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Properties

Label 1.6.N.4.1b
  
Name \(A(1,4)_2\)
Weight $1$
Degree $6$
Real dimension $1$
Components $4$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)_3\)
Component group \(C_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$1$
Components:$4$
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:$C_4$
Order:$4$
Abelian:yes
Generators:$\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{7} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0\\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{5} \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$A(1,2)$
Minimal supergroups:$J_s(A(1,4)_2)$, $A(1,8)_1$, $A(1,12)$, $B(3,2;2)$, $B(1,4;2)_2$${}^{\times 3}$, $A(2,4)$, $J(A(1,4)_2)$, $J_n(A(1,4)_2)$, $B(1,4)_2$, $E(36)$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $6$ $0$ $126$ $0$ $3660$ $0$ $114870$ $0$ $3720276$ $0$ $122763564$
$a_2$ $1$ $3$ $27$ $309$ $3963$ $53073$ $727101$ $10105875$ $141907059$ $2008185033$ $28592728257$ $409122034335$ $5877905052117$
$a_3$ $1$ $0$ $44$ $0$ $11820$ $0$ $3864140$ $0$ $1340497564$ $0$ $479997013104$ $0$ $175364875326036$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $3$ $6$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $27$ $14$ $56$ $126$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $44$ $309$ $170$ $694$ $380$ $1586$ $3660$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $522$ $3963$ $2168$ $1194$ $9126$ $4994$ $21160$ $11550$ $49230$ $114870$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $6828$ $53073$ $3732$ $28940$ $15798$ $123602$ $67340$ $36732$ $288732$ $157146$ $675852$ $367444$ $1584562$ $3720276$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $11820$ $92150$ $727101$ $50232$ $395302$ $215070$ $1703772$ $117068$ $925550$ $503072$ $3999062$ $2170706$ $1178940$ $9398810$
$$ $5098072$ $22114106$ $11987052$ $52081512$ $122763564$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&3&0&6&10&0&8&0&15&0&0&40\\0&6&0&8&0&36&0&0&64&0&28&0&96&158&0\\2&0&22&0&26&0&84&102&0&86&0&198&0&0&500\\0&8&0&22&0&76&0&0&132&0&56&0&224&338&0\\3&0&26&0&41&0&114&146&0&132&0&285&0&0&720\\0&36&0&76&0&320&0&0&576&0&256&0&936&1480&0\\6&0&84&0&114&0&370&444&0&384&0&896&0&0&2272\\10&0&102&0&146&0&444&570&0&498&0&1114&0&0&2844\\0&64&0&132&0&576&0&0&1054&0&472&0&1708&2726&0\\8&0&86&0&132&0&384&498&0&456&0&988&0&0&2528\\0&28&0&56&0&256&0&0&472&0&216&0&764&1224&0\\15&0&198&0&285&0&896&1114&0&988&0&2259&0&0&5760\\0&96&0&224&0&936&0&0&1708&0&764&0&2872&4488&0\\0&158&0&338&0&1480&0&0&2726&0&1224&0&4488&7146&0\\40&0&500&0&720&0&2272&2844&0&2528&0&5760&0&0&14784\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&6&22&22&41&320&370&570&1054&456&216&2259&2872&7146&14784&7706&7891&19640&17460&5178\end{bmatrix}$

Event probabilities

$\mathrm{Pr}[a_i=n]=0$ for $i=1,2,3$ and $n\in\mathbb{Z}$.