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Properties

Label 1.6.N.72.41a
  
Name \(E(72)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $72$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(\PSU(3,2)\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$72$
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:$\PSU(3,2)$
Order:$72$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \frac{1}{\zeta_{3}^{1}-\zeta_{3}^{2}}\begin{bmatrix}1 & 1 & 1 & 0 & 0 & 0 \\1 & \zeta_{3}^{1} & \zeta_{3}^{2} & 0 & 0 & 0 \\1 & \zeta_{3}^{2} & \zeta_{3}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 1 & 1 \\0 & 0 & 0 & 1 & \zeta_{3}^{2} & \zeta_{3}^{1} \\0 & 0 & 0 & 1 & \zeta_{3}^{1} & \zeta_{3}^{2} \\\end{bmatrix}, \frac{1}{\zeta_{3}^{1}-\zeta_{3}^{2}}\begin{bmatrix}1 & 1 & \zeta_{3}^{2} & 0 & 0 & 0 \\1 & \zeta_{3}^{1} & \zeta_{3}^{1} & 0 & 0 & 0 \\\zeta_{3}^{1} & 1 & \zeta_{3}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 1 & \zeta_{3}^{1} \\0 & 0 & 0 & 1 & \zeta_{3}^{2} & \zeta_{3}^{2} \\0 & 0 & 0 & \zeta_{3}^{2} & 1 & \zeta_{3}^{2} \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$E(36)$${}^{\times 3}$, $B(1,4;2)_2$
Minimal supergroups:$J(E(72))$, $E(216)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $12$ $0$ $220$ $0$ $6440$ $0$ $206892$ $0$ $6820968$
$a_2$ $1$ $1$ $4$ $23$ $236$ $2991$ $40512$ $561765$ $7884648$ $111568451$ $1588492364$ $22729023285$ $326550342164$
$a_3$ $1$ $0$ $6$ $0$ $694$ $0$ $215160$ $0$ $74478838$ $0$ $26666597676$ $0$ $9742494493468$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $4$ $2$ $6$ $12$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $6$ $23$ $14$ $46$ $26$ $98$ $220$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $36$ $236$ $132$ $78$ $526$ $294$ $1202$ $660$ $2770$ $6440$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $400$ $2991$ $222$ $1640$ $904$ $6918$ $3784$ $2084$ $16110$ $8792$ $37644$ $20482$ $88158$ $206892$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $694$ $5176$ $40512$ $2842$ $22050$ $12026$ $94794$ $6560$ $51540$ $28048$ $222360$ $120756$ $65660$ $522414$
$$ $283458$ $1228920$ $666204$ $2893884$ $6820968$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&1&0&0&1&0&1&0&0&0&0&2\\0&2&0&0&0&2&0&0&4&0&2&0&4&8&0\\0&0&3&0&1&0&5&4&0&5&0&11&0&0&28\\0&0&0&4&0&4&0&0&6&0&2&0&16&16&0\\1&0&1&0&4&0&5&9&0&10&0&15&0&0&40\\0&2&0&4&0&18&0&0&32&0&14&0&52&82&0\\0&0&5&0&5&0&25&22&0&17&0&49&0&0&126\\1&0&4&0&9&0&22&36&0&30&0&61&0&0&158\\0&4&0&6&0&32&0&0&62&0&28&0&90&152&0\\1&0&5&0&10&0&17&30&0&37&0&54&0&0&140\\0&2&0&2&0&14&0&0&28&0&14&0&38&70&0\\0&0&11&0&15&0&49&61&0&54&0&128&0&0&320\\0&4&0&16&0&52&0&0&90&0&38&0&172&244&0\\0&8&0&16&0&82&0&0&152&0&70&0&244&404&0\\2&0&28&0&40&0&126&158&0&140&0&320&0&0&822\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&3&4&4&18&25&36&62&37&14&128&172&404&822&450&446&1118&990&332\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$1/9$$0$$1/9$$0$$0$$0$
$a_1=0$$1/9$$1/9$$0$$1/9$$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$