# Properties

 Label 1.6.N.72.49a Name $$J(A(6,6))$$ Weight $1$ Degree $6$ Real dimension $1$ Components $72$ Contained in $$\mathrm{USp}(6)$$ Identity component $$\mathrm{U}(1)_3$$ Component group $$C_2^2\times C_3:S_3$$

## Invariants

 Weight: $1$ Degree: $6$ $\mathbb{R}$-dimension: $1$ Components: $72$ Contained in: $\mathrm{USp}(6)$ Rational: yes

## Identity component

 Name: $\mathrm{U}(1)_3$ $\mathbb{R}$-dimension: $1$ Description: $\left\{\begin{bmatrix}\alpha I_3&0,\\ 0&\bar \alpha I_3\end{bmatrix}: \alpha\bar\alpha=1,\ \alpha\in\mathbb{C}\right\}$ Symplectic form: $\begin{bmatrix} 0 & I_3\\ -I_3 & 0\end{bmatrix}$ Hodge circle: $u\mapsto \mathrm{diag}(u,u, u, \bar u, \bar u, \bar u)$

## Component group

 Name: $C_2^2\times C_3:S_3$ Order: $72$ Abelian: no Generators: $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{18}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{18}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{9}^{8} & 0 & 0 & 0 \\0 & 0 & 0 &\zeta_{18}^{17} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{18}^{17} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{9}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0& 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

## Subgroups and supergroups

 Maximal subgroups: $J(A(2,6))$${}^{\times 3}, J(A(3,6))$${}^{\times 6}$, $A(6,6)$, $J(A(6,2))$ Minimal supergroups: $J(B(6,6))$, $J(C(6,6))$

## Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $45$ $0$ $930$ $0$ $22365$ $0$ $586278$ $0$ $16250850$
$a_2$ $1$ $3$ $15$ $105$ $963$ $10233$ $117432$ $1409670$ $17449479$ $221132301$ $2857056750$ $37531830240$ $500300094966$
$a_3$ $1$ $0$ $16$ $0$ $2460$ $0$ $554620$ $0$ $147080780$ $0$ $43065258516$ $0$ $13554887266140$