Properties

Label 1.4.F.6.2c
  
Name \(C_6\)
Weight $1$
Degree $4$
Real dimension $1$
Components $6$
Contained in \(\mathrm{USp}(4)\)
Identity component \(\mathrm{U}(1)_2\)
Component group \(C_6\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$1$
Components:$6$
Contained in:$\mathrm{USp}(4)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)_2$
$\mathbb{R}$-dimension:$1$
Description:$\left\{\begin{bmatrix}\alpha I_2&0\\0&\bar\alpha I_2\end{bmatrix}: \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ Symplectic form:$\begin{bmatrix}0&I_2\\-I_2&0\end{bmatrix}$
Hodge circle:$u\mapsto\mathrm{diag}(u, u,\bar u,\bar u)$

Component group

Name:$C_6$
Order:$6$
Abelian:yes
Generators:$\begin{bmatrix}\zeta_{12}&0&0&0\\0&\zeta_{12}^{11}&0&0\\0&0&\zeta_{12}^{11}&0\\0&0&0&\zeta_{12}\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$C_2$, $C_3$
Minimal supergroups:$D_6$, $D_{6,2}$, $J(C_6)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $36$ $0$ $400$ $0$ $4900$ $0$ $63504$ $0$ $855624$
$a_2$ $1$ $2$ $8$ $32$ $148$ $712$ $3586$ $18524$ $97796$ $524744$ $2854258$ $15701644$ $87215618$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ $2$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ $8$ $16$ $36$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ $32$ $72$ $168$ $400$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ $148$ $344$ $824$ $2000$ $4900$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ $712$ $1712$ $4176$ $10280$ $25480$ $63504$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ $3586$ $8772$ $21684$ $53960$ $134988$ $339192$ $855624$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&2&0&2&0&3&0&2\\0&4&0&0&8&0&8&0&12&0\\1&0&5&6&0&6&0&15&0&10\\2&0&6&12&0&12&0&26&0&16\\0&8&0&0&24&0&24&0&40&0\\2&0&6&12&0&16&0&30&0&20\\0&8&0&0&24&0&28&0&44&0\\3&0&15&26&0&30&0&73&0&50\\0&12&0&0&40&0&44&0&80&0\\2&0&10&16&0&20&0&50&0&42\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&5&12&24&16&28&73&80&42\end{bmatrix}$

Event probabilities

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