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Properties

Label 1.6.N.216.99a
  
Name \(J(C(6,6))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $216$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_6^2:C_6\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$216$
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:$C_6^2:C_6$
Order:$216$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{18}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{18}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{9}^{8} & 0 & 0 & 0 \\0 & 0 & 0 &\zeta_{18}^{17} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{18}^{17} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{9}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0& 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$J(C(6,2))$, $J(C(3,3))$, $J(A(6,6))$, $C(6,6)$
Minimal supergroups:$J(D(6,6))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $15$ $0$ $310$ $0$ $7455$ $0$ $195426$ $0$ $5416950$
$a_2$ $1$ $1$ $5$ $35$ $321$ $3411$ $39144$ $469890$ $5816493$ $73710767$ $952352250$ $12510610080$ $166766698322$
$a_3$ $1$ $0$ $6$ $0$ $822$ $0$ $184880$ $0$ $49026950$ $0$ $14355086256$ $0$ $4518295755688$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $5$ $2$ $7$ $15$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $6$ $35$ $18$ $65$ $38$ $141$ $310$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $48$ $321$ $178$ $104$ $675$ $390$ $1495$ $860$ $3330$ $7455$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $500$ $3411$ $288$ $1926$ $1104$ $7533$ $4294$ $2456$ $16917$ $9620$ $38130$ $21630$ $86205$ $195426$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $822$ $5580$ $39144$ $3180$ $22074$ $12516$ $88245$ $7112$ $49818$ $28184$ $200157$ $112780$ $63680$ $455130$
$$ $255990$ $1037169$ $582372$ $2368170$ $5416950$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&1&2&0&0&0&1&0&0&2\\0&1&0&1&0&4&0&0&5&0&3&0&7&11&0\\0&0&4&0&2&0&6&10&0&4&0&14&0&0&30\\0&1&0&3&0&6&0&0&9&0&5&0&15&21&0\\0&0&2&0&6&0&10&8&0&10&0&22&0&0&40\\0&4&0&6&0&26&0&0&36&0&18&0&56&82&0\\1&0&6&0&10&0&27&26&0&22&0&55&0&0&114\\2&0&10&0&8&0&26&40&0&22&0&60&0&0&136\\0&5&0&9&0&36&0&0&57&0&27&0&87&131&0\\0&0&4&0&10&0&22&22&0&26&0&50&0&0&108\\0&3&0&5&0&18&0&0&27&0&16&0&42&65&0\\1&0&14&0&22&0&55&60&0&50&0&125&0&0&264\\0&7&0&15&0&56&0&0&87&0&42&0&142&207&0\\0&11&0&21&0&82&0&0&131&0&65&0&207&315&0\\2&0&30&0&40&0&114&136&0&108&0&264&0&0&602\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&4&3&6&26&27&40&57&26&16&125&142&315&602&281&299&662&544&151\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$91/108$$0$$73/108$$0$$0$$1/6$
$a_1=0$$91/108$$91/108$$0$$73/108$$0$$0$$1/6$
$a_3=0$$1/2$$1/2$$0$$1/3$$0$$0$$1/6$
$a_1=a_3=0$$1/2$$1/2$$0$$1/3$$0$$0$$1/6$