Properties

Label 10.2.3.c6
  
Name \(\mathrm{SU}(2)[C_{6}]\)
Weight $10$
Degree $2$
Real dimension $3$
Components $6$
Contained in \(\mathrm{O}(2)\)
Identity component \(\mathrm{SU}(2)\)
Component group \(C_6\)

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Invariants

Weight:$10$
Degree:$2$
$\mathbb{R}$-dimension:$3$
Components:$6$
Contained in:$\mathrm{O}(2)$
Rational:yes

Identity component

Name:$\mathrm{SU}(2)$
$\mathbb{R}$-dimension:$3$
Description:$\left\{\begin{bmatrix}\alpha&\beta\\-\bar\beta&\bar\alpha\end{bmatrix}:\alpha\bar\alpha+\beta\bar\beta = 1,\ \alpha,\beta\in\mathbb{C}\right\}$ Symplectic form:$\begin{bmatrix}0&1\\-1&0\end{bmatrix}$
Hodge circle:$u\mapsto\mathrm{diag}(u,\bar u)$

Component group

Name:$C_6$
Order:$6$
Abelian:yes
Generators:$\begin{bmatrix}1&0\\0&\zeta_{6}\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$\mathrm{SU}(2)[C_{3}]$, $\mathrm{SU}(2)[C_{2}]$
Minimal supergroups:$\mathrm{SU}(2)[C_{12}]$, $\mathrm{SU}(2)[C_{18}]$, $\mathrm{SU}(2)[C_{30}]$, $\cdots$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $0$ $0$ $0$ $0$ $5$ $0$ $0$ $0$ $0$ $0$ $132$