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Properties

Label 1.6.N.8.2b
  
Name \(A(2,4)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $8$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_2\times C_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$8$
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:$C_2\times C_4$
Order:$8$
Abelian:yes
Generators:$\begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0& 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{5} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{7} & 0 \\0 & 0 & 0 & 0 & 0& \zeta_{12}^{7} \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $6$ $0$ $102$ $0$ $2460$ $0$ $68390$ $0$ $2057076$ $0$ $64991388$
$a_2$ $1$ $3$ $23$ $225$ $2555$ $31393$ $405617$ $5422567$ $74234275$ $1033433145$ $14560332773$ $206947339987$ $2960611564037$
$a_3$ $1$ $0$ $36$ $0$ $7244$ $0$ $2086980$ $0$ $689430812$ $0$ $242452141776$ $0$ $88002696123188$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $23$ $12$ $46$ $102$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $36$ $225$ $128$ $490$ $276$ $1090$ $2460$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $364$ $2555$ $1432$ $814$ $5746$ $3222$ $13052$ $7280$ $29800$ $68390$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $4280$ $31393$ $2388$ $17416$ $9690$ $71890$ $39804$ $22116$ $165456$ $91412$ $382184$ $210644$ $885458$ $2057076$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $7244$ $53496$ $405617$ $29624$ $223100$ $122962$ $939514$ $67872$ $515850$ $283708$ $2182790$ $1196408$ $656900$ $5083828$
$$ $2782192$ $11865826$ $6484296$ $27746964$ $64991388$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&3&0&4&8&0&6&0&11&0&0&24\\0&6&0&6&0&28&0&0&42&0&22&0&58&98&0\\2&0&18&0&20&0&58&66&0&48&0&126&0&0&284\\0&6&0&18&0&52&0&0&78&0&34&0&140&192&0\\3&0&20&0&33&0&72&94&0&80&0&177&0&0&400\\0&28&0&52&0&208&0&0&336&0&160&0&536&832&0\\4&0&58&0&72&0&230&256&0&204&0&516&0&0&1232\\8&0&66&0&94&0&256&334&0&280&0&626&0&0&1524\\0&42&0&78&0&336&0&0&590&0&268&0&918&1468&0\\6&0&48&0&80&0&204&280&0&264&0&524&0&0&1328\\0&22&0&34&0&160&0&0&268&0&136&0&416&676&0\\11&0&126&0&177&0&516&626&0&524&0&1259&0&0&3040\\0&58&0&140&0&536&0&0&918&0&416&0&1568&2372&0\\0&98&0&192&0&832&0&0&1468&0&676&0&2372&3770&0\\24&0&284&0&400&0&1232&1524&0&1328&0&3040&0&0&7664\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&6&18&18&33&208&230&334&590&264&136&1259&1568&3770&7664&3990&4051&10048&8876&2702\end{bmatrix}$

Event probabilities

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