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Properties

Label 1.6.N.8.1c
  
Name \(A(1,8)_2\)
Weight $1$
Degree $6$
Real dimension $1$
Components $8$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_8\)

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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:$C_8$
Order:$8$
Abelian:yes
Generators:$\begin{bmatrix}\zeta_{12}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{24}^{5} & 0 & 0& 0 & 0 \\0 & 0 & \zeta_{24}^{17} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{12}^{11} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{24}^{19} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{24}^{7} \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$A(1,4)_1$
Minimal supergroups:$B(3,4;4)$, $B(2,4;4)$${}^{\times 2}$, $J(A(1,8)_2)$, $J_s(A(1,8)_2)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $6$ $0$ $102$ $0$ $2460$ $0$ $68390$ $0$ $2057076$ $0$ $64991388$
$a_2$ $1$ $3$ $23$ $225$ $2555$ $31393$ $405617$ $5422567$ $74234275$ $1033433145$ $14560332773$ $206947339987$ $2960611564037$
$a_3$ $1$ $0$ $36$ $0$ $7236$ $0$ $2086800$ $0$ $689427676$ $0$ $242452091376$ $0$ $88002695338668$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $23$ $12$ $46$ $102$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $36$ $225$ $128$ $490$ $276$ $1090$ $2460$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $364$ $2555$ $1432$ $812$ $5746$ $3220$ $13050$ $7280$ $29800$ $68390$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $4280$ $31393$ $2388$ $17416$ $9688$ $71890$ $39800$ $22104$ $165450$ $91400$ $382172$ $210644$ $885458$ $2057076$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $7236$ $53496$ $405617$ $29616$ $223100$ $122952$ $939514$ $67872$ $515836$ $283696$ $2182770$ $1196384$ $656844$ $5083792$
$$ $2782136$ $11865770$ $6484296$ $27746964$ $64991388$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&3&0&4&8&0&6&0&11&0&0&24\\0&6&0&6&0&28&0&0&42&0&22&0&58&98&0\\2&0&18&0&20&0&58&66&0&48&0&126&0&0&284\\0&6&0&18&0&52&0&0&78&0&34&0&138&194&0\\3&0&20&0&33&0&72&94&0&78&0&177&0&0&400\\0&28&0&52&0&208&0&0&336&0&160&0&536&832&0\\4&0&58&0&72&0&228&258&0&208&0&516&0&0&1232\\8&0&66&0&94&0&258&332&0&278&0&626&0&0&1524\\0&42&0&78&0&336&0&0&590&0&266&0&922&1466&0\\6&0&48&0&78&0&208&278&0&254&0&524&0&0&1328\\0&22&0&34&0&160&0&0&266&0&136&0&420&674&0\\11&0&126&0&177&0&516&626&0&524&0&1259&0&0&3040\\0&58&0&138&0&536&0&0&922&0&420&0&1556&2378&0\\0&98&0&194&0&832&0&0&1466&0&674&0&2378&3766&0\\24&0&284&0&400&0&1232&1524&0&1328&0&3040&0&0&7664\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&6&18&18&33&208&228&332&590&254&136&1259&1556&3766&7664&3962&4047&10012&8856&2630\end{bmatrix}$

Event probabilities

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