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Properties

Label 1.4.F.24.14a
  
Name \(J(D_6)\)
Weight $1$
Degree $4$
Real dimension $1$
Components $24$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{U}(1)_2\)
Component group \(C_2\times D_6\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$1$
Components:$24$
Contained in:$\mathrm{USp}(4)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)_2$
$\mathbb{R}$-dimension:$1$
Description:$\left\{\begin{bmatrix}\alpha I_2&0\\0&\bar\alpha I_2\end{bmatrix}: \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ Symplectic form:$\begin{bmatrix}0&I_2\\-I_2&0\end{bmatrix}$
Hodge circle:$u\mapsto\mathrm{diag}(u, u,\bar u,\bar u)$

Component group

Name:$C_2\times D_6$
Order:$24$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{12}&0&0&0\\0&\zeta_{12}^{11}&0&0\\0&0&\zeta_{12}^{11}&0\\0&0&0&\zeta_{12}\end{bmatrix}, \begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}, \begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(D_2)$, $D_6$, $J(D_3)$${}^{\times 2}$, $D_{6,1}$, $D_{6,2}$${}^{\times 2}$, $J(C_6)$
Minimal supergroups:

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $9$ $0$ $100$ $0$ $1225$ $0$ $15876$ $0$ $213906$
$a_2$ $1$ $1$ $4$ $10$ $44$ $186$ $923$ $4663$ $24552$ $131314$ $713969$ $3925923$ $21805501$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $4$ $4$ $9$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $10$ $18$ $42$ $100$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $44$ $86$ $206$ $500$ $1225$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $186$ $428$ $1044$ $2570$ $6370$ $15876$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $923$ $2193$ $5421$ $13490$ $33747$ $84798$ $213906$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&2&0&0&0&0\\0&1&0&0&2&0&2&0&3&0\\0&0&3&0&0&0&0&5&0&6\\0&0&0&5&0&3&0&6&0&1\\0&2&0&0&6&0&6&0&10&0\\2&0&0&3&0&9&0&4&0&1\\0&2&0&0&6&0&7&0&11&0\\0&0&5&6&0&4&0&21&0&16\\0&3&0&0&10&0&11&0&20&0\\0&0&6&1&0&1&0&16&0&20\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&3&5&6&9&7&21&20&20\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-2$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$
$-$$1$$1/2$$1/24$$1/12$$0$$1/12$$7/24$
$a_1=0$$19/24$$1/2$$1/24$$1/12$$0$$1/12$$7/24$