Properties

Label 1.6.N.54.5a
  
Name \(J(C(3,3))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $54$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(He_3:C_2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$54$
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:$He_3:C_2$
Order:$54$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{9}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{9}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{9}^{7} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{9}^{8} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{9}^{8} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{9}^{2} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0 & 0& 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 &0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 &-1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(C(3,1))$, $C(3,3)$, $J(A(3,3))$
Minimal supergroups:$J(D(3,3))$, $J(C(6,6))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $15$ $0$ $330$ $0$ $9135$ $0$ $282366$ $0$ $9168390$
$a_2$ $1$ $1$ $5$ $36$ $357$ $4266$ $55734$ $759501$ $10576773$ $149149026$ $2120393700$ $30320202771$ $435492394110$
$a_3$ $1$ $0$ $6$ $0$ $966$ $0$ $290620$ $0$ $99524166$ $0$ $35567544816$ $0$ $12990657192744$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $5$ $2$ $7$ $15$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $6$ $36$ $18$ $67$ $38$ $147$ $330$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $50$ $357$ $192$ $108$ $765$ $426$ $1731$ $960$ $3960$ $9135$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $572$ $4266$ $318$ $2328$ $1286$ $9675$ $5316$ $2928$ $22329$ $12240$ $51822$ $28350$ $120771$ $282366$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $966$ $7220$ $55734$ $3966$ $30378$ $16618$ $129285$ $9105$ $70500$ $38493$ $301761$ $164313$ $89574$ $706401$
$$ $384174$ $1657341$ $900396$ $3895290$ $9168390$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&1&2&0&0&0&1&0&0&2\\0&1&0&1&0&4&0&0&5&0&3&0&7&13&0\\0&0&4&0&2&0&6&11&0&5&0&15&0&0&38\\0&1&0&3&0&6&0&0&11&0&5&0&17&27&0\\0&0&2&0&6&0&10&9&0&11&0&25&0&0&52\\0&4&0&6&0&28&0&0&44&0&20&0&70&112&0\\1&0&6&0&10&0&31&33&0&31&0&68&0&0&170\\2&0&11&0&9&0&33&50&0&33&0&81&0&0&212\\0&5&0&11&0&44&0&0&81&0&33&0&129&203&0\\0&0&5&0&11&0&31&33&0&38&0&74&0&0&188\\0&3&0&5&0&20&0&0&33&0&20&0&56&93&0\\1&0&15&0&25&0&68&81&0&74&0&173&0&0&424\\0&7&0&17&0&70&0&0&129&0&56&0&214&333&0\\0&13&0&27&0&112&0&0&203&0&93&0&333&533&0\\2&0&38&0&52&0&170&212&0&188&0&424&0&0&1104\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&4&3&6&28&31&50&81&38&20&173&214&533&1104&573&605&1456&1302&385\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$47/54$$0$$19/27$$0$$0$$1/6$
$a_1=0$$47/54$$47/54$$0$$19/27$$0$$0$$1/6$
$a_3=0$$1/2$$1/2$$0$$1/3$$0$$0$$1/6$
$a_1=a_3=0$$1/2$$1/2$$0$$1/3$$0$$0$$1/6$