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Properties

Label 1.6.N.24.6c
  
Name \(J(A(1,12))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $24$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(D_{12}\)

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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:$D_{12}$
Order:$24$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{9}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{36}^{7} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{36}^{25} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{9}^{8} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{36}^{29} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{36}^{11} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(A(1,6)_1)$${}^{\times 2}$, $A(1,12)$, $J(A(1,4)_2)$
Minimal supergroups:$J_s(B(1,12))$, $J_s(B(1,12;2))$, $J(B(3,6;2))$, $J(B(1,12))$, $J(B(1,12;2))$${}^{\times 3}$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $51$ $0$ $1230$ $0$ $33635$ $0$ $978138$ $0$ $29575854$
$a_2$ $1$ $3$ $16$ $126$ $1310$ $15458$ $193293$ $2495937$ $32942370$ $442189518$ $6018260711$ $82869440327$ $1152477637731$
$a_3$ $1$ $0$ $18$ $0$ $3570$ $0$ $967300$ $0$ $294577906$ $0$ $96268641048$ $0$ $33028029167892$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $3$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $16$ $6$ $23$ $51$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $18$ $126$ $64$ $245$ $138$ $545$ $1230$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $182$ $1310$ $714$ $406$ $2851$ $1604$ $6461$ $3620$ $14710$ $33635$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $2120$ $15458$ $1188$ $8562$ $4790$ $34985$ $19502$ $10900$ $80139$ $44600$ $184102$ $102270$ $423913$ $978138$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $3570$ $25976$ $193293$ $14482$ $107002$ $59442$ $444177$ $33061$ $246012$ $136445$ $1025085$ $566973$ $314044$ $2370259$
$$ $1309322$ $5489757$ $3028788$ $12733602$ $29575854$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&7&0&1&0&4&0&0&12\\0&3&0&3&0&14&0&0&21&0&11&0&29&49&0\\2&0&11&0&7&0&27&39&0&20&0&60&0&0&140\\0&3&0&9&0&26&0&0&39&0&17&0&69&95&0\\0&0&7&0&21&0&39&38&0&45&0&93&0&0&196\\0&14&0&26&0&104&0&0&166&0&80&0&264&404&0\\1&0&27&0&39&0&116&121&0&104&0&257&0&0&588\\7&0&39&0&38&0&121&180&0&121&0&296&0&0&716\\0&21&0&39&0&166&0&0&283&0&131&0&441&689&0\\1&0&20&0&45&0&104&121&0&127&0&256&0&0&604\\0&11&0&17&0&80&0&0&131&0&68&0&204&325&0\\4&0&60&0&93&0&257&296&0&256&0&614&0&0&1420\\0&29&0&69&0&264&0&0&441&0&204&0&742&1107&0\\0&49&0&95&0&404&0&0&689&0&325&0&1107&1727&0\\12&0&140&0&196&0&588&716&0&604&0&1420&0&0&3426\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&11&9&21&104&116&180&283&127&68&614&742&1727&3426&1699&1768&4172&3572&1027\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$1/2$$0$$0$$0$$0$$1/2$
$a_1=0$$1/2$$1/2$$0$$0$$0$$0$$1/2$
$a_3=0$$1/2$$1/2$$0$$0$$0$$0$$1/2$
$a_1=a_3=0$$1/2$$1/2$$0$$0$$0$$0$$1/2$