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Properties

Label 1.6.N.216.153a
  
Name \(E(216)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $216$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(\PGU(3,2)\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$216$
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:$\PGU(3,2)$
Order:$216$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \frac{1}{\zeta_{3}^{1}-\zeta_{3}^{2}}\begin{bmatrix}1 & 1 & 1 & 0 & 0 & 0 \\1 & \zeta_{3}^{1} & \zeta_{3}^{2} & 0 & 0 & 0 \\1 & \zeta_{3}^{2} & \zeta_{3}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 1 & 1 \\0 & 0 & 0 & 1 & \zeta_{3}^{2} & \zeta_{3}^{1} \\0 & 0 & 0 & 1 & \zeta_{3}^{1} & \zeta_{3}^{2} \\\end{bmatrix}, \begin{bmatrix}\zeta_{9}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{9}^{2} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{9}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{9}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{9}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{9}^{4} \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $12$ $0$ $140$ $0$ $2800$ $0$ $75852$ $0$ $2348808$
$a_2$ $1$ $1$ $4$ $19$ $136$ $1311$ $15312$ $197961$ $2692692$ $37582003$ $531904924$ $7591188441$ $108942074056$
$a_3$ $1$ $0$ $6$ $0$ $334$ $0$ $76240$ $0$ $25055870$ $0$ $8901095196$ $0$ $3248164830020$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $4$ $2$ $6$ $12$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $6$ $19$ $12$ $34$ $20$ $66$ $140$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $26$ $136$ $80$ $50$ $272$ $160$ $574$ $330$ $1250$ $2800$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $208$ $1311$ $120$ $744$ $426$ $2846$ $1604$ $916$ $6336$ $3550$ $14340$ $7980$ $32830$ $75852$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $334$ $2142$ $15312$ $1210$ $8482$ $4722$ $34734$ $2636$ $19162$ $10608$ $79720$ $43822$ $24180$ $184414$
$$ $101094$ $429016$ $234612$ $1002204$ $2348808$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&1&0&0&1&0&1&0&0&0&0&0\\0&2&0&0&0&2&0&0&2&0&2&0&0&4&0\\0&0&3&0&1&0&3&2&0&1&0&5&0&0&10\\0&0&0&4&0&2&0&0&2&0&0&0&8&6&0\\1&0&1&0&4&0&1&5&0&4&0&7&0&0&12\\0&2&0&2&0&10&0&0&12&0&6&0&18&30&0\\0&0&3&0&1&0&13&8&0&5&0&17&0&0&44\\1&0&2&0&5&0&8&16&0&12&0&21&0&0&54\\0&2&0&2&0&12&0&0&26&0&8&0&30&52&0\\1&0&1&0&4&0&5&12&0&17&0&16&0&0&48\\0&2&0&0&0&6&0&0&8&0&10&0&10&26&0\\0&0&5&0&7&0&17&21&0&16&0&48&0&0&104\\0&0&0&8&0&18&0&0&30&0&10&0&64&80&0\\0&4&0&6&0&30&0&0&52&0&26&0&80&140&0\\0&0&10&0&12&0&44&54&0&48&0&104&0&0&282\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&3&4&4&10&13&16&26&17&10&48&64&140&282&158&158&382&338&120\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$7/27$$0$$7/27$$0$$0$$0$
$a_1=0$$7/27$$7/27$$0$$7/27$$0$$0$$0$
$a_3=0$$0$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$0$$0$$0$$0$$0$$0$$0$