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Properties

Label 1.6.N.8.3b
  
Name \(B(1,4)_2\)
Weight $1$
Degree $6$
Real dimension $1$
Components $8$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(D_4\)

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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:$D_4$
Order:$8$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{7} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0\\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{5} \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0\\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 &0 & 0 & 0 & -1 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$A(2,2)$${}^{\times 2}$, $A(1,4)_2$
Minimal supergroups:$B(2,4)$${}^{\times 3}$, $B(1,12)$, $B(1,8)_1$, $D(2,2)$, $B(6,2)$, $J(B(1,4)_2)$, $J_s(B(1,4)_2)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $66$ $0$ $1840$ $0$ $57470$ $0$ $1860264$ $0$ $61382244$
$a_2$ $1$ $2$ $15$ $158$ $1991$ $26562$ $363621$ $5053134$ $70954083$ $1004094086$ $14296368605$ $204561029994$ $2938952562953$
$a_3$ $1$ $0$ $24$ $0$ $5932$ $0$ $1932360$ $0$ $670252828$ $0$ $239998564704$ $0$ $87682438514836$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $15$ $6$ $28$ $66$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $24$ $158$ $86$ $348$ $186$ $792$ $1840$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $260$ $1991$ $1084$ $604$ $4564$ $2500$ $10582$ $5760$ $24610$ $57470$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $3416$ $26562$ $1854$ $14472$ $7894$ $61804$ $33668$ $18392$ $144366$ $78584$ $337932$ $183666$ $792260$ $1860264$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $5932$ $46076$ $363621$ $25126$ $197654$ $107540$ $851892$ $58488$ $462778$ $251516$ $1999534$ $1085344$ $589568$ $4699402$
$$ $2549078$ $11057074$ $5993316$ $26040672$ $61382244$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&7&0&6&0&7&0&0&20\\0&4&0&2&0&18&0&0&34&0&16&0&42&82&0\\1&0&12&0&11&0&44&50&0&38&0&99&0&0&250\\0&2&0&16&0&38&0&0&60&0&22&0&128&162&0\\2&0&11&0&25&0&51&77&0&78&0&142&0&0&360\\0&18&0&38&0&160&0&0&288&0&128&0&468&740&0\\1&0&44&0&51&0&198&210&0&172&0&451&0&0&1136\\7&0&50&0&77&0&210&298&0&266&0&553&0&0&1422\\0&34&0&60&0&288&0&0&536&0&244&0&832&1372&0\\6&0&38&0&78&0&172&266&0&266&0&490&0&0&1264\\0&16&0&22&0&128&0&0&244&0&116&0&362&620&0\\7&0&99&0&142&0&451&553&0&490&0&1131&0&0&2880\\0&42&0&128&0&468&0&0&832&0&362&0&1492&2220&0\\0&82&0&162&0&740&0&0&1372&0&620&0&2220&3584&0\\20&0&250&0&360&0&1136&1422&0&1264&0&2880&0&0&7392\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&12&16&25&160&198&298&536&266&116&1131&1492&3584&7392&3932&3967&9940&8794&2764\end{bmatrix}$

Event probabilities

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