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Properties

Label 1.6.N.54.5b
  
Name \(D(3,3)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $54$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_3^2:C_6\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$54$
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:$C_3^2:C_6$
Order:$54$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{9}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{9}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{9}^{7} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{9}^{8} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{9}^{8} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{9}^{2} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0 & 0& 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$D(3,1)$, $C(3,3)$, $B(3,3)$
Minimal supergroups:$E(216)$, $J(D(3,3))$, $D(6,6)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $18$ $0$ $340$ $0$ $9170$ $0$ $282492$ $0$ $9168852$
$a_2$ $1$ $1$ $5$ $35$ $353$ $4251$ $55683$ $759333$ $10576233$ $149147315$ $2120388335$ $30320186073$ $435492342431$
$a_3$ $1$ $0$ $8$ $0$ $984$ $0$ $290820$ $0$ $99526616$ $0$ $35567576568$ $0$ $12990657619852$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $5$ $2$ $8$ $18$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $8$ $35$ $20$ $70$ $38$ $150$ $340$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $52$ $353$ $196$ $114$ $772$ $432$ $1740$ $960$ $3970$ $9170$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $580$ $4251$ $318$ $2340$ $1292$ $9694$ $5328$ $2948$ $22350$ $12260$ $51852$ $28350$ $120806$ $282492$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $984$ $7240$ $55683$ $3984$ $30410$ $16642$ $129336$ $9104$ $70536$ $38512$ $301818$ $164352$ $89644$ $706470$
$$ $384244$ $1657446$ $900396$ $3895416$ $9168852$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&1&0&0&2&0&2&0&1&0&0&2\\0&2&0&0&0&4&0&0&6&0&4&0&4&14&0\\0&0&4&0&2&0&8&8&0&4&0&16&0&0&38\\0&0&0&6&0&6&0&0&8&0&2&0&24&24&0\\1&0&2&0&7&0&6&14&0&14&0&23&0&0&52\\0&4&0&6&0&28&0&0&44&0&20&0&70&112&0\\0&0&8&0&6&0&36&30&0&22&0&68&0&0&170\\2&0&8&0&14&0&30&50&0&44&0&82&0&0&212\\0&6&0&8&0&44&0&0&86&0&36&0&120&206&0\\2&0&4&0&14&0&22&44&0&50&0&70&0&0&188\\0&4&0&2&0&20&0&0&36&0&24&0&48&96&0\\1&0&16&0&23&0&68&82&0&70&0&173&0&0&424\\0&4&0&24&0&70&0&0&120&0&48&0&236&324&0\\0&14&0&24&0&112&0&0&206&0&96&0&324&538&0\\2&0&38&0&52&0&170&212&0&188&0&424&0&0&1104\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&6&7&28&36&50&86&50&24&173&236&538&1104&602&601&1500&1320&446\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$10/27$$0$$10/27$$0$$0$$0$
$a_1=0$$10/27$$10/27$$0$$10/27$$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$