Properties

Label 1.4.1.48.48a
  
Name \(J(O)\)
Weight 1
Degree 4
Real dimension 1
Components 48
Contained in \(\mathrm{USp}(4)\)
Identity Component \(\mathrm{U}(1)_2\)
Component group \(C_2\times S_4\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$1$
Components:$48$
Contained in:$\mathrm{USp}(4)$
Rational:$\mathrm{True}$

Identity Component

Name:$\mathrm{U}(1)_2$
Index:$48$
$\mathbb{R}$-dimension:$1$
Description:$\left\{\begin{bmatrix}\mathrm{diag}_2(\alpha)&0\\0&\mathrm{diag}_2(\bar\alpha)\end{bmatrix}: \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\},\ \mathrm{diag}_2(\alpha):=\begin{bmatrix}\alpha&0\\0&\alpha\end{bmatrix}$

Component Group

Name:$C_2\times S_4$
Order:$48$
Abelian:$\mathrm{False}$

Subgroups and Supergroups

Maximal Subgroups:$J(D_3)$, $J(T)$, $J(D_4)$, $O_1$, $O$
Minimal Supergroups:

Moment Statistics

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $6$ $0$ $50$ $0$ $525$ $0$ $6426$ $0$ $86394$
$a_2$ $1$ $1$ $3$ $7$ $26$ $96$ $432$ $2045$ $10432$ $55144$ $300548$ $1669515$ $9406817$

Event Probabilities

$\mathrm{P}[a_1=0]=\frac{33}{48}$
$\mathrm{P}[a_2=-2]=\frac{1}{48}$ $\mathrm{P}[a_2=0]=\frac{6}{48}$ $\mathrm{P}[a_2=1]=\frac{8}{48}$ $\mathrm{P}[a_2=2]=\frac{9}{48}$