Properties

Label 1.4.1.12.4c
  
Name \(D_{6,1}\)
Weight 1
Degree 4
Real dimension 1
Components 12
Contained in \(\mathrm{USp}(4)\)
Identity Component \(\mathrm{U}(1)_2\)
Component group \(D_6\)

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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:$D_6$
Order:$12$
Abelian:$\mathrm{False}$

Subgroups and Supergroups

Maximal Subgroups:$D_{2,1}$, $D_3$, $D_{3,2}$, $C_{6,1}$
Minimal Supergroups:$J(D_6)$

Moment Statistics

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $1$ $0$ $9$ $0$ $110$ $0$ $1505$ $0$ $21546$ $0$ $315546$
$a_2$ $1$ $1$ $4$ $11$ $48$ $221$ $1153$ $6224$ $34576$ $194525$ $1104699$ $6311834$ $36235785$

Event Probabilities

$\mathrm{P}[a_1=0]=\frac{9}{12}$
$\mathrm{P}[a_2=-1]=\frac{2}{12}$ $\mathrm{P}[a_2=2]=\frac{4}{12}$