I_3&0,\\ 0&\bar \alpha I_3\end{bmatrix}: \alpha\bar\alpha=1,\ \alpha\in\mathbb{C}\right\}$ Symplectic form:$\begin{bmatrix} 0 & I_3\\ -I_3 & 0\end{bmatrix}$
Hodge circle:$u\mapsto \mathrm{diag}(u,u, u, \bar u, \bar u, \bar u)$

Component group

Name:$C_4\wr C_2$
Order:$32$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & i & 0 & 0 & 0 & 0 \\0 & 0 & -i & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i\\\end{bmatrix}, \begin{bmatrix}\zeta_{12}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 &\zeta_{12}^{11} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0& 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$A(4,4)$, $B(2,4)$, $B(2,4;4)$
Minimal supergroups:$D(4,4)$, $J(B(4,4))$, $B(O,2)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $48$ $0$ $940$ $0$ $22610$ $0$ $612864$ $0$ $18053112$
$a_2$ $1$ $2$ $12$ $95$ $935$ $10302$ $122508$ $1540653$ $20199495$ $273030170$ $3772071992$ $52930009233$ $750982446893$
$a_3$ $1$ $0$ $18$ $0$ $2496$ $0$ $599400$ $0$ $181951196$ $0$ $61839886008$ $0$ $22160803720440$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $12$ $6$ $22$ $48$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $18$ $95$ $56$ $198$ $114$ $426$ $940$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $146$ $935$ $538$ $318$ $2038$ $1176$ $4512$ $2580$ $10060$ $22610$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1514$ $10302$ $864$ $5844$ $3330$ $23044$ $13038$ $7424$ $51942$ $29300$ $117720$ $66150$ $268016$ $612864$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $2496$ $17114$ $122508$ $9684$ $68546$ $38490$ $278766$ $21660$ $155556$ $87056$ $637416$ $354700$ $198010$ $1463284$
$$ $812224$ $3371004$ $1866564$ $7790076$ $18053112$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&4&0&3&0&4&0&0&8\\0&4&0&2&0&12&0&0&16&0&10&0&18&34&0\\1&0&9&0&8&0&23&23&0&14&0&45&0&0&88\\0&2&0&10&0&20&0&0&24&0&10&0&52&58&0\\2&0&8&0&16&0&24&35&0&29&0&61&0&0&120\\0&12&0&20&0&76&0&0&108&0&56&0&168&254&0\\1&0&23&0&24&0&83&79&0&54&0&163&0&0&356\\4&0&23&0&35&0&79&111&0&88&0&190&0&0&432\\0&16&0&24&0&108&0&0&182&0&84&0&258&420&0\\3&0&14&0&29&0&54&88&0&90&0&145&0&0&364\\0&10&0&10&0&56&0&0&84&0&50&0&118&202&0\\4&0&45&0&61&0&163&190&0&145&0&381&0&0&840\\0&18&0&52&0&168&0&0&258&0&118&0&466&654&0\\0&34&0&58&0&254&0&0&420&0&202&0&654&1044&0\\8&0&88&0&120&0&356&432&0&364&0&840&0&0&2052\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&9&10&16&76&83&111&182&90&50&381&466&1044&2052&1074&1070&2638&2301&742\end{bmatrix}$

Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.