# 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 $$D_4$$

## 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

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