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Properties

Label 1.6.N.144.154a
  
Name \(J(B(6,6))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $144$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(D_6:D_6\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$144$
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:$D_6:D_6$
Order:$144$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{18}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{18}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{9}^{8} & 0 & 0 & 0 \\0 & 0 & 0 &\zeta_{18}^{17} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{18}^{17} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{9}^{1} \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0& 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0& 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0& 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(B(3,6;2))$, $B(6,6)$, $J(B(6,2))$, $J_s(B(3,6;2))$, $J(B(1,12))$, $J(A(6,6))$, $J_s(A(6,6))$, $J(B(3,6))$, $J_s(B(3,6))$
Minimal supergroups:$J(D(6,6))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $24$ $0$ $470$ $0$ $11200$ $0$ $293202$ $0$ $8125656$
$a_2$ $1$ $2$ $9$ $56$ $491$ $5142$ $58787$ $705035$ $8725311$ $110567798$ $1428533159$ $18765929085$ $250150088415$
$a_3$ $1$ $0$ $9$ $0$ $1239$ $0$ $277410$ $0$ $73541615$ $0$ $21532645134$ $0$ $6777443846624$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $9$ $3$ $11$ $24$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $9$ $56$ $28$ $99$ $57$ $213$ $470$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $73$ $491$ $269$ $159$ $1016$ $588$ $2247$ $1290$ $5000$ $11200$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $754$ $5142$ $432$ $2895$ $1659$ $11309$ $6447$ $3694$ $25386$ $14440$ $57210$ $32445$ $129325$ $293202$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1239$ $8380$ $58787$ $4779$ $33127$ $18786$ $132393$ $10668$ $74745$ $42286$ $300264$ $169190$ $95555$ $682730$
$$ $384020$ $1555806$ $873558$ $3552318$ $8125656$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&0&0&1&0&0&4\\0&2&0&1&0&6&0&0&8&0&5&0&9&17&0\\1&0&6&0&2&0&10&15&0&5&0&21&0&0&44\\0&1&0&5&0&10&0&0&12&0&5&0&26&29&0\\0&0&2&0&11&0&14&12&0&18&0&33&0&0&60\\0&6&0&10&0&38&0&0&54&0&28&0&84&124&0\\0&0&10&0&14&0&43&36&0&29&0&83&0&0&172\\4&0&15&0&12&0&36&65&0&35&0&88&0&0&204\\0&8&0&12&0&54&0&0&88&0&42&0&126&198&0\\0&0&5&0&18&0&29&35&0&46&0&75&0&0&164\\0&5&0&5&0&28&0&0&42&0&25&0&59&98&0\\1&0&21&0&33&0&83&88&0&75&0&189&0&0&396\\0&9&0&26&0&84&0&0&126&0&59&0&224&306&0\\0&17&0&29&0&124&0&0&198&0&98&0&306&474&0\\4&0&44&0&60&0&172&204&0&164&0&396&0&0&900\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&5&11&38&43&65&88&46&25&189&224&474&900&435&454&1013&831&248\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$37/72$$1/24$$7/72$$0$$1/12$$7/24$
$a_1=0$$37/72$$37/72$$1/24$$7/72$$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$