Properties

Label 1.2.1.d4
  
Name \(\mathrm{U}(1)[D_{4}]\)
Weight $1$
Degree $2$
Real dimension $1$
Components $8$
Contained in \(\mathrm{USp}(2)\)
Identity component \(\mathrm{U}(1)\)
Component group \(D_4\)

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Invariants

Weight:$1$
Degree:$2$
$\mathbb{R}$-dimension:$1$
Components:$8$
Contained in:$\mathrm{USp}(2)$
Rational:yes

Identity component

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

Component group

Name:$D_4$
Order:$8$
Abelian:no
Generators:$\begin{bmatrix}0&1\\-1&0\end{bmatrix}, \begin{bmatrix}1&0\\0&\zeta_{4}\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:1.2.B.D2
Minimal supergroups:1.2.B.D8, 1.2.B.D12, 1.2.B.D20, $\cdots$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $0$ $0$ $3$ $0$ $0$ $0$ $35$ $0$ $0$ $0$ $462$