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Properties

Label 1.6.N.72.30b
  
Name \(B(6,6)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $72$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_6\wr C_2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$72$
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:$C_6\wr C_2$
Order:$72$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{18}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{18}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{9}^{8} & 0 & 0 & 0 \\0 & 0 & 0 &\zeta_{18}^{17} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{18}^{17} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{9}^{1} \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0& 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$B(1,12)$, $B(3,6)$, $A(6,6)$, $B(3,6;2)$, $B(6,2)$
Minimal supergroups:$J(B(6,6))$, $D(6,6)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $48$ $0$ $940$ $0$ $22400$ $0$ $586404$ $0$ $16251312$
$a_2$ $1$ $2$ $12$ $95$ $932$ $10137$ $117138$ $1408773$ $17446752$ $221124029$ $2857031702$ $37531754493$ $500299866140$
$a_3$ $1$ $0$ $18$ $0$ $2478$ $0$ $554820$ $0$ $147083230$ $0$ $43065290268$ $0$ $13554887693248$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $12$ $6$ $22$ $48$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $18$ $95$ $56$ $198$ $114$ $426$ $940$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $146$ $932$ $538$ $318$ $2032$ $1176$ $4494$ $2580$ $10000$ $22400$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1508$ $10137$ $864$ $5790$ $3318$ $22618$ $12894$ $7388$ $50772$ $28880$ $114420$ $64890$ $258650$ $586404$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $2478$ $16760$ $117138$ $9558$ $66254$ $37572$ $264786$ $21336$ $149490$ $84572$ $600528$ $338380$ $191110$ $1365460$
$$ $768040$ $3111612$ $1747116$ $7104636$ $16251312$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&4&0&3&0&4&0&0&8\\0&4&0&2&0&12&0&0&16&0&10&0&18&34&0\\1&0&9&0&8&0&23&23&0&14&0&45&0&0&88\\0&2&0&10&0&20&0&0&24&0&10&0&52&58&0\\2&0&8&0&16&0&24&35&0&29&0&61&0&0&120\\0&12&0&20&0&76&0&0&108&0&56&0&168&248&0\\1&0&23&0&24&0&83&79&0&54&0&163&0&0&344\\4&0&23&0&35&0&79&108&0&85&0&187&0&0&408\\0&16&0&24&0&108&0&0&176&0&84&0&252&396&0\\3&0&14&0&29&0&54&85&0&81&0&142&0&0&328\\0&10&0&10&0&56&0&0&84&0&50&0&118&196&0\\4&0&45&0&61&0&163&187&0&142&0&372&0&0&792\\0&18&0&52&0&168&0&0&252&0&118&0&448&612&0\\0&34&0&58&0&248&0&0&396&0&196&0&612&948&0\\8&0&88&0&120&0&344&408&0&328&0&792&0&0&1800\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&9&10&16&76&83&108&176&81&50&372&448&948&1800&870&860&2026&1638&496\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$1/36$$0$$1/36$$0$$0$$0$
$a_1=0$$1/36$$1/36$$0$$1/36$$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$