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Properties

Label 1.6.N.192.956a
  
Name \(J(D(4,4))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $192$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_4^2:D_6\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$192$
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:$C_4^2:D_6$
Order:$192$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & i & 0 & 0 & 0 & 0 \\0 & 0 & -i & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i\\\end{bmatrix}, \begin{bmatrix}\zeta_{12}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 &\zeta_{12}^{11} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0& 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 &0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $9$ $0$ $160$ $0$ $3780$ $0$ $102186$ $0$ $3009006$
$a_2$ $1$ $1$ $4$ $21$ $171$ $1761$ $20548$ $257160$ $3367724$ $45508422$ $628688769$ $8821698315$ $125163830949$
$a_3$ $1$ $0$ $4$ $0$ $423$ $0$ $99970$ $0$ $30326037$ $0$ $10306658694$ $0$ $3693467440080$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $4$ $1$ $4$ $9$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $4$ $21$ $10$ $34$ $19$ $72$ $160$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $25$ $171$ $91$ $55$ $342$ $198$ $755$ $430$ $1680$ $3780$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $255$ $1761$ $144$ $978$ $557$ $3847$ $2177$ $1244$ $8664$ $4890$ $19630$ $11025$ $44681$ $102186$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $423$ $2859$ $20548$ $1620$ $11435$ $6423$ $46478$ $3610$ $25938$ $14516$ $106255$ $59130$ $33025$ $243904$
$$ $135394$ $561869$ $311094$ $1298388$ $3009006$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&0&2&0&0&0&0&0&0&1\\0&1&0&0&0&2&0&0&3&0&2&0&2&6&0\\0&0&3&0&0&0&3&5&0&1&0&7&0&0&15\\0&0&0&3&0&3&0&0&3&0&1&0&11&9&0\\0&0&0&0&5&0&4&4&0&8&0&11&0&0&20\\0&2&0&3&0&13&0&0&18&0&9&0&28&42&0\\0&0&3&0&4&0&17&10&0&7&0&28&0&0&59\\2&0&5&0&4&0&10&26&0&13&0&28&0&0&72\\0&3&0&3&0&18&0&0&32&0&15&0&40&71&0\\0&0&1&0&8&0&7&13&0&24&0&26&0&0&60\\0&2&0&1&0&9&0&0&15&0&10&0&17&35&0\\0&0&7&0&11&0&28&28&0&26&0&66&0&0&140\\0&2&0&11&0&28&0&0&40&0&17&0&85&106&0\\0&6&0&9&0&42&0&0&71&0&35&0&106&176&0\\1&0&15&0&20&0&59&72&0&60&0&140&0&0&343\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&3&3&5&13&17&26&32&24&10&66&85&176&343&188&195&454&399&143\end{bmatrix}$

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$$1/16$$1/3$$1/8$$0$$7/48$
$a_1=0$$2/3$$2/3$$1/16$$1/3$$1/8$$0$$7/48$
$a_3=0$$1/2$$1/2$$1/16$$1/6$$1/8$$0$$7/48$
$a_1=a_3=0$$1/2$$1/2$$1/16$$1/6$$1/8$$0$$7/48$