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Properties

Label 1.6.N.24.11b
  
Name \(B(1,12;2)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $24$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_3\times Q_8\)

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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:$C_3\times Q_8$
Order:$24$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{9}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{36}^{7} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{36}^{25} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{9}^{8} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{36}^{29} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{36}^{11} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 & 0 & 0 \\0 & i & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -i \\0 & 0& 0 & 0 & -i & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $54$ $0$ $1240$ $0$ $33670$ $0$ $978264$ $0$ $29576316$
$a_2$ $1$ $2$ $13$ $116$ $1279$ $15362$ $192999$ $2495040$ $32939643$ $442181246$ $6018235663$ $82869364580$ $1152477408905$
$a_3$ $1$ $0$ $20$ $0$ $3588$ $0$ $967500$ $0$ $294580356$ $0$ $96268672800$ $0$ $33028029595000$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $13$ $6$ $24$ $54$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $20$ $116$ $66$ $248$ $138$ $548$ $1240$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $184$ $1279$ $718$ $412$ $2858$ $1610$ $6470$ $3620$ $14720$ $33670$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $2128$ $15362$ $1188$ $8574$ $4796$ $35004$ $19514$ $10920$ $80160$ $44620$ $184132$ $102270$ $423948$ $978264$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $3588$ $25996$ $192999$ $14500$ $107034$ $59466$ $444228$ $33062$ $246048$ $136466$ $1025142$ $567014$ $314114$ $2370330$
$$ $1309392$ $5489862$ $3028788$ $12733728$ $29576316$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&5&0&4&0&5&0&0&12\\0&4&0&2&0&14&0&0&22&0&12&0&26&50&0\\1&0&10&0&9&0&30&32&0&22&0&63&0&0&140\\0&2&0&12&0&26&0&0&36&0&14&0&76&92&0\\2&0&9&0&19&0&33&49&0&44&0&88&0&0&196\\0&14&0&26&0&104&0&0&166&0&80&0&264&404&0\\1&0&30&0&33&0&120&122&0&92&0&255&0&0&588\\5&0&32&0&49&0&122&168&0&140&0&303&0&0&716\\0&22&0&36&0&166&0&0&288&0&134&0&432&692&0\\4&0&22&0&44&0&92&140&0&134&0&248&0&0&604\\0&12&0&14&0&80&0&0&134&0&72&0&196&328&0\\5&0&63&0&88&0&255&303&0&248&0&611&0&0&1420\\0&26&0&76&0&264&0&0&432&0&196&0&764&1098&0\\0&50&0&92&0&404&0&0&692&0&328&0&1098&1732&0\\12&0&140&0&196&0&588&716&0&604&0&1420&0&0&3426\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&10&12&19&104&120&168&288&134&72&611&764&1732&3426&1724&1737&4212&3578&1072\end{bmatrix}$

Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.