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Properties

Label 1.6.N.432.734a
  
Name \(J(E(216))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $432$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_3^2:\GL(2,3)\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$432$
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:$C_3^2:\GL(2,3)$
Order:$432$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \frac{1}{\zeta_{3}^{1}-\zeta_{3}^{2}}\begin{bmatrix}1 & 1 & 1 & 0 & 0 & 0 \\1 & \zeta_{3}^{1} & \zeta_{3}^{2} & 0 & 0 & 0 \\1 & \zeta_{3}^{2} & \zeta_{3}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 1 & 1 \\0 & 0 & 0 & 1 & \zeta_{3}^{2} & \zeta_{3}^{1} \\0 & 0 & 0 & 1 & \zeta_{3}^{1} & \zeta_{3}^{2} \\\end{bmatrix}, \begin{bmatrix}\zeta_{9}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{9}^{2} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{9}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{9}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{9}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{9}^{4} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0& 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(E(72))$, $J(D(3,3))$, $E(216)$, $J_s(B(T,1;1))$
Minimal supergroups:

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $6$ $0$ $70$ $0$ $1400$ $0$ $37926$ $0$ $1174404$
$a_2$ $1$ $1$ $3$ $12$ $75$ $676$ $7717$ $99163$ $1346893$ $18792642$ $265957383$ $3795608983$ $54471081315$
$a_3$ $1$ $0$ $3$ $0$ $167$ $0$ $38120$ $0$ $12527935$ $0$ $4450547598$ $0$ $1624082415010$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $3$ $1$ $3$ $6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $3$ $12$ $6$ $17$ $10$ $33$ $70$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $13$ $75$ $40$ $25$ $136$ $80$ $287$ $165$ $625$ $1400$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $104$ $676$ $60$ $372$ $213$ $1423$ $802$ $458$ $3168$ $1775$ $7170$ $3990$ $16415$ $37926$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $167$ $1071$ $7717$ $605$ $4241$ $2361$ $17367$ $1318$ $9581$ $5304$ $39860$ $21911$ $12090$ $92207$
$$ $50547$ $214508$ $117306$ $501102$ $1174404$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&0&1&0&0&0&0&0&0&0\\0&1&0&0&0&1&0&0&1&0&1&0&0&2&0\\0&0&2&0&0&0&1&2&0&0&0&2&0&0&5\\0&0&0&2&0&1&0&0&1&0&0&0&4&3&0\\0&0&0&0&3&0&1&1&0&3&0&4&0&0&6\\0&1&0&1&0&5&0&0&6&0&3&0&9&15&0\\0&0&1&0&1&0&7&3&0&3&0&9&0&0&22\\1&0&2&0&1&0&3&11&0&4&0&9&0&0&27\\0&1&0&1&0&6&0&0&13&0&4&0&15&26&0\\0&0&0&0&3&0&3&4&0&10&0&9&0&0&24\\0&1&0&0&0&3&0&0&4&0&5&0&5&13&0\\0&0&2&0&4&0&9&9&0&9&0&25&0&0&52\\0&0&0&4&0&9&0&0&15&0&5&0&32&40&0\\0&2&0&3&0&15&0&0&26&0&13&0&40&70&0\\0&0&5&0&6&0&22&27&0&24&0&52&0&0&141\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&2&3&5&7&11&13&10&5&25&32&70&141&79&86&191&172&60\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$17/27$$0$$8/27$$1/4$$0$$1/12$
$a_1=0$$17/27$$17/27$$0$$8/27$$1/4$$0$$1/12$
$a_3=0$$1/2$$1/2$$0$$1/6$$1/4$$0$$1/12$
$a_1=a_3=0$$1/2$$1/2$$0$$1/6$$1/4$$0$$1/12$