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Properties

Label 1.6.N.12.2c
  
Name \(A(1,12)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $12$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_{12}\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$12$
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:$C_{12}$
Order:$12$
Abelian:yes
Generators:$\begin{bmatrix}\zeta_{9}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{36}^{7} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{36}^{25} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{9}^{8} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{36}^{29} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{36}^{11} \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $6$ $0$ $102$ $0$ $2460$ $0$ $67270$ $0$ $1956276$ $0$ $59151708$
$a_2$ $1$ $3$ $23$ $225$ $2539$ $30673$ $385857$ $4989687$ $65878179$ $884359353$ $12036462373$ $165738703507$ $2304954744021$
$a_3$ $1$ $0$ $36$ $0$ $7140$ $0$ $1934600$ $0$ $589155812$ $0$ $192537282096$ $0$ $66056058335784$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $23$ $12$ $46$ $102$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $36$ $225$ $128$ $490$ $276$ $1090$ $2460$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $364$ $2539$ $1428$ $812$ $5702$ $3208$ $12922$ $7240$ $29420$ $67270$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $4240$ $30673$ $2376$ $17124$ $9580$ $69970$ $39004$ $21800$ $160278$ $89200$ $368204$ $204540$ $847826$ $1956276$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $7140$ $51952$ $385857$ $28964$ $214004$ $118884$ $888354$ $66122$ $492024$ $272890$ $2050170$ $1133946$ $628088$ $4740518$
$$ $2618644$ $10979514$ $6057576$ $25467204$ $59151708$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&3&0&4&8&0&6&0&11&0&0&24\\0&6&0&6&0&28&0&0&42&0&22&0&58&98&0\\2&0&18&0&20&0&58&66&0&48&0&126&0&0&280\\0&6&0&18&0&52&0&0&78&0&34&0&138&190&0\\3&0&20&0&33&0&72&94&0&78&0&177&0&0&392\\0&28&0&52&0&208&0&0&332&0&160&0&528&808&0\\4&0&58&0&72&0&228&254&0&200&0&508&0&0&1176\\8&0&66&0&94&0&254&324&0&266&0&610&0&0&1432\\0&42&0&78&0&332&0&0&566&0&262&0&882&1378&0\\6&0&48&0&78&0&200&266&0&238&0&500&0&0&1208\\0&22&0&34&0&160&0&0&262&0&136&0&408&650&0\\11&0&126&0&177&0&508&610&0&500&0&1219&0&0&2840\\0&58&0&138&0&528&0&0&882&0&408&0&1484&2214&0\\0&98&0&190&0&808&0&0&1378&0&650&0&2214&3454&0\\24&0&280&0&392&0&1176&1432&0&1208&0&2840&0&0&6852\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&6&18&18&33&208&228&324&566&238&136&1219&1484&3454&6852&3398&3455&8344&7108&2054\end{bmatrix}$

Event probabilities

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