Properties

Label 1.4.1.16.11a
  
Name \(J(D_4)\)
Weight 1
Degree 4
Real dimension 1
Components 16
Contained in \(\mathrm{USp}(4)\)
Identity Component \(\mathrm{U}(1)_2\)
Component group \(C_2\times D_4\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$1$
Components:$16$
Contained in:$\mathrm{USp}(4)$
Rational:$\mathrm{True}$

Identity Component

Name:$\mathrm{U}(1)_2$
Index:$16$
$\mathbb{R}$-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:$C_2\times D_4$
Order:$16$
Abelian:$\mathrm{False}$

Subgroups and Supergroups

Maximal Subgroups:$J(C_4)$, $D_4$, $D_{4,1}$, $D_{4,2}$, $J(D_2)$
Minimal Supergroups:$J(O)$

Moment Statistics

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $9$ $0$ $100$ $0$ $1260$ $0$ $17136$ $0$ $243936$
$a_2$ $1$ $1$ $4$ $10$ $45$ $191$ $973$ $5048$ $27443$ $151579$ $850659$ $4818848$ $27506262$

Event Probabilities

$\mathrm{P}[a_1=0]=\frac{13}{16}$
$\mathrm{P}[a_2=-2]=\frac{1}{16}$ $\mathrm{P}[a_2=0]=\frac{2}{16}$ $\mathrm{P}[a_2=2]=\frac{5}{16}$