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Properties

Label 1.6.N.144.182a
  
Name \(J(E(72))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $144$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(F_9:C_2\)

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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:$F_9:C_2$
Order:$144$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \frac{1}{\zeta_{3}^{1}-\zeta_{3}^{2}}\begin{bmatrix}1 & 1 & 1 & 0 & 0 & 0 \\1 & \zeta_{3}^{1} & \zeta_{3}^{2} & 0 & 0 & 0 \\1 & \zeta_{3}^{2} & \zeta_{3}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 1 & 1 \\0 & 0 & 0 & 1 & \zeta_{3}^{2} & \zeta_{3}^{1} \\0 & 0 & 0 & 1 & \zeta_{3}^{1} & \zeta_{3}^{2} \\\end{bmatrix}, \frac{1}{\zeta_{3}^{1}-\zeta_{3}^{2}}\begin{bmatrix}1 & 1 & \zeta_{3}^{2} & 0 & 0 & 0 \\1 & \zeta_{3}^{1} & \zeta_{3}^{1} & 0 & 0 & 0 \\\zeta_{3}^{1} & 1 & \zeta_{3}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 1 & \zeta_{3}^{1} \\0 & 0 & 0 & 1 & \zeta_{3}^{2} & \zeta_{3}^{2} \\0 & 0 & 0 & \zeta_{3}^{2} & 1 & \zeta_{3}^{2} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1& 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0\\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(E(36))$, $E(72)$, $J_n(E(36))$, $J_s(B(1,4;2)_2)$
Minimal supergroups:$J(E(216))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $6$ $0$ $110$ $0$ $3220$ $0$ $103446$ $0$ $3410484$
$a_2$ $1$ $1$ $3$ $14$ $125$ $1516$ $20317$ $281065$ $3942871$ $55785866$ $794251103$ $11364526405$ $163275215369$
$a_3$ $1$ $0$ $3$ $0$ $347$ $0$ $107580$ $0$ $37239419$ $0$ $13333298838$ $0$ $4871247246734$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $3$ $1$ $3$ $6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $3$ $14$ $7$ $23$ $13$ $49$ $110$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $18$ $125$ $66$ $39$ $263$ $147$ $601$ $330$ $1385$ $3220$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $200$ $1516$ $111$ $820$ $452$ $3459$ $1892$ $1042$ $8055$ $4396$ $18822$ $10241$ $44079$ $103446$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $347$ $2588$ $20317$ $1421$ $11025$ $6013$ $47397$ $3280$ $25770$ $14024$ $111180$ $60378$ $32830$ $261207$
$$ $141729$ $614460$ $333102$ $1446942$ $3410484$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&0&1&0&0&0&0&0&0&1\\0&1&0&0&0&1&0&0&2&0&1&0&2&4&0\\0&0&2&0&0&0&2&3&0&2&0&5&0&0&14\\0&0&0&2&0&2&0&0&3&0&1&0&8&8&0\\0&0&0&0&3&0&3&3&0&6&0&8&0&0&20\\0&1&0&2&0&9&0&0&16&0&7&0&26&41&0\\0&0&2&0&3&0&13&10&0&9&0&25&0&0&63\\1&0&3&0&3&0&10&21&0&13&0&29&0&0&79\\0&2&0&3&0&16&0&0&31&0&14&0&45&76&0\\0&0&2&0&6&0&9&13&0&20&0&28&0&0&70\\0&1&0&1&0&7&0&0&14&0&7&0&19&35&0\\0&0&5&0&8&0&25&29&0&28&0&65&0&0&160\\0&2&0&8&0&26&0&0&45&0&19&0&86&122&0\\0&4&0&8&0&41&0&0&76&0&35&0&122&202&0\\1&0&14&0&20&0&63&79&0&70&0&160&0&0&411\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&2&3&9&13&21&31&20&7&65&86&202&411&225&230&559&498&166\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$5/9$$0$$2/9$$1/4$$0$$1/12$
$a_1=0$$5/9$$5/9$$0$$2/9$$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$