Properties

Label 3.2.3.c5
  
Name \(\mathrm{SU}(2)[C_{5}]\)
Weight 3
Degree 2
Real dimension 3
Components 5
Contained in \(\mathrm{U}(2)\)
Identity Component \(\mathrm{SU}(2)\)
Component group \(C_{5}\)

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Invariants

Weight:$3$
Degree:$2$
$\mathbb{R}$-dimension:$3$
Components:$5$
Contained in:$\mathrm{U}(2)$
Rational:$\mathrm{False}$

Identity Component

Name:$\mathrm{SU}(2)$
Index:$5$
$\mathbb{R}$-dimension:$3$
Description:$\left\{\begin{bmatrix}\alpha&\beta\\-\bar\beta&\bar\alpha\end{bmatrix}:\alpha\bar\alpha+\beta\bar\beta = 1,\ \alpha,\beta\in\mathbb{C}\right\}$

Component Group

Name:$C_{5}$
Order:$5$
Abelian:$\mathrm{True}$
Generators:$\begin{bmatrix} 1 & 0 \\ 0 & \zeta_{5}\end{bmatrix}$

Subgroups and Supergroups

Maximal Subgroups:$\mathrm{SU}(2)$
Minimal Supergroups:$\mathrm{SU}(2)[C_{10}]$, $\mathrm{SU}(2)[C_{15}]$, $\mathrm{SU}(2)[C_{25}]$, $\ldots$

Moment Statistics

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $42$ $0$ $0$