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Properties

Label 1.6.N.72.44a
  
Name \(J(C(6,2))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $72$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(S_3\times A_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$72$
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:$S_3\times A_4$
Order:$72$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0 & 0& 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 &0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 &-1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $15$ $0$ $310$ $0$ $7875$ $0$ $228186$ $0$ $7135590$
$a_2$ $1$ $1$ $5$ $35$ $327$ $3661$ $45444$ $598410$ $8159379$ $113643569$ $1604260590$ $22849425160$ $327473387828$
$a_3$ $1$ $0$ $6$ $0$ $862$ $0$ $231350$ $0$ $75919158$ $0$ $26796119736$ $0$ $9754337319562$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $5$ $2$ $7$ $15$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $6$ $35$ $18$ $65$ $38$ $141$ $310$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $48$ $327$ $180$ $105$ $693$ $397$ $1548$ $880$ $3480$ $7875$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $516$ $3661$ $294$ $2036$ $1151$ $8209$ $4598$ $2590$ $18726$ $10446$ $42920$ $23842$ $98777$ $228186$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $862$ $6116$ $45444$ $3426$ $25094$ $13943$ $104505$ $7772$ $57687$ $31938$ $241875$ $133132$ $73486$ $561570$
$$ $308308$ $1307397$ $716100$ $3051090$ $7135590$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&1&2&0&0&0&1&0&0&2\\0&1&0&1&0&4&0&0&5&0&3&0&7&11&0\\0&0&4&0&2&0&6&10&0&4&0&14&0&0&32\\0&1&0&3&0&6&0&0&9&0&5&0&16&22&0\\0&0&2&0&6&0&10&8&0&11&0&22&0&0&44\\0&4&0&6&0&26&0&0&38&0&18&0&60&90&0\\1&0&6&0&10&0&28&27&0&24&0&59&0&0&134\\2&0&10&0&8&0&27&43&0&27&0&66&0&0&166\\0&5&0&9&0&38&0&0&65&0&30&0&101&160&0\\0&0&4&0&11&0&24&27&0&33&0&60&0&0&144\\0&3&0&5&0&18&0&0&30&0&16&0&46&74&0\\1&0&14&0&22&0&59&66&0&60&0&139&0&0&332\\0&7&0&16&0&60&0&0&101&0&46&0&172&258&0\\0&11&0&22&0&90&0&0&160&0&74&0&258&409&0\\2&0&32&0&44&0&134&166&0&144&0&332&0&0&840\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&4&3&6&26&28&43&65&33&16&139&172&409&840&439&457&1104&986&303\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$31/36$$0$$25/36$$0$$0$$1/6$
$a_1=0$$31/36$$31/36$$0$$25/36$$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$