Properties

Label 1.6.N.144.183a
  
Name \(J(D(6,2))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $144$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(S_3\times S_4\)

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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:$S_3\times S_4$
Order:$144$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0 & 0& 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $9$ $0$ $160$ $0$ $3955$ $0$ $114156$ $0$ $3568026$
$a_2$ $1$ $1$ $4$ $21$ $173$ $1856$ $22793$ $299405$ $4080261$ $56823432$ $802135079$ $11424726545$ $163736734846$
$a_3$ $1$ $0$ $4$ $0$ $442$ $0$ $115820$ $0$ $37961602$ $0$ $13398088944$ $0$ $4877169085690$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $4$ $1$ $4$ $9$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $4$ $21$ $10$ $34$ $19$ $72$ $160$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $25$ $173$ $92$ $56$ $350$ $202$ $779$ $440$ $1745$ $3955$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $262$ $1856$ $147$ $1024$ $579$ $4114$ $2306$ $1308$ $9375$ $5236$ $21478$ $11921$ $49406$ $114156$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $442$ $3068$ $22793$ $1724$ $12563$ $6986$ $52278$ $3886$ $28865$ $15982$ $120971$ $66592$ $36792$ $280829$
$$ $154203$ $653765$ $358050$ $1525608$ $3568026$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&0&2&0&0&0&0&0&0&1\\0&1&0&0&0&2&0&0&3&0&2&0&2&6&0\\0&0&3&0&0&0&3&5&0&1&0&7&0&0&16\\0&0&0&3&0&3&0&0&3&0&1&0&12&9&0\\0&0&0&0&5&0&4&4&0&9&0&11&0&0&22\\0&2&0&3&0&13&0&0&19&0&9&0&30&45&0\\0&0&3&0&4&0&18&10&0&7&0&30&0&0&67\\2&0&5&0&4&0&10&27&0&16&0&31&0&0&83\\0&3&0&3&0&19&0&0&35&0&17&0&45&82&0\\0&0&1&0&9&0&7&16&0&28&0&30&0&0&72\\0&2&0&1&0&9&0&0&17&0&10&0&18&39&0\\0&0&7&0&11&0&30&31&0&30&0&71&0&0&166\\0&2&0&12&0&30&0&0&45&0&18&0&100&123&0\\0&6&0&9&0&45&0&0&82&0&39&0&123&208&0\\1&0&16&0&22&0&67&83&0&72&0&166&0&0&420\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&3&3&5&13&18&27&35&28&10&71&100&208&420&240&238&582&513&196\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$49/72$$1/24$$31/72$$0$$1/12$$1/8$
$a_1=0$$49/72$$49/72$$1/24$$31/72$$0$$1/12$$1/8$
$a_3=0$$1/2$$1/2$$1/24$$1/4$$0$$1/12$$1/8$
$a_1=a_3=0$$1/2$$1/2$$1/24$$1/4$$0$$1/12$$1/8$