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Properties

Label 1.6.N.32.7a
  
Name \(J_s(B(2,4;4))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $32$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(\OD_{16}:C_2\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$32$
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:$\OD_{16}:C_2$
Order:$32$
Abelian:no
Generators:$\begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0& 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{5} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{7} & 0 \\0 & 0 & 0 & 0 & 0& \zeta_{12}^{7} \\\end{bmatrix}, \begin{bmatrix}i & 0 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & \zeta_{8}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -i& 0 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0\\0 & 0 & 0 & 0 & 0 & -1 \\-i & 0 & 0 & 0 & 0 & 0 \\0 & -i & 0 & 0 & 0 & 0 \\0 &0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J_n(A(2,4))$, $J(A(2,4))$, $B(2,4;4)$
Minimal supergroups:$J(B(4,4))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $27$ $0$ $620$ $0$ $17115$ $0$ $514332$ $0$ $16248078$
$a_2$ $1$ $2$ $9$ $65$ $664$ $7922$ $101622$ $1356287$ $18560486$ $258363992$ $3640100194$ $51736885697$ $740153042317$
$a_3$ $1$ $0$ $10$ $0$ $1818$ $0$ $521800$ $0$ $172358158$ $0$ $60613039320$ $0$ $22000674064446$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $9$ $3$ $12$ $27$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $10$ $65$ $33$ $124$ $69$ $274$ $620$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $92$ $664$ $360$ $206$ $1440$ $808$ $3267$ $1820$ $7455$ $17115$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1074$ $7922$ $597$ $4360$ $2425$ $17982$ $9956$ $5536$ $41373$ $22860$ $95558$ $52661$ $221382$ $514332$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1818$ $13384$ $101622$ $7413$ $55791$ $30750$ $234904$ $16968$ $128977$ $70934$ $545721$ $299116$ $164246$ $1270983$
$$ $695569$ $2966495$ $1621074$ $6936804$ $16248078$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&1&0&2&0&0&6\\0&2&0&1&0&7&0&0&11&0&6&0&13&25&0\\1&0&6&0&3&0&14&19&0&9&0&30&0&0&71\\0&1&0&6&0&13&0&0&18&0&7&0&38&47&0\\0&0&3&0&12&0&18&20&0&25&0&46&0&0&100\\0&7&0&13&0&52&0&0&84&0&40&0&134&208&0\\0&0&14&0&18&0&61&59&0&50&0&131&0&0&308\\4&0&19&0&20&0&59&95&0&67&0&151&0&0&381\\0&11&0&18&0&84&0&0&150&0&68&0&226&368&0\\1&0&9&0&25&0&50&67&0&75&0&133&0&0&332\\0&6&0&7&0&40&0&0&68&0&36&0&101&170&0\\2&0&30&0&46&0&131&151&0&133&0&318&0&0&760\\0&13&0&38&0&134&0&0&226&0&101&0&400&590&0\\0&25&0&47&0&208&0&0&368&0&170&0&590&944&0\\6&0&71&0&100&0&308&381&0&332&0&760&0&0&1916\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&6&12&52&61&95&150&75&36&318&400&944&1916&1004&1037&2524&2235&684\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$1/2$$0$$0$$1/4$$0$$1/4$
$a_1=0$$1/2$$1/2$$0$$0$$1/4$$0$$1/4$
$a_3=0$$1/2$$1/2$$0$$0$$1/4$$0$$1/4$
$a_1=a_3=0$$1/2$$1/2$$0$$0$$1/4$$0$$1/4$