# Properties

 Label 1.4.F.12.4d Name $$D_{6,2}$$ Weight $1$ Degree $4$ Real dimension $1$ Components $12$ Contained in $$\mathrm{USp}(4)$$ Identity component $$\mathrm{U}(1)_2$$ Component group $$D_6$$

## Invariants

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

## Identity component

 Name: $\mathrm{U}(1)_2$ $\mathbb{R}$-dimension: $1$ Description: $\left\{\begin{bmatrix}\alpha I_2&0\\0&\bar\alpha I_2\end{bmatrix}: \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ Symplectic form: $\begin{bmatrix}0&I_2\\-I_2&0\end{bmatrix}$ Hodge circle: $u\mapsto\mathrm{diag}(u, u,\bar u,\bar u)$

## Component group

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

 Maximal subgroups: $D_{2,1}$, $D_{3,2}$${}^{\times 2}, C_6 Minimal supergroups: J(D_6) ## Moment sequences x \mathrm{E}[x^{0}] \mathrm{E}[x^{1}] \mathrm{E}[x^{2}] \mathrm{E}[x^{3}] \mathrm{E}[x^{4}] \mathrm{E}[x^{5}] \mathrm{E}[x^{6}] \mathrm{E}[x^{7}] \mathrm{E}[x^{8}] \mathrm{E}[x^{9}] \mathrm{E}[x^{10}] \mathrm{E}[x^{11}] \mathrm{E}[x^{12}] a_1 1 0 2 0 18 0 200 0 2450 0 31752 0 427812 a_2 1 2 6 20 82 372 1825 9326 49026 262628 1427641 7851846 43609857 ## Moment simplex  \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon 2 2 6 8 18 20 36 84 200 82 172 412 1000 2450 372 856 2088 5140 12740 31752 1825 4386 10842 26980 67494 169596 427812 ## Moment matrix \mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&2&0&1&0&2\\0&2&0&0&4&0&4&0&6&0\\1&0&3&2&0&4&0&7&0&6\\0&0&2&8&0&4&0&14&0&6\\0&4&0&0&12&0&12&0&20&0\\2&0&4&4&0&10&0&14&0&12\\0&4&0&0&12&0&14&0&22&0\\1&0&7&14&0&14&0&37&0&24\\0&6&0&0&20&0&22&0&40&0\\2&0&6&6&0&12&0&24&0&23\end{bmatrix} \ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&3&8&12&10&14&37&40&23\end{bmatrix} ## Event probabilities -$$a_2\in\mathbb{Z}$$a_2=-2$$a_2=-1$$a_2=0$$a_2=1$$a_2=2 -$$1$$1/2$$0$$0$$0$$0$$1/2$
$a_1=0$$7/12$$1/2$$0$$0$$0$$0$$1/2$