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Properties

Label 1.6.N.336.208a
  
Name \(J(E(168))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $336$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(\PGL(2,7)\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$336$
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:$\PGL(2,7)$
Order:$336$
Abelian:no
Generators:$\begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 &0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}\zeta_{7}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{7}^{2} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{7}^{4} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{7}^{6} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{7}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{7}^{3} \\\end{bmatrix}, \sqrt{-7}\begin{bmatrix}\zeta_{7}^{4}-\zeta_{7}^{3} & \zeta_{7}^{2}-\zeta_{7}^{5} & \zeta_{7}^{1}-\zeta_{7}^{6} & 0 & 0 & 0 \\\zeta_{7}^{2}-\zeta_{7}^{5} & \zeta_{7}^{1}-\zeta_{7}^{6} & \zeta_{7}^{4}-\zeta_{7}^{3} & 0 & 0 & 0 \\\zeta_{7}^{1}-\zeta_{7}^{6} & \zeta_{7}^{4}-\zeta_{7}^{3} & \zeta_{7}^{2}-\zeta_{7}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{7}^{3}-\zeta_{7}^{4} & \zeta_{7}^{5}-\zeta_{7}^{2} & \zeta_{7}^{6}-\zeta_{7}^{1} \\0 & 0 & 0 & \zeta_{7}^{5}-\zeta_{7}^{2} & \zeta_{7}^{6}-\zeta_{7}^{1} & \zeta_{7}^{3}-\zeta_{7}^{4} \\0 & 0 & 0 & \zeta_{7}^{6}-\zeta_{7}^{1} & \zeta_{7}^{3}-\zeta_{7}^{4} & \zeta_{7}^{5}-\zeta_{7}^{2} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{1} & 0 &0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{3}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{2} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 &1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 &0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(C(1,7))$, $J(B(3,1))$, $J_s(B(1,4)_2)$, $E(168)$
Minimal supergroups:

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $6$ $0$ $70$ $0$ $1540$ $0$ $45486$ $0$ $1470084$
$a_2$ $1$ $1$ $3$ $12$ $77$ $746$ $9117$ $122243$ $1697735$ $23943936$ $340555703$ $4871255423$ $69978522737$
$a_3$ $1$ $0$ $3$ $0$ $183$ $0$ $47040$ $0$ $15987279$ $0$ $5715109398$ $0$ $2087703441318$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $3$ $1$ $3$ $6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $3$ $12$ $6$ $17$ $10$ $33$ $70$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $13$ $77$ $41$ $26$ $143$ $84$ $308$ $175$ $680$ $1540$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $110$ $746$ $63$ $408$ $232$ $1615$ $902$ $512$ $3669$ $2036$ $8432$ $4641$ $19523$ $45486$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $183$ $1218$ $9117$ $684$ $4978$ $2751$ $20902$ $1522$ $11451$ $6291$ $48604$ $26542$ $14545$ $113529$
$$ $61859$ $266028$ $144648$ $624792$ $1470084$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&0&1&0&0&0&0&0&0&0\\0&1&0&0&0&1&0&0&1&0&1&0&0&2&0\\0&0&2&0&0&0&1&2&0&0&0&2&0&0&6\\0&0&0&2&0&1&0&0&1&0&0&0&5&3&0\\0&0&0&0&3&0&1&1&0&4&0&4&0&0&8\\0&1&0&1&0&5&0&0&7&0&3&0&11&17&0\\0&0&1&0&1&0&8&3&0&3&0&11&0&0&27\\1&0&2&0&1&0&3&12&0&6&0&11&0&0&34\\0&1&0&1&0&7&0&0&15&0&6&0&18&33&0\\0&0&0&0&4&0&3&6&0&13&0&12&0&0&30\\0&1&0&0&0&3&0&0&6&0&5&0&6&16&0\\0&0&2&0&4&0&11&11&0&12&0&29&0&0&68\\0&0&0&5&0&11&0&0&18&0&6&0&42&50&0\\0&2&0&3&0&17&0&0&33&0&16&0&50&88&0\\0&0&6&0&8&0&27&34&0&30&0&68&0&0&178\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&2&3&5&8&12&15&13&5&29&42&88&178&103&106&249&221&86\end{bmatrix}$

Event probabilities

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