Properties

Label 1.6.N.24.13a
  
Name \(J(C(2,2))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $24$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_2\times A_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$24$
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:$C_2\times A_4$
Order:$24$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0& 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0 & 0& 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 &0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 &-1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(A(2,2))$, $C(2,2)$, $J_s(A(3,1))$
Minimal supergroups:$J(C(4,4))$, $J(C(6,2))$, $J(D(2,2))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $21$ $0$ $610$ $0$ $19145$ $0$ $620046$ $0$ $20460594$
$a_2$ $1$ $1$ $6$ $56$ $674$ $8886$ $121305$ $1684677$ $23652270$ $334700786$ $4765464551$ $68187035247$ $979650930593$
$a_3$ $1$ $0$ $8$ $0$ $1972$ $0$ $644030$ $0$ $223416284$ $0$ $79999502268$ $0$ $29227479221314$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $6$ $2$ $9$ $21$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $8$ $56$ $28$ $115$ $62$ $263$ $610$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $86$ $674$ $360$ $199$ $1519$ $831$ $3524$ $1920$ $8200$ $19145$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $1136$ $8886$ $618$ $4820$ $2629$ $20595$ $11218$ $6122$ $48114$ $26186$ $112632$ $61222$ $264075$ $620046$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $1972$ $15352$ $121305$ $8368$ $65874$ $35837$ $283947$ $19496$ $154245$ $83830$ $666489$ $361764$ $196490$ $1566438$
$$ $849660$ $3685647$ $1997772$ $8680182$ $20460594$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&1&3&0&1&0&2&0&0&6\\0&1&0&1&0&6&0&0&11&0&5&0&15&27&0\\0&0&5&0&3&0&13&19&0&12&0&32&0&0&84\\0&1&0&5&0&12&0&0&21&0&9&0&40&56&0\\0&0&3&0&9&0&19&22&0&26&0&49&0&0&120\\0&6&0&12&0&54&0&0&96&0&42&0&156&246&0\\1&0&13&0&19&0&65&70&0&62&0&151&0&0&378\\3&0&19&0&22&0&70&103&0&82&0&182&0&0&474\\0&11&0&21&0&96&0&0&177&0&80&0&281&456&0\\1&0&12&0&26&0&62&82&0&86&0&166&0&0&420\\0&5&0&9&0&42&0&0&80&0&38&0&124&206&0\\2&0&32&0&49&0&151&182&0&166&0&378&0&0&960\\0&15&0&40&0&156&0&0&281&0&124&0&488&744&0\\0&27&0&56&0&246&0&0&456&0&206&0&744&1193&0\\6&0&84&0&120&0&378&474&0&420&0&960&0&0&2466\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&5&5&9&54&65&103&177&86&38&378&488&1193&2466&1297&1334&3294&2926&895\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$5/6$$0$$2/3$$0$$0$$1/6$
$a_1=0$$5/6$$5/6$$0$$2/3$$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$