# Properties

 Label 1.4.E.8.3a Name $$J(E_4)$$ Weight $1$ Degree $4$ Real dimension $3$ Components $8$ Contained in $$\mathrm{USp}(4)$$ Identity component $$\mathrm{SU}(2)_2$$ Component group $$D_4$$

## Invariants

 Weight: $1$ Degree: $4$ $\mathbb{R}$-dimension: $3$ Components: $8$ 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: $D_4$ Order: $8$ Abelian: no Generators: $\begin{bmatrix}\zeta_8&0&0&0\\0&\zeta_8&0&0\\0&0&\zeta_8^7&0\\0&0&0&\zeta_8^7\end{bmatrix}, \begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}$

 Maximal subgroups: $E_4$, $J(E_2)$${}^{\times 2} Minimal supergroups: ## Moment sequences x \mathrm{E}[x^{0}] \mathrm{E}[x^{1}] \mathrm{E}[x^{2}] \mathrm{E}[x^{3}] \mathrm{E}[x^{4}] \mathrm{E}[x^{5}] \mathrm{E}[x^{6}] \mathrm{E}[x^{7}] \mathrm{E}[x^{8}] \mathrm{E}[x^{9}] \mathrm{E}[x^{10}] \mathrm{E}[x^{11}] \mathrm{E}[x^{12}] a_1 1 0 1 0 6 0 50 0 504 0 5712 0 69696 a_2 1 1 3 7 26 96 422 1926 9326 46402 236810 1229394 6474228 ## Moment simplex  \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon 1 1 3 3 6 7 11 23 50 26 45 98 220 504 96 198 446 1028 2408 5712 422 918 2124 5000 11920 28704 69696 ## Moment matrix \mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&1&0&0&0&0\\0&1&0&0&1&0&1&0&1&0\\0&0&2&0&0&0&0&2&0&3\\0&0&0&3&0&1&0&2&0&0\\0&1&0&0&3&0&2&0&4&0\\1&0&0&1&0&5&0&1&0&0\\0&1&0&0&2&0&3&0&3&0\\0&0&2&2&0&1&0&8&0&6\\0&1&0&0&4&0&3&0&8&0\\0&0&3&0&0&0&0&6&0&9\end{bmatrix} \ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&3&3&5&3&8&8&9\end{bmatrix} ## Event probabilities -$$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$$5/8$$0$$0$$0$$0$$0$$0$