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Properties

Label 1.6.N.64.134a
  
Name \(J(B(4,4))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $64$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(D_4:D_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$64$
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:$D_4:D_4$
Order:$64$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & i & 0 & 0 & 0 & 0 \\0 & 0 & -i & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i\\\end{bmatrix}, \begin{bmatrix}\zeta_{12}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 &\zeta_{12}^{11} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0& 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0& 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0& 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$B(4,4)$, $J_s(B(2,4))$, $J(B(2,4;4))$, $J(B(2,4))$, $J_s(A(4,4))$, $J(A(4,4))$, $J_s(B(2,4;4))$
Minimal supergroups:$J(D(4,4))$, $J(B(O,2))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $24$ $0$ $470$ $0$ $11305$ $0$ $306432$ $0$ $9026556$
$a_2$ $1$ $2$ $9$ $56$ $493$ $5227$ $61482$ $771010$ $10101798$ $136521236$ $1886054449$ $26465059975$ $375491389522$
$a_3$ $1$ $0$ $9$ $0$ $1248$ $0$ $299700$ $0$ $90975598$ $0$ $30919943004$ $0$ $11080401860220$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $9$ $3$ $11$ $24$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $9$ $56$ $28$ $99$ $57$ $213$ $470$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $73$ $493$ $269$ $159$ $1019$ $588$ $2256$ $1290$ $5030$ $11305$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $757$ $5227$ $432$ $2922$ $1665$ $11522$ $6519$ $3712$ $25971$ $14650$ $58860$ $33075$ $134008$ $306432$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1248$ $8557$ $61482$ $4842$ $34273$ $19245$ $139383$ $10830$ $77778$ $43528$ $318708$ $177350$ $99005$ $731642$
$$ $406112$ $1685502$ $933282$ $3895038$ $9026556$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&0&0&1&0&0&4\\0&2&0&1&0&6&0&0&8&0&5&0&9&17&0\\1&0&6&0&2&0&10&15&0&5&0&21&0&0&44\\0&1&0&5&0&10&0&0&12&0&5&0&26&29&0\\0&0&2&0&11&0&14&12&0&18&0&33&0&0&60\\0&6&0&10&0&38&0&0&54&0&28&0&84&127&0\\0&0&10&0&14&0&43&36&0&29&0&83&0&0&178\\4&0&15&0&12&0&36&67&0&36&0&89&0&0&216\\0&8&0&12&0&54&0&0&91&0&42&0&129&210&0\\0&0&5&0&18&0&29&36&0&51&0&77&0&0&182\\0&5&0&5&0&28&0&0&42&0&25&0&59&101&0\\1&0&21&0&33&0&83&89&0&77&0&194&0&0&420\\0&9&0&26&0&84&0&0&129&0&59&0&233&327&0\\0&17&0&29&0&127&0&0&210&0&101&0&327&522&0\\4&0&44&0&60&0&178&216&0&182&0&420&0&0&1026\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&5&11&38&43&67&91&51&25&194&233&522&1026&537&561&1319&1164&371\end{bmatrix}$

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$$1/16$$0$$1/8$$0$$5/16$
$a_1=0$$1/2$$1/2$$1/16$$0$$1/8$$0$$5/16$
$a_3=0$$1/2$$1/2$$1/16$$0$$1/8$$0$$5/16$
$a_1=a_3=0$$1/2$$1/2$$1/16$$0$$1/8$$0$$5/16$