Properties

Label 1.6.N.18.3a
  
Name \(J(C(3,1))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $18$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_3\times S_3\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$18$
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:$C_3\times S_3$
Order:$18$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{3}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{2} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0 & 0& 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 &0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 &-1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(A(3,1))$, $C(3,1)$, $J_s(A(3,1))$
Minimal supergroups:$J(C(6,2))$, $J_s(C(6,2))$, $J(C(3,3))$, $J(D(3,1))$${}^{\times 2}$, $J_s(C(3,3))$${}^{\times 3}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $27$ $0$ $810$ $0$ $25515$ $0$ $826686$ $0$ $27280638$
$a_2$ $1$ $1$ $7$ $72$ $891$ $11826$ $161676$ $2246049$ $31535811$ $446266098$ $6353947962$ $90916032879$ $1306201198968$
$a_3$ $1$ $0$ $10$ $0$ $2622$ $0$ $858610$ $0$ $297887030$ $0$ $106665983640$ $0$ $38969972011146$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $7$ $3$ $12$ $27$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $10$ $72$ $37$ $153$ $84$ $351$ $810$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $115$ $891$ $480$ $263$ $2025$ $1107$ $4698$ $2565$ $10935$ $25515$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $1514$ $11826$ $828$ $6426$ $3507$ $27459$ $14958$ $8154$ $64152$ $34911$ $150174$ $81648$ $352107$ $826686$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $2622$ $20469$ $161676$ $11154$ $87831$ $47781$ $378594$ $26010$ $205659$ $111780$ $888651$ $482355$ $261954$ $2088585$
$$ $1132866$ $4914189$ $2663766$ $11573604$ $27280638$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&2&3&0&1&0&3&0&0&8\\0&1&0&2&0&8&0&0&14&0&6&0&22&35&0\\0&0&6&0&5&0&17&25&0&18&0&43&0&0&112\\0&2&0&5&0&16&0&0&30&0&14&0&48&77&0\\0&0&5&0&10&0&27&29&0&30&0&65&0&0&160\\0&8&0&16&0&72&0&0&128&0&56&0&208&328&0\\2&0&17&0&27&0&82&98&0&89&0&200&0&0&504\\3&0&25&0&29&0&98&131&0&105&0&245&0&0&632\\0&14&0&30&0&128&0&0&233&0&104&0&382&605&0\\1&0&18&0&30&0&89&105&0&101&0&222&0&0&560\\0&6&0&14&0&56&0&0&104&0&48&0&172&272&0\\3&0&43&0&65&0&200&245&0&222&0&503&0&0&1280\\0&22&0&48&0&208&0&0&382&0&172&0&632&1000&0\\0&35&0&77&0&328&0&0&605&0&272&0&1000&1587&0\\8&0&112&0&160&0&504&632&0&560&0&1280&0&0&3288\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&6&5&10&72&82&131&233&101&48&503&632&1587&3288&1703&1767&4352&3878&1135\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$17/18$$0$$7/9$$0$$0$$1/6$
$a_1=0$$17/18$$17/18$$0$$7/9$$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$