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Properties

Label 1.6.N.48.15b
  
Name \(J_s(B(1,12))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $48$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_3:D_8\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$48$
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:$C_3:D_8$
Order:$48$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{9}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{36}^{7} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{36}^{25} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{9}^{8} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{36}^{29} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{36}^{11} \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0\\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 &0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & -i & 0 & 0 \\0 &0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\-i & 0 & 0 & 0 & 0 & 0 \\0 & -i & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(A(1,12))$, $B(1,12)$, $J_n(A(1,12))$, $J_s(B(1,4)_2)$
Minimal supergroups:

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $27$ $0$ $620$ $0$ $16835$ $0$ $489132$ $0$ $14788158$
$a_2$ $1$ $2$ $9$ $65$ $660$ $7742$ $96682$ $1248067$ $16471462$ $221095544$ $3009132594$ $41434726577$ $576238837313$
$a_3$ $1$ $0$ $10$ $0$ $1794$ $0$ $483750$ $0$ $147290178$ $0$ $48134336400$ $0$ $16514014797500$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $9$ $3$ $12$ $27$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $10$ $65$ $33$ $124$ $69$ $274$ $620$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $92$ $660$ $359$ $206$ $1429$ $805$ $3235$ $1810$ $7360$ $16835$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1064$ $7742$ $594$ $4287$ $2398$ $17502$ $9757$ $5460$ $40080$ $22310$ $92066$ $51135$ $211974$ $489132$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1794$ $12998$ $96682$ $7250$ $53517$ $29733$ $222114$ $16530$ $123024$ $68232$ $512571$ $283506$ $157057$ $1185164$
$$ $654696$ $2744931$ $1514394$ $6366864$ $14788158$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&1&0&2&0&0&6\\0&2&0&1&0&7&0&0&11&0&6&0&13&25&0\\1&0&6&0&3&0&14&19&0&9&0&30&0&0&70\\0&1&0&6&0&13&0&0&18&0&7&0&38&46&0\\0&0&3&0&12&0&18&20&0&25&0&46&0&0&98\\0&7&0&13&0&52&0&0&83&0&40&0&132&202&0\\0&0&14&0&18&0&61&58&0&48&0&129&0&0&294\\4&0&19&0&20&0&58&93&0&64&0&147&0&0&358\\0&11&0&18&0&83&0&0&144&0&67&0&216&346&0\\1&0&9&0&25&0&48&64&0&71&0&127&0&0&302\\0&6&0&7&0&40&0&0&67&0&36&0&98&164&0\\2&0&30&0&46&0&129&147&0&127&0&308&0&0&710\\0&13&0&38&0&132&0&0&216&0&98&0&382&549&0\\0&25&0&46&0&202&0&0&346&0&164&0&549&866&0\\6&0&70&0&98&0&294&358&0&302&0&710&0&0&1713\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&6&12&52&61&93&144&71&36&308&382&866&1713&864&889&2108&1798&544\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$1/2$$0$$0$$1/4$$0$$1/4$
$a_1=0$$1/2$$1/2$$0$$0$$1/4$$0$$1/4$
$a_3=0$$1/2$$1/2$$0$$0$$1/4$$0$$1/4$
$a_1=a_3=0$$1/2$$1/2$$0$$0$$1/4$$0$$1/4$