↑

Properties

Label 1.6.N.42.1a
  
Name \(J(C(1,7))\)
Weight $1$
Degree $6$
Real dimension $1$
Components $42$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(F_7\)

Downloads

Learn more

Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$42$
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:$F_7$
Order:$42$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{21}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{21}^{16} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{21}^{4} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{21}^{20} & 0& 0 \\0 & 0 & 0 & 0 & \zeta_{21}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{21}^{17} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 &0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$C(1,7)$, $J_s(A(3,1))$, $J(A(1,7))$
Minimal supergroups:$J(E(168))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $15$ $0$ $370$ $0$ $11095$ $0$ $355446$ $0$ $11700150$
$a_2$ $1$ $1$ $5$ $38$ $409$ $5176$ $69734$ $964483$ $13523593$ $191293544$ $2723285740$ $38964766633$ $559803969586$
$a_3$ $1$ $0$ $6$ $0$ $1158$ $0$ $368910$ $0$ $127693398$ $0$ $45714831456$ $0$ $16701442626066$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $1$ $1$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $5$ $2$ $7$ $15$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $6$ $38$ $19$ $73$ $41$ $163$ $370$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $55$ $409$ $219$ $122$ $899$ $496$ $2065$ $1135$ $4775$ $11095$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $674$ $5176$ $372$ $2812$ $1543$ $11903$ $6504$ $3560$ $27714$ $15116$ $64730$ $35252$ $151543$ $355446$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $1158$ $8884$ $69734$ $4857$ $37899$ $20655$ $162850$ $11270$ $88552$ $48193$ $381815$ $207398$ $112741$ $896704$
$$ $486640$ $2108785$ $1143534$ $4964820$ $11700150$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&1&2&0&0&0&1&0&0&3\\0&1&0&1&0&4&0&0&6&0&3&0&9&15&0\\0&0&4&0&2&0&7&12&0&7&0&18&0&0&48\\0&1&0&3&0&7&0&0&13&0&6&0&21&33&0\\0&0&2&0&6&0&12&11&0&14&0&29&0&0&68\\0&4&0&7&0&32&0&0&55&0&24&0&89&140&0\\1&0&7&0&12&0&37&41&0&39&0&86&0&0&216\\2&0&12&0&11&0&41&60&0&43&0&103&0&0&271\\0&6&0&13&0&55&0&0&101&0&44&0&164&259&0\\0&0&7&0&14&0&39&43&0&46&0&96&0&0&240\\0&3&0&6&0&24&0&0&44&0&22&0&73&117&0\\1&0&18&0&29&0&86&103&0&96&0&217&0&0&548\\0&9&0&21&0&89&0&0&164&0&73&0&272&428&0\\0&15&0&33&0&140&0&0&259&0&117&0&428&681&0\\3&0&48&0&68&0&216&271&0&240&0&548&0&0&1411\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&4&3&6&32&37&60&101&46&22&217&272&681&1411&731&767&1866&1667&489\end{bmatrix}$

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$