# Properties

 Label 1.6.H.4.2c Name $$H_{a,b}$$ Weight $1$ Degree $6$ Real dimension $3$ Components $4$ Contained in $$\mathrm{USp}(6)$$ Identity component $$\mathrm{U}(1)^3$$ Component group $$C_2^2$$

## Invariants

 Weight: $1$ Degree: $6$ $\mathbb{R}$-dimension: $3$ Components: $4$ Contained in: $\mathrm{USp}(6)$ Rational: yes

## Identity component

 Name: $\mathrm{U}(1)^3$ $\mathbb{R}$-dimension: $3$ Description: $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}: A,B,C\in\mathrm{U}(1)\subseteq\mathrm{SU}(2)\right\}$ Symplectic form: $\begin{bmatrix}J_2&0&0\\0&J_2&0\\0&0&J_2\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ Hodge circle: $u\mapsto\mathrm{diag}(u,\bar u, u, \bar u, u, \bar u)$

## Component group

 Name: $C_2^2$ Order: $4$ Abelian: yes Generators: $\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0& 0 & 0 & 0 & 1 \\\end{bmatrix}$

## Subgroups and supergroups

 Maximal subgroups: $H_{ab}$, $H_{a}$${}^{\times 2} Minimal supergroups: H_{a,b,c}$${}^{\times 3}$

## Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $42$ $0$ $670$ $0$ $13650$ $0$ $324954$ $0$ $8551158$
$a_2$ $1$ $3$ $14$ $84$ $648$ $6048$ $64139$ $737187$ $8915132$ $111482364$ $1426839309$ $18576367857$ $245064424185$
$a_3$ $1$ $0$ $18$ $0$ $1542$ $0$ $290400$ $0$ $73958710$ $0$ $21243414528$ $0$ $6501562803120$