# Properties

 Label 1.6.G.4.2a Name $$F_{a,b}\times \mathrm{SU}(2)$$ Weight $1$ Degree $6$ Real dimension $5$ Components $4$ Contained in $$\mathrm{USp}(6)$$ Identity component $$\mathrm{U}(1)^2\times\mathrm{SU}(2)$$ Component group $$C_2^2$$

## Invariants

 Weight: $1$ Degree: $6$ $\mathbb{R}$-dimension: $5$ Components: $4$ Contained in: $\mathrm{USp}(6)$ Rational: yes

## Identity component

 Name: $\mathrm{U}(1)^2\times\mathrm{SU}(2)$ $\mathbb{R}$-dimension: $5$ Description: $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&C\end{bmatrix}: A,B\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),C\in\mathrm{SU}(2)\right\}$ 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: $\begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0& 0 & 0 & 0 & 1 \\\end{bmatrix}$