Properties

Label 1.6.N.96.72a
  
Name \(J(C(4,4))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $96$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_4^2:C_6\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$96$
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:$C_4^2:C_6$
Order:$96$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & i & 0 & 0 & 0 & 0 \\0 & 0 & -i & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & -i & 0 \\0 & 0 & 0 & 0 & 0 & i\\\end{bmatrix}, \begin{bmatrix}\zeta_{12}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 &\zeta_{12}^{11} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0& 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$C(4,4)$, $J(A(4,4))$, $J(C(2,2))$
Minimal supergroups:$J(D(4,4))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $15$ $0$ $310$ $0$ $7525$ $0$ $204246$ $0$ $6017550$
$a_2$ $1$ $1$ $5$ $35$ $322$ $3466$ $40934$ $513850$ $6734074$ $91012814$ $1257365680$ $17643361660$ $250327558573$
$a_3$ $1$ $0$ $6$ $0$ $828$ $0$ $199740$ $0$ $60649596$ $0$ $20613284436$ $0$ $7386934420602$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $5$ $2$ $7$ $15$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $6$ $35$ $18$ $65$ $38$ $141$ $310$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $48$ $322$ $178$ $104$ $677$ $390$ $1501$ $860$ $3350$ $7525$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $502$ $3466$ $288$ $1944$ $1108$ $7675$ $4342$ $2468$ $17307$ $9760$ $39230$ $22050$ $89327$ $204246$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $828$ $5698$ $40934$ $3222$ $22838$ $12822$ $92905$ $7220$ $51840$ $29012$ $212453$ $118220$ $65980$ $487738$
$$ $270718$ $1123633$ $622188$ $2596650$ $6017550$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&1&2&0&0&0&1&0&0&2\\0&1&0&1&0&4&0&0&5&0&3&0&7&11&0\\0&0&4&0&2&0&6&10&0&4&0&14&0&0&30\\0&1&0&3&0&6&0&0&9&0&5&0&15&21&0\\0&0&2&0&6&0&10&8&0&10&0&22&0&0&40\\0&4&0&6&0&26&0&0&36&0&18&0&56&84&0\\1&0&6&0&10&0&27&26&0&22&0&55&0&0&118\\2&0&10&0&8&0&26&41&0&23&0&61&0&0&144\\0&5&0&9&0&36&0&0&59&0&27&0&89&139&0\\0&0&4&0&10&0&22&23&0&29&0&51&0&0&120\\0&3&0&5&0&18&0&0&27&0&16&0&42&67&0\\1&0&14&0&22&0&55&61&0&51&0&128&0&0&280\\0&7&0&15&0&56&0&0&89&0&42&0&148&221&0\\0&11&0&21&0&84&0&0&139&0&67&0&221&347&0\\2&0&30&0&40&0&118&144&0&120&0&280&0&0&686\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&4&3&6&26&27&41&59&29&16&128&148&347&686&349&369&866&765&233\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$5/6$$0$$2/3$$0$$0$$1/6$
$a_1=0$$5/6$$5/6$$0$$2/3$$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$