Properties

Label 1.6.N.36.11a
  
Name \(C(6,2)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $36$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_3\times A_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$36$
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:$C_3\times A_4$
Order:$36$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0 & 0& 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $30$ $0$ $620$ $0$ $15750$ $0$ $456372$ $0$ $14271180$
$a_2$ $1$ $1$ $7$ $61$ $627$ $7241$ $90645$ $1196091$ $16316571$ $227280577$ $3208501497$ $45698791271$ $654946598509$
$a_3$ $1$ $0$ $12$ $0$ $1724$ $0$ $462700$ $0$ $151838316$ $0$ $53592239472$ $0$ $19508674639124$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $7$ $4$ $14$ $30$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $12$ $61$ $36$ $130$ $76$ $282$ $620$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $96$ $627$ $360$ $210$ $1386$ $794$ $3096$ $1760$ $6960$ $15750$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $1032$ $7241$ $588$ $4072$ $2302$ $16418$ $9196$ $5180$ $37452$ $20892$ $85840$ $47684$ $197554$ $456372$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $1724$ $12232$ $90645$ $6852$ $50188$ $27886$ $209010$ $15544$ $115374$ $63876$ $483750$ $266264$ $146972$ $1123140$
$$ $616616$ $2614794$ $1432200$ $6102180$ $14271180$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&1&0&2&2&0&2&0&3&0&0&4\\0&2&0&2&0&8&0&0&10&0&6&0&14&22&0\\0&0&6&0&6&0&14&16&0&10&0&30&0&0&64\\0&2&0&6&0&12&0&0&18&0&10&0&32&44&0\\1&0&6&0&9&0&18&22&0&18&0&41&0&0&88\\0&8&0&12&0&52&0&0&76&0&36&0&120&180&0\\2&0&14&0&18&0&54&58&0&46&0&116&0&0&268\\2&0&16&0&22&0&58&74&0&62&0&138&0&0&332\\0&10&0&18&0&76&0&0&130&0&60&0&202&320&0\\2&0&10&0&18&0&46&62&0&60&0&116&0&0&288\\0&6&0&10&0&36&0&0&60&0&32&0&92&148&0\\3&0&30&0&41&0&116&138&0&116&0&275&0&0&664\\0&14&0&32&0&120&0&0&202&0&92&0&344&516&0\\0&22&0&44&0&180&0&0&320&0&148&0&516&818&0\\4&0&64&0&88&0&268&332&0&288&0&664&0&0&1680\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&6&9&52&54&74&130&60&32&275&344&818&1680&878&887&2208&1960&606\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$13/18$$0$$13/18$$0$$0$$0$
$a_1=0$$13/18$$13/18$$0$$13/18$$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$