Properties

Label 1.4.1.12.5a
  
Name \(J(C_6)\)
Weight 1
Degree 4
Real dimension 1
Components 12
Contained in \(\mathrm{USp}(4)\)
Identity Component \(\mathrm{U}(1)_2\)
Component group \(C_2\times C_6\)

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Invariants

Weight:$1$
Degree:$4$
$\mathbb{R}$-dimension:$1$
Components:$12$
Contained in:$\mathrm{USp}(4)$
Rational:$\mathrm{True}$

Identity Component

Name:$\mathrm{U}(1)_2$
Index:$12$
$\mathbb{R}$-dimension:$1$
Description:$\left\{\begin{bmatrix}\mathrm{diag}_2(\alpha)&0\\0&\mathrm{diag}_2(\bar\alpha)\end{bmatrix}: \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\},\ \mathrm{diag}_2(\alpha):=\begin{bmatrix}\alpha&0\\0&\alpha\end{bmatrix}$

Component Group

Name:$C_2\times C_6$
Order:$12$
Abelian:$\mathrm{True}$

Subgroups and Supergroups

Maximal Subgroups:$J(C_2)$, $C_6$, $J(C_3)$, $C_{6,1}$
Minimal Supergroups:$J(D_6)$

Moment Statistics

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $18$ $0$ $200$ $0$ $2450$ $0$ $31752$ $0$ $427812$
$a_2$ $1$ $1$ $5$ $16$ $77$ $356$ $1804$ $9262$ $48941$ $262372$ $1427300$ $7850822$ $43608492$

Event Probabilities

$\mathrm{P}[a_1=0]=\frac{7}{12}$
$\mathrm{P}[a_2=-2]=\frac{1}{12}$ $\mathrm{P}[a_2=-1]=\frac{2}{12}$ $\mathrm{P}[a_2=1]=\frac{2}{12}$ $\mathrm{P}[a_2=2]=\frac{1}{12}$