# Properties

 Label 1.4.F.4.2c Name $$D_{2,1}$$ Weight $1$ Degree $4$ Real dimension $1$ Components $4$ Contained in $$\mathrm{USp}(4)$$ Identity component $$\mathrm{U}(1)_2$$ Component group $$C_2^2$$

## Invariants

 Weight: $1$ Degree: $4$ $\mathbb{R}$-dimension: $1$ Components: $4$ 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: $C_2^2$ Order: $4$ Abelian: yes Generators: $\begin{bmatrix}0&0&0&i\\0&0&i&0\\0&i&0&0\\i&0&0&0\end{bmatrix}, \begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}$

 Maximal subgroups: $C_2$, $C_{2,1}$${}^{\times 2} Minimal supergroups: D_{4,1}, D_{4,2}, J(D_2), D_{6,1}, D_{6,2} ## Moment sequences x \mathrm{E}[x^{0}] \mathrm{E}[x^{1}] \mathrm{E}[x^{2}] \mathrm{E}[x^{3}] \mathrm{E}[x^{4}] \mathrm{E}[x^{5}] \mathrm{E}[x^{6}] \mathrm{E}[x^{7}] \mathrm{E}[x^{8}] \mathrm{E}[x^{9}] \mathrm{E}[x^{10}] \mathrm{E}[x^{11}] \mathrm{E}[x^{12}] a_1 1 0 2 0 24 0 320 0 4480 0 64512 0 946176 a_2 1 2 7 26 123 622 3346 18412 103011 581750 3308982 18921652 108667950 ## Moment simplex  \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon \left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon 2 2 7 10 24 26 52 128 320 123 278 696 1760 4480 622 1516 3840 9792 25088 64512 3346 8388 21424 54976 141568 365568 946176 ## Moment matrix \mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&3&0&2&0&3\\0&2&0&0&6&0&6&0&12&0\\1&0&4&3&0&6&0&13&0&13\\0&0&3&11&0&9&0&25&0&16\\0&6&0&0&20&0&22&0&42&0\\3&0&6&9&0&18&0&29&0&25\\0&6&0&0&22&0&26&0&48&0\\2&0&13&25&0&29&0&77&0&60\\0&12&0&0&42&0&48&0&92&0\\3&0&13&16&0&25&0&60&0&54\end{bmatrix} \ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&11&20&18&26&77&92&54\end{bmatrix} ## Event probabilities -$$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$$3/4$$1/2$$0$$0$$0$$0$$1/2$