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Properties

Label 1.6.N.24.14a
  
Name \(J(B(3,2))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $24$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_2\times D_6\)

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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:$C_2\times D_6$
Order:$24$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0& 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1& 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0\\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

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

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $30$ $0$ $720$ $0$ $20650$ $0$ $641592$ $0$ $20776140$
$a_2$ $1$ $2$ $10$ $75$ $784$ $9607$ $126378$ $1721715$ $23928108$ $336779043$ $4781241730$ $68307462775$ $980574050046$
$a_3$ $1$ $0$ $11$ $0$ $2181$ $0$ $660790$ $0$ $224864661$ $0$ $80129140506$ $0$ $29239323750582$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $10$ $3$ $13$ $30$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $11$ $75$ $37$ $142$ $78$ $316$ $720$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $105$ $784$ $420$ $240$ $1706$ $952$ $3898$ $2155$ $8945$ $20650$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1274$ $9607$ $702$ $5250$ $2902$ $21940$ $12076$ $6680$ $50802$ $27907$ $118010$ $64638$ $274820$ $641592$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $2181$ $16351$ $126378$ $9008$ $69011$ $37836$ $293805$ $20754$ $160518$ $87818$ $686202$ $374305$ $204512$ $1605861$
$$ $874762$ $3764502$ $2047878$ $8837850$ $20776140$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&5&0&1&0&2&0&0&7\\0&2&0&1&0&8&0&0&13&0&7&0&15&30&0\\1&0&7&0&3&0&16&23&0&10&0&35&0&0&87\\0&1&0&7&0&15&0&0&21&0&8&0&48&56&0\\0&0&3&0&14&0&21&23&0&33&0&55&0&0&124\\0&8&0&15&0&61&0&0&102&0&47&0&164&255&0\\0&0&16&0&21&0&75&69&0&59&0&161&0&0&387\\5&0&23&0&23&0&69&118&0&86&0&185&0&0&482\\0&13&0&21&0&102&0&0&186&0&86&0&281&467&0\\1&0&10&0&33&0&59&86&0&102&0&170&0&0&426\\0&7&0&8&0&47&0&0&86&0&43&0&122&213&0\\2&0&35&0&55&0&161&185&0&170&0&392&0&0&972\\0&15&0&48&0&164&0&0&281&0&122&0&516&746&0\\0&30&0&56&0&255&0&0&467&0&213&0&746&1207&0\\7&0&87&0&124&0&387&482&0&426&0&972&0&0&2481\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&7&7&14&61&75&118&186&102&43&392&516&1207&2481&1330&1355&3339&2960&953\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$7/12$$0$$1/4$$0$$0$$1/3$
$a_1=0$$7/12$$7/12$$0$$1/4$$0$$0$$1/3$
$a_3=0$$1/2$$1/2$$0$$1/6$$0$$0$$1/3$
$a_1=a_3=0$$1/2$$1/2$$0$$1/6$$0$$0$$1/3$