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Properties

Label 1.6.N.72.39a
  
Name \(J_n(E(36))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $72$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(F_9\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$72$
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:$F_9$
Order:$72$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \frac{1}{\zeta_{3}^{1}-\zeta_{3}^{2}}\begin{bmatrix}1 & 1 & 1 & 0 & 0 & 0 \\1 & \zeta_{3}^{1} & \zeta_{3}^{2} & 0 & 0 & 0 \\1 & \zeta_{3}^{2} & \zeta_{3}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 1 & 1 \\0 & 0 & 0 & 1 & \zeta_{3}^{2} & \zeta_{3}^{1} \\0 & 0 & 0 & 1 & \zeta_{3}^{1} & \zeta_{3}^{2} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 1 & \zeta_{3}^{1} \\0 & 0 & 0 &1 & \zeta_{3}^{2} & \zeta_{3}^{2} \\0 & 0 & 0 & \zeta_{3}^{2} & 1 & \zeta_{3}^{2} \\-1 & -1 & -\zeta_{3}^{2} & 0 & 0 & 0 \\-1 & -\zeta_{3}^{1} & -\zeta_{3}^{1} & 0 & 0 & 0 \\-\zeta_{3}^{1} & -1 & -\zeta_{3}^{1} & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$E(36)$, $J_n(A(1,4)_2)$
Minimal supergroups:$J(E(72))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $9$ $0$ $210$ $0$ $6405$ $0$ $206766$ $0$ $6820506$
$a_2$ $1$ $1$ $3$ $20$ $227$ $2966$ $40442$ $561569$ $7884095$ $111566882$ $1588487888$ $22729010459$ $326550305270$
$a_3$ $1$ $0$ $4$ $0$ $672$ $0$ $214870$ $0$ $74474792$ $0$ $26666539524$ $0$ $9742493641650$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $3$ $1$ $4$ $9$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $4$ $20$ $11$ $41$ $22$ $91$ $210$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $31$ $227$ $124$ $71$ $513$ $283$ $1184$ $645$ $2745$ $6405$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $386$ $2966$ $210$ $1618$ $885$ $6883$ $3754$ $2058$ $16062$ $8751$ $37578$ $20426$ $88067$ $206766$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $672$ $5137$ $40442$ $2808$ $21989$ $11973$ $94698$ $6514$ $51457$ $27976$ $222229$ $120643$ $65562$ $522235$
$$ $283304$ $1228675$ $665994$ $2893548$ $6820506$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&0&1&0&1&0&1&0&0&2\\0&1&0&0&0&2&0&0&4&0&2&0&4&9&0\\0&0&2&0&1&0&5&5&0&4&0&11&0&0&28\\0&0&0&3&0&4&0&0&6&0&2&0&16&17&0\\0&0&1&0&4&0&5&9&0&10&0&15&0&0&40\\0&2&0&4&0&18&0&0&32&0&14&0&52&82&0\\0&0&5&0&5&0&24&22&0&17&0&50&0&0&126\\1&0&5&0&9&0&22&35&0&31&0&61&0&0&158\\0&4&0&6&0&32&0&0&61&0&28&0&90&153&0\\1&0&4&0&10&0&17&31&0&35&0&54&0&0&140\\0&2&0&2&0&14&0&0&28&0&14&0&38&70&0\\1&0&11&0&15&0&50&61&0&54&0&127&0&0&320\\0&4&0&16&0&52&0&0&90&0&38&0&172&244&0\\0&9&0&17&0&82&0&0&153&0&70&0&244&401&0\\2&0&28&0&40&0&126&158&0&140&0&320&0&0&822\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&3&4&18&24&35&61&35&14&127&172&401&822&447&445&1118&986&329\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$11/18$$0$$1/9$$1/2$$0$$0$
$a_1=0$$11/18$$11/18$$0$$1/9$$1/2$$0$$0$
$a_3=0$$1/2$$1/2$$0$$0$$1/2$$0$$0$
$a_1=a_3=0$$1/2$$1/2$$0$$0$$1/2$$0$$0$