Properties

Label 1.6.N.48.38a
  
Name \(J(B(6,2))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $48$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(S_3\times D_4\)

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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:$S_3\times D_4$
Order:$48$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0& 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1& 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0\\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(B(3,2))$, $J_s(B(3,2;2))$, $J_s(B(3,2))$, $B(6,2)$, $J(B(1,4)_2)$, $J(A(6,2))$, $J(B(3,2;2))$, $J_s(A(6,2))$
Minimal supergroups:$J(B(6,6))$, $J(D(6,2))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $24$ $0$ $470$ $0$ $11830$ $0$ $342342$ $0$ $10703616$
$a_2$ $1$ $2$ $9$ $56$ $500$ $5517$ $68237$ $897815$ $12239640$ $170467001$ $2406395669$ $34274151705$ $491210122674$
$a_3$ $1$ $0$ $9$ $0$ $1301$ $0$ $347160$ $0$ $113880725$ $0$ $40194208554$ $0$ $14631506404790$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $9$ $3$ $11$ $24$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $9$ $56$ $28$ $99$ $57$ $213$ $470$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $73$ $500$ $272$ $161$ $1043$ $599$ $2327$ $1320$ $5225$ $11830$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $778$ $5517$ $441$ $3060$ $1730$ $12323$ $6904$ $3898$ $28101$ $15682$ $64398$ $35763$ $148183$ $342342$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $1301$ $9184$ $68237$ $5150$ $37657$ $20929$ $156783$ $11658$ $86552$ $47920$ $362846$ $199724$ $110278$ $842399$
$$ $462511$ $1961162$ $1074150$ $4576698$ $10703616$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&0&0&1&0&0&4\\0&2&0&1&0&6&0&0&8&0&5&0&9&17&0\\1&0&6&0&2&0&10&15&0&5&0&21&0&0&47\\0&1&0&5&0&10&0&0&12&0&5&0&28&30&0\\0&0&2&0&11&0&14&12&0&20&0&33&0&0&66\\0&6&0&10&0&38&0&0&57&0&28&0&90&136&0\\0&0&10&0&14&0&45&37&0&31&0&89&0&0&202\\4&0&15&0&12&0&37&70&0&43&0&97&0&0&249\\0&8&0&12&0&57&0&0&100&0&47&0&146&242&0\\0&0&5&0&20&0&31&43&0&59&0&90&0&0&218\\0&5&0&5&0&28&0&0&47&0&25&0&64&112&0\\1&0&21&0&33&0&89&97&0&90&0&210&0&0&498\\0&9&0&28&0&90&0&0&146&0&64&0&272&381&0\\0&17&0&30&0&136&0&0&242&0&112&0&381&616&0\\4&0&47&0&66&0&202&249&0&218&0&498&0&0&1257\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&5&11&38&45&70&100&59&25&210&272&616&1257&679&692&1685&1499&494\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$13/24$$1/24$$1/8$$0$$1/12$$7/24$
$a_1=0$$13/24$$13/24$$1/24$$1/8$$0$$1/12$$7/24$
$a_3=0$$1/2$$1/2$$1/24$$1/12$$0$$1/12$$7/24$
$a_1=a_3=0$$1/2$$1/2$$1/24$$1/12$$0$$1/12$$7/24$