# Properties

 Label 1.6.J.12.5a Name $$J_2(E_6)$$ Weight $1$ Degree $6$ Real dimension $4$ Components $12$ Contained in $$\mathrm{USp}(6)$$ Identity component $$\mathrm{U}(1)\times\mathrm{SU}(2)_2$$ Component group $$C_2\times C_6$$

## Invariants

 Weight: $1$ Degree: $6$ $\mathbb{R}$-dimension: $4$ Components: $12$ Contained in: $\mathrm{USp}(6)$ Rational: yes

## Identity component

 Name: $\mathrm{U}(1)\times\mathrm{SU}(2)_2$ $\mathbb{R}$-dimension: $4$ Description: $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&\overline{B}\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),B\in\mathrm{SU}(2)\right\}$ Symplectic form: $\begin{bmatrix}J_2&0&0\\0&0&I_2\\0&-I_2&0\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ Hodge circle: $\mathrm{diag}(u,\bar u, u,\bar u,\bar u,u)$

## Component group

 Name: $C_2\times C_6$ Order: $12$ Abelian: yes Generators: $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{12}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{11} \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

## Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $27$ $0$ $380$ $0$ $6895$ $0$ $146160$ $0$ $3425070$
$a_2$ $1$ $2$ $9$ $51$ $369$ $3158$ $30343$ $315122$ $3453969$ $39373182$ $462516147$ $5565186465$ $68304727147$
$a_3$ $1$ $0$ $12$ $0$ $852$ $0$ $127700$ $0$ $26130300$ $0$ $6248485404$ $0$ $1646686964604$