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Properties

Label 1.6.N.108.22a
  
Name \(C(6,6)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $108$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_3^2:A_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$108$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)_3$
$\mathbb{R}$-dimension:$1$
Description:$\left\{\begin{bmatrix}\alpha I_3&0,\\ 0&\bar \alpha I_3\end{bmatrix}: \alpha\bar\alpha=1,\ \alpha\in\mathbb{C}\right\}$ Symplectic form:$\begin{bmatrix} 0 & I_3\\ -I_3 & 0\end{bmatrix}$
Hodge circle:$u\mapsto \mathrm{diag}(u,u, u, \bar u, \bar u, \bar u)$

Component group

Name:$C_3^2:A_4$
Order:$108$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{18}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{18}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{9}^{8} & 0 & 0 & 0 \\0 & 0 & 0 &\zeta_{18}^{17} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{18}^{17} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{9}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0& 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$C(6,2)$${}^{\times 3}$, $A(6,6)$, $C(3,3)$
Minimal supergroups:$J_s(C(6,6))$, $D(6,6)$, $J(C(6,6))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $30$ $0$ $620$ $0$ $14910$ $0$ $390852$ $0$ $10833900$
$a_2$ $1$ $1$ $7$ $61$ $615$ $6741$ $78045$ $939051$ $11630799$ $147414973$ $1904684817$ $25021161111$ $333533219497$
$a_3$ $1$ $0$ $12$ $0$ $1644$ $0$ $369760$ $0$ $98053900$ $0$ $28710172512$ $0$ $9036591511376$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $7$ $4$ $14$ $30$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $12$ $61$ $36$ $130$ $76$ $282$ $620$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $96$ $615$ $356$ $208$ $1350$ $780$ $2990$ $1720$ $6660$ $14910$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $1000$ $6741$ $576$ $3852$ $2208$ $15066$ $8588$ $4912$ $33834$ $19240$ $76260$ $43260$ $172410$ $390852$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1644$ $11160$ $78045$ $6360$ $44148$ $25032$ $176490$ $14224$ $99636$ $56368$ $400314$ $225560$ $127360$ $910260$
$$ $511980$ $2074338$ $1164744$ $4736340$ $10833900$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&1&0&2&2&0&2&0&3&0&0&4\\0&2&0&2&0&8&0&0&10&0&6&0&14&22&0\\0&0&6&0&6&0&14&16&0&10&0&30&0&0&60\\0&2&0&6&0&12&0&0&18&0&10&0&30&42&0\\1&0&6&0&9&0&18&22&0&16&0&41&0&0&80\\0&8&0&12&0&52&0&0&72&0&36&0&112&164&0\\2&0&14&0&18&0&52&56&0&42&0&108&0&0&228\\2&0&16&0&22&0&56&68&0&52&0&126&0&0&272\\0&10&0&18&0&72&0&0&114&0&54&0&174&262&0\\2&0&10&0&16&0&42&52&0&46&0&96&0&0&216\\0&6&0&10&0&36&0&0&54&0&32&0&84&130&0\\3&0&30&0&41&0&108&126&0&96&0&247&0&0&528\\0&14&0&30&0&112&0&0&174&0&84&0&284&414&0\\0&22&0&42&0&164&0&0&262&0&130&0&414&630&0\\4&0&60&0&80&0&228&272&0&216&0&528&0&0&1204\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&6&9&52&52&68&114&46&32&247&284&630&1204&562&571&1324&1076&302\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$37/54$$0$$37/54$$0$$0$$0$
$a_1=0$$37/54$$37/54$$0$$37/54$$0$$0$$0$
$a_3=0$$0$$0$$0$$0$$0$$0$$0$
$a_1=a_3=0$$0$$0$$0$$0$$0$$0$$0$