# Properties

 Label 1.6.I.2.1a Name $$\mathrm{SU}(2)\times E_2$$ Weight $1$ Degree $6$ Real dimension $6$ Components $2$ Contained in $$\mathrm{USp}(6)$$ Identity component $$\mathrm{SU}(2)\times\mathrm{SU}(2)_2$$ Component group $$C_2$$

## Invariants

 Weight: $1$ Degree: $6$ $\mathbb{R}$-dimension: $6$ Components: $2$ Contained in: $\mathrm{USp}(6)$ Rational: yes

## Identity component

 Name: $\mathrm{SU}(2)\times\mathrm{SU}(2)_2$ $\mathbb{R}$-dimension: $6$ Description: $\left\{\begin{bmatrix}A&0&0\\0&B&0\\0&0&\overline{B}\end{bmatrix}: A,B\in\mathrm{SU}(2)\right\}$ Symplectic form: $\begin{bmatrix}J_2&0&0\\0&0&I_2\\0&-I_2&0\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$ Hodge circle: $\mathrm{diag}(u,\bar u, u,\bar u,\bar u,u)$

## Component group

 Name: $C_2$ Order: $2$ Abelian: yes Generators: $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 &0 & 0 \\0 & 0 & 0 & i & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & -i \\\end{bmatrix}$

## Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $3$ $0$ $30$ $0$ $465$ $0$ $8806$ $0$ $187446$ $0$ $4319436$
$a_2$ $1$ $2$ $11$ $69$ $531$ $4628$ $44045$ $446288$ $4739171$ $52197030$ $592008193$ $6878489621$ $81554916013$
$a_3$ $1$ $0$ $14$ $0$ $1228$ $0$ $183420$ $0$ $35005460$ $0$ $7734397416$ $0$ $1884962825328$

## Moment simplex

 $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$  $2$ $3$ $11$ $5$ $16$ $30$ $14$ $69$ $39$ $121$ $70$ $232$ $465$ $95$ $531$ $306$ $182$ $1019$ $600$ $2052$ $1205$ $4218$ $8806$ $794$ $4628$ $462$ $2692$ $1576$ $9389$ $5488$ $3222$ $19516$ $11399$ $41113$ $23954$ $87444$ $187446$ $1228$ $7217$ $44045$ $4218$ $25565$ $14892$ $92418$ $8685$ $53652$ $31203$ $196554$ $113898$ $66130$ $421711$ $243880$ $910978$ $525714$ $1978920$ $4319436$

## Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&1&4&0&1&0&1&0&0&4\\0&3&0&2&0&8&0&0&10&0&3&0&7&17&0\\1&0&8&0&4&0&13&15&0&6&0&18&0&0&36\\0&2&0&7&0&12&0&0&12&0&3&0&20&26&0\\1&0&4&0&9&0&13&16&0&13&0&24&0&0&44\\0&8&0&12&0&38&0&0&48&0&20&0&60&92&0\\1&0&13&0&13&0&38&35&0&21&0&59&0&0&116\\4&0&15&0&16&0&35&52&0&30&0&60&0&0&132\\0&10&0&12&0&48&0&0&71&0&28&0&79&130&0\\1&0&6&0&13&0&21&30&0&30&0&46&0&0&96\\0&3&0&3&0&20&0&0&28&0&19&0&39&56&0\\1&0&18&0&24&0&59&60&0&46&0&119&0&0&224\\0&7&0&20&0&60&0&0&79&0&39&0&135&172&0\\0&17&0&26&0&92&0&0&130&0&56&0&172&269&0\\4&0&36&0&44&0&116&132&0&96&0&224&0&0&470\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&8&7&9&38&38&52&71&30&19&119&135&269&470&209&231&463&334&97\end{bmatrix}$

## Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.