Properties

Label 1.6.N.24.1b
  
Name \(B(3,4;4)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $24$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_3:C_8\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$24$
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:$C_3:C_8$
Order:$24$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{5} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{7} & 0 \\0 & 0 & 0 & 0 & 0& \zeta_{12}^{7} \\\end{bmatrix}, \begin{bmatrix}i & 0 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & \zeta_{8}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -i& 0 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$A(1,8)_2$, $A(3,4)$
Minimal supergroups:$B(O,2)$, $J(B(3,4;4))$, $J_s(B(3,4;4))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $48$ $0$ $960$ $0$ $24500$ $0$ $708624$ $0$ $21989352$
$a_2$ $1$ $2$ $12$ $96$ $974$ $11252$ $140618$ $1846224$ $25028578$ $346593780$ $4869399242$ $69103707224$ $987797537174$
$a_3$ $1$ $0$ $18$ $0$ $2658$ $0$ $713160$ $0$ $231278698$ $0$ $80947591308$ $0$ $29346093106308$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $12$ $6$ $22$ $48$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $18$ $96$ $56$ $200$ $114$ $432$ $960$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $148$ $974$ $554$ $322$ $2140$ $1218$ $4782$ $2700$ $10780$ $24500$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1596$ $11252$ $900$ $6304$ $3546$ $25464$ $14224$ $7988$ $58074$ $32340$ $133156$ $73892$ $306600$ $708624$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $2658$ $18962$ $140618$ $10592$ $77796$ $43166$ $323696$ $24006$ $178642$ $98832$ $748304$ $411992$ $227392$ $1735624$
$$ $953540$ $4036844$ $2213400$ $9411696$ $21989352$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&4&0&3&0&4&0&0&8\\0&4&0&2&0&12&0&0&16&0&10&0&18&36&0\\1&0&9&0&8&0&23&24&0&15&0&46&0&0&98\\0&2&0&10&0&20&0&0&26&0&10&0&54&66&0\\2&0&8&0&16&0&24&36&0&30&0&64&0&0&136\\0&12&0&20&0&78&0&0&118&0&58&0&186&286&0\\1&0&23&0&24&0&87&88&0&67&0&180&0&0&420\\4&0&24&0&36&0&88&120&0&100&0&214&0&0&516\\0&16&0&26&0&118&0&0&208&0&92&0&310&498&0\\3&0&15&0&30&0&67&100&0&95&0&176&0&0&448\\0&10&0&10&0&58&0&0&92&0&54&0&138&232&0\\4&0&46&0&64&0&180&214&0&176&0&434&0&0&1024\\0&18&0&54&0&186&0&0&310&0&138&0&544&796&0\\0&36&0&66&0&286&0&0&498&0&232&0&796&1270&0\\8&0&98&0&136&0&420&516&0&448&0&1024&0&0&2574\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&9&10&16&78&87&120&208&95&54&434&544&1270&2574&1340&1360&3362&2972&894\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$1/12$$0$$1/12$$0$$0$$0$
$a_1=0$$1/12$$1/12$$0$$1/12$$0$$0$$0$
$a_3=0$$0$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$0$$0$$0$$0$$0$$0$$0$