Properties

Label 1.4.1.4.1a
  
Name \(C_4\)
Weight 1
Degree 4
Real dimension 1
Components 4
Contained in \(\mathrm{USp}(4)\)
Identity Component \(\mathrm{U}(1)_2\)
Component group \(C_4\)

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Invariants

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

Identity Component

Name:$\mathrm{U}(1)_2$
Index:$4$
$\mathbb{R}$-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:$C_4$
Order:$4$
Abelian:$\mathrm{True}$

Subgroups and Supergroups

Maximal Subgroups:$C_2$
Minimal Supergroups:$J(C_4)$, $D_4$, $D_{4,2}$

Moment Statistics

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $36$ $0$ $400$ $0$ $5040$ $0$ $68544$ $0$ $975744$
$a_2$ $1$ $2$ $8$ $32$ $150$ $732$ $3776$ $20064$ $109318$ $605804$ $3400848$ $19273344$ $110017980$

Event Probabilities

$\mathrm{P}[a_1=0]=\frac{1}{4}$