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Properties

Label 1.6.N.24.12b
  
Name \(D(2,2)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $24$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(S_4\)

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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:$S_4$
Order:$24$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0& 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0 & 0& 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$B(3,1)$, $B(1,4)_2$, $C(2,2)$
Minimal supergroups:$D(4,4)$, $D(6,2)$${}^{\times 3}$, $E(168)$${}^{\times 2}$, $J(D(2,2))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $24$ $0$ $620$ $0$ $19180$ $0$ $620172$ $0$ $20461056$
$a_2$ $1$ $1$ $6$ $55$ $670$ $8871$ $121254$ $1684509$ $23651730$ $334699075$ $4765459186$ $68187018549$ $979650878914$
$a_3$ $1$ $0$ $10$ $0$ $1994$ $0$ $644320$ $0$ $223420330$ $0$ $79999560420$ $0$ $29227480073132$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $6$ $2$ $10$ $24$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $10$ $55$ $30$ $118$ $62$ $266$ $620$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $88$ $670$ $364$ $206$ $1526$ $838$ $3534$ $1920$ $8210$ $19180$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1144$ $8871$ $618$ $4832$ $2636$ $20614$ $11232$ $6148$ $48138$ $26212$ $112668$ $61222$ $264110$ $620172$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1994$ $15372$ $121254$ $8390$ $65906$ $35866$ $283998$ $19496$ $154288$ $83856$ $666556$ $361816$ $196588$ $1566526$
$$ $849758$ $3685780$ $1997772$ $8680308$ $20461056$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&1&0&0&3&0&3&0&2&0&0&6\\0&2&0&0&0&6&0&0&12&0&6&0&12&28&0\\0&0&5&0&3&0&15&16&0&11&0&33&0&0&84\\0&0&0&8&0&12&0&0&18&0&6&0&48&52&0\\1&0&3&0&10&0&15&27&0&30&0&47&0&0&120\\0&6&0&12&0&54&0&0&96&0&42&0&156&246&0\\0&0&15&0&15&0&71&66&0&51&0&151&0&0&378\\3&0&16&0&27&0&66&104&0&94&0&183&0&0&474\\0&12&0&18&0&96&0&0&182&0&84&0&270&460&0\\3&0&11&0&30&0&51&94&0&103&0&162&0&0&420\\0&6&0&6&0&42&0&0&84&0&42&0&114&210&0\\2&0&33&0&47&0&151&183&0&162&0&378&0&0&960\\0&12&0&48&0&156&0&0&270&0&114&0&516&732&0\\0&28&0&52&0&246&0&0&460&0&210&0&732&1200&0\\6&0&84&0&120&0&378&474&0&420&0&960&0&0&2466\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&8&10&54&71&104&182&103&42&378&516&1200&2466&1338&1332&3354&2954&984\end{bmatrix}$

Event probabilities

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