# Properties

 Label 1.4.E.12.4a Name $$J(E_6)$$ Weight $1$ Degree $4$ Real dimension $3$ Components $12$ Contained in $$\mathrm{USp}(4)$$ Identity component $$\mathrm{SU}(2)_2$$ Component group $$D_6$$

## Invariants

 Weight: $1$ Degree: $4$ $\mathbb{R}$-dimension: $3$ Components: $12$ Contained in: $\mathrm{USp}(4)$ Rational: yes

## Identity component

 Name: $\mathrm{SU}(2)_2$ $\mathbb{R}$-dimension: $3$ Description: $\left\{\begin{bmatrix}A&0\\0&\bar{A}\end{bmatrix}: A\in \mathrm{SU}(2)\right\}$ Symplectic form: $\begin{bmatrix}0&I_2\\-I_2&0\end{bmatrix}$ Hodge circle: $u\mapsto\mathrm{diag}(u,\bar u,\bar u,u)$

## Component group

 Name: $D_6$ Order: $12$ Abelian: no Generators: $\begin{bmatrix}\zeta_{12}&0&0&0\\0&\zeta_{12}&0&0\\0&0&\zeta_{12}^{11}&0\\0&0&0&\zeta_{12}^{11}\end{bmatrix}, \begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}$

 Maximal subgroups: $J(E_2)$, $J(E_3)$${}^{\times 2}, E_6 Minimal supergroups: ## Moment sequences x \mathrm{E}[x^{0}] \mathrm{E}[x^{1}] \mathrm{E}[x^{2}] \mathrm{E}[x^{3}] \mathrm{E}[x^{4}] \mathrm{E}[x^{5}] \mathrm{E}[x^{6}] \mathrm{E}[x^{7}] \mathrm{E}[x^{8}] \mathrm{E}[x^{9}] \mathrm{E}[x^{10}] \mathrm{E}[x^{11}] \mathrm{E}[x^{12}] a_1 1 0 1 0 6 0 50 0 490 0 5292 0 61116 a_2 1 1 3 7 25 91 387 1716 8045 38821 192415 972544 4999447 ## Moment simplex  \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon 1 1 3 3 6 7 11 23 50 25 44 96 215 490 91 188 423 970 2254 5292 387 843 1942 4537 10710 25494 61116 ## Moment matrix \mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&1&0&0&0&0\\0&1&0&0&1&0&1&0&1&0\\0&0&2&0&0&0&0&2&0&2\\0&0&0&3&0&1&0&2&0&0\\0&1&0&0&3&0&2&0&3&0\\1&0&0&1&0&4&0&1&0&0\\0&1&0&0&2&0&3&0&3&0\\0&0&2&2&0&1&0&7&0&4\\0&1&0&0&3&0&3&0&6&0\\0&0&2&0&0&0&0&4&0&7\end{bmatrix} \ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&3&3&4&3&7&6&7\end{bmatrix} ## Event probabilities -$$a_2\in\mathbb{Z}$$a_2=-2$$a_2=-1$$a_2=0$$a_2=1$$a_2=2 -$$1$$0$$0$$0$$0$$0$$0$
$a_1=0$$7/12$$0$$0$$0$$0$$0$$0$