Properties

Label 1.4.1.24.12b
  
Name \(O_1\)
Weight 1
Degree 4
Real dimension 1
Components 24
Contained in \(\mathrm{USp}(4)\)
Identity Component \(\mathrm{U}(1)_2\)
Component group \(S_4\)

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Invariants

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

Identity Component

Name:$\mathrm{U}(1)_2$
Index:$24$
$\mathbb{R}$-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:$S_4$
Order:$24$
Abelian:$\mathrm{False}$

Subgroups and Supergroups

Maximal Subgroups:$D_{3,2}$, $D_{4,1}$, $T$
Minimal Supergroups:$J(O)$

Moment Statistics

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $6$ $0$ $60$ $0$ $770$ $0$ $10836$ $0$ $158004$
$a_2$ $1$ $1$ $3$ $8$ $30$ $126$ $617$ $3221$ $17586$ $98090$ $554673$ $3162501$ $18136681$

Event Probabilities

$\mathrm{P}[a_1=0]=\frac{15}{24}$
$\mathrm{P}[a_2=0]=\frac{6}{24}$ $\mathrm{P}[a_2=2]=\frac{6}{24}$