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Properties

Label 1.6.N.12.1a
  
Name \(B(3,2;2)\)
Weight $1$
Degree $6$
Real dimension $1$
Components $12$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_3:C_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$12$
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:$C_3:C_4$
Order:$12$
Abelian:no
Generators:$\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & i& 0 & 0 & 0 \\0 & i & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -i \\0 & 0 & 0 & 0 & -i & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$A(1,4)_2$, $A(3,2)$
Minimal supergroups:$J_s(B(3,2;2))$, $B(3,6;2)$, $B(6,2)$, $B(3,4)$, $J(B(3,2;2))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $4$ $0$ $60$ $0$ $1440$ $0$ $41300$ $0$ $1283184$ $0$ $41552280$
$a_2$ $1$ $2$ $14$ $132$ $1514$ $19052$ $252270$ $3441972$ $47851842$ $673544964$ $9562444094$ $136614807452$ $1961147745798$
$a_3$ $1$ $0$ $22$ $0$ $4362$ $0$ $1321580$ $0$ $449729322$ $0$ $160258281012$ $0$ $58478647501164$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $4$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $14$ $8$ $28$ $60$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $22$ $132$ $76$ $288$ $164$ $640$ $1440$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $216$ $1514$ $848$ $480$ $3424$ $1912$ $7812$ $4340$ $17920$ $41300$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $2560$ $19052$ $1428$ $10520$ $5828$ $43912$ $24184$ $13360$ $101652$ $55844$ $236080$ $129388$ $549752$ $1283184$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $4362$ $32740$ $252270$ $18040$ $138080$ $75720$ $587700$ $41600$ $321116$ $175728$ $1372532$ $748732$ $409024$ $3211904$
$$ $1749636$ $7529228$ $4096176$ $17676120$ $41552280$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&3&4&0&3&0&6&0&0&14\\0&4&0&4&0&16&0&0&24&0&12&0&36&56&0\\1&0&11&0&12&0&33&38&0&31&0&74&0&0&174\\0&4&0&10&0&30&0&0&48&0&22&0&80&118&0\\2&0&12&0&18&0&44&54&0&46&0&104&0&0&248\\0&16&0&30&0&122&0&0&204&0&94&0&328&510&0\\3&0&33&0&44&0&135&158&0&133&0&314&0&0&774\\4&0&38&0&54&0&158&200&0&170&0&386&0&0&964\\0&24&0&48&0&204&0&0&364&0&164&0&584&924&0\\3&0&31&0&46&0&133&170&0&157&0&336&0&0&852\\0&12&0&22&0&94&0&0&164&0&78&0&264&418&0\\6&0&74&0&104&0&314&386&0&336&0&778&0&0&1944\\0&36&0&80&0&328&0&0&584&0&264&0&976&1516&0\\0&56&0&118&0&510&0&0&924&0&418&0&1516&2406&0\\14&0&174&0&248&0&774&964&0&852&0&1944&0&0&4962\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&11&10&18&122&135&200&364&157&78&778&976&2406&4962&2584&2636&6558&5836&1734\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$1/6$$0$$1/6$$0$$0$$0$
$a_1=0$$1/6$$1/6$$0$$1/6$$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$