# Properties

 Label 1.4.1.8.2a Name $$J(C_4)$$ Weight 1 Degree 4 Real dimension 1 Components 8 Contained in $$\mathrm{USp}(4)$$ Identity Component $$\mathrm{U}(1)_2$$ Component group $$C_2\times C_4$$

## 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: $C_2\times C_4$ Order: $8$ Abelian: $\mathrm{True}$

## Moment Statistics

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $18$ $0$ $200$ $0$ $2520$ $0$ $34272$ $0$ $487872$
$a_2$ $1$ $1$ $5$ $16$ $79$ $366$ $1904$ $10032$ $54723$ $302902$ $1700680$ $9636672$ $55010014$

## Event Probabilities

 $\mathrm{P}[a_1=0]=\frac{5}{8}$ $\mathrm{P}[a_2=-2]=\frac{1}{8}$ $\mathrm{P}[a_2=0]=\frac{2}{8}$ $\mathrm{P}[a_2=2]=\frac{1}{8}$