# Properties

 Label 1.4.1.8.3b Name $$D_{4,1}$$ Weight 1 Degree 4 Real dimension 1 Components 8 Contained in $$\mathrm{USp}(4)$$ Identity Component $$\mathrm{U}(1)_2$$ Component group $$D_4$$

## Invariants

 Weight: $1$ Degree: $4$ $\mathbb{R}$-dimension: $1$ Components: $8$ Contained in: $\mathrm{USp}(4)$ Rational: $\mathrm{True}$

## Identity Component

 Name: $\mathrm{U}(1)_2$ Index: $8$ $\mathbb{R}$-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: $D_4$ Order: $8$ Abelian: $\mathrm{False}$

## Moment Statistics

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $12$ $0$ $160$ $0$ $2240$ $0$ $32256$ $0$ $473088$
$a_2$ $1$ $1$ $4$ $13$ $63$ $311$ $1678$ $9206$ $51523$ $290875$ $1654554$ $9460826$ $54334206$

## Event Probabilities

 $\mathrm{P}[a_1=0]=\frac{7}{8}$ $\mathrm{P}[a_2=0]=\frac{2}{8}$ $\mathrm{P}[a_2=2]=\frac{2}{8}$