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Properties

Label 1.6.N.48.48a
  
Name \(J(D(2,2))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $48$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_2\times S_4\)

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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:$C_2\times S_4$
Order:$48$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0& 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0 & 0& 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $12$ $0$ $310$ $0$ $9590$ $0$ $310086$ $0$ $10230528$
$a_2$ $1$ $1$ $5$ $33$ $352$ $4486$ $60779$ $842710$ $11827232$ $167353638$ $2382741895$ $34093546180$ $489825550174$
$a_3$ $1$ $0$ $5$ $0$ $997$ $0$ $322160$ $0$ $111710165$ $0$ $39999780210$ $0$ $14613740036566$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $5$ $1$ $5$ $12$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $5$ $33$ $15$ $59$ $31$ $133$ $310$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $44$ $352$ $182$ $103$ $763$ $419$ $1767$ $960$ $4105$ $9590$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $572$ $4486$ $309$ $2416$ $1318$ $10307$ $5616$ $3074$ $24069$ $13106$ $56334$ $30611$ $132055$ $310086$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $997$ $7686$ $60779$ $4195$ $32953$ $17933$ $141999$ $9748$ $77144$ $41928$ $333278$ $180908$ $98294$ $783263$
$$ $424879$ $1842890$ $998886$ $4340154$ $10230528$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&0&3&0&0&0&0&0&0&3\\0&1&0&0&0&3&0&0&6&0&3&0&6&14&0\\0&0&4&0&0&0&6&10&0&5&0&16&0&0&42\\0&0&0&4&0&6&0&0&9&0&3&0&24&26&0\\0&0&0&0&7&0&9&10&0&17&0&25&0&0&60\\0&3&0&6&0&27&0&0&48&0&21&0&78&123&0\\0&0&6&0&9&0&37&31&0&26&0&76&0&0&189\\3&0&10&0&10&0&31&60&0&41&0&87&0&0&237\\0&6&0&9&0&48&0&0&91&0&42&0&135&230&0\\0&0&5&0&17&0&26&41&0&57&0&85&0&0&210\\0&3&0&3&0&21&0&0&42&0&21&0&57&105&0\\0&0&16&0&25&0&76&87&0&85&0&192&0&0&480\\0&6&0&24&0&78&0&0&135&0&57&0&258&366&0\\0&14&0&26&0&123&0&0&230&0&105&0&366&600&0\\3&0&42&0&60&0&189&237&0&210&0&480&0&0&1233\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&4&4&7&27&37&60&91&57&21&192&258&600&1233&669&684&1677&1489&492\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$2/3$$1/8$$1/3$$0$$0$$5/24$
$a_1=0$$2/3$$2/3$$1/8$$1/3$$0$$0$$5/24$
$a_3=0$$1/2$$1/2$$1/8$$1/6$$0$$0$$5/24$
$a_1=a_3=0$$1/2$$1/2$$1/8$$1/6$$0$$0$$5/24$