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Properties

Label 1.6.N.48.17a
  
Name \(J_s(B(3,4;4))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $48$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(Q_8:S_3\)

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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:$Q_8:S_3$
Order:$48$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{5} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{7} & 0 \\0 & 0 & 0 & 0 & 0& \zeta_{12}^{7} \\\end{bmatrix}, \begin{bmatrix}i & 0 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & \zeta_{8}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -i& 0 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(A(3,4))$, $B(3,4;4)$, $J_n(A(3,4))$, $J_s(A(1,8)_2)$
Minimal supergroups:$J(B(O,2))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $24$ $0$ $480$ $0$ $12250$ $0$ $354312$ $0$ $10994676$
$a_2$ $1$ $2$ $9$ $56$ $510$ $5692$ $70502$ $923680$ $12515972$ $173301896$ $2434714554$ $34551898240$ $493898902130$
$a_3$ $1$ $0$ $9$ $0$ $1329$ $0$ $356580$ $0$ $115639349$ $0$ $40473795654$ $0$ $14673046553154$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $9$ $3$ $11$ $24$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $9$ $56$ $28$ $100$ $57$ $216$ $480$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $74$ $510$ $277$ $161$ $1070$ $609$ $2391$ $1350$ $5390$ $12250$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $798$ $5692$ $450$ $3152$ $1773$ $12732$ $7112$ $3994$ $29037$ $16170$ $66578$ $36946$ $153300$ $354312$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1329$ $9481$ $70502$ $5296$ $38898$ $21583$ $161848$ $12003$ $89321$ $49416$ $374152$ $205996$ $113696$ $867812$
$$ $476770$ $2018422$ $1106700$ $4705848$ $10994676$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&0&0&1&0&0&4\\0&2&0&1&0&6&0&0&8&0&5&0&9&18&0\\1&0&6&0&2&0&10&15&0&6&0&22&0&0&49\\0&1&0&5&0&10&0&0&13&0&5&0&27&33&0\\0&0&2&0&11&0&14&13&0&18&0&34&0&0&68\\0&6&0&10&0&39&0&0&59&0&29&0&93&143&0\\0&0&10&0&14&0&45&41&0&35&0&91&0&0&210\\4&0&15&0&13&0&41&70&0&43&0&102&0&0&258\\0&8&0&13&0&59&0&0&104&0&46&0&155&249&0\\0&0&6&0&18&0&35&43&0&53&0&92&0&0&224\\0&5&0&5&0&29&0&0&46&0&27&0&69&116&0\\1&0&22&0&34&0&91&102&0&92&0&220&0&0&512\\0&9&0&27&0&93&0&0&155&0&69&0&272&398&0\\0&18&0&33&0&143&0&0&249&0&116&0&398&635&0\\4&0&49&0&68&0&210&258&0&224&0&512&0&0&1287\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&5&11&39&45&70&104&53&27&220&272&635&1287&670&701&1681&1498&447\end{bmatrix}$

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$$1/12$$1/24$$0$$1/6$$1/4$
$a_1=0$$13/24$$13/24$$1/12$$1/24$$0$$1/6$$1/4$
$a_3=0$$1/2$$1/2$$1/12$$0$$0$$1/6$$1/4$
$a_1=a_3=0$$1/2$$1/2$$1/12$$0$$0$$1/6$$1/4$