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Properties

Label 1.6.L.48.48d
  
Name \(L(J(O),O)\)
Weight $1$
Degree $6$
Real dimension $6$
Components $48$
Contained in \(\mathrm{USp}(6)\)
Identity component \(\mathrm{U}(1)\times\mathrm{U}(1)_2\)
Component group \(C_2\times S_4\)

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Invariants

Weight:$1$
Degree:$6$
$\mathbb{R}$-dimension:$6$
Components:$48$
Contained in:$\mathrm{USp}(6)$
Rational:yes

Identity component

Name:$\mathrm{U}(1)\times\mathrm{U}(1)_2$
$\mathbb{R}$-dimension:$6$
Description:$\left\{\begin{bmatrix}A&0&0\\0&\alpha I_2&0\\0&0&\bar\alpha I_2\end{bmatrix}: A\in\mathrm{U}(1)\subseteq\mathrm{SU}(2),\ \alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ Symplectic form:$\begin{bmatrix}J_2&0&0\\0&0&I_2\\0&-I_2&0\end{bmatrix},\ J_2:=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$
Hodge circle:$u\mapsto\mathrm{diag}(u,\bar u, u, u,\bar u, \bar u)$

Component group

Name:$C_2\times S_4$
Order:$48$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \frac{1+i}{2} & \frac{1+i}{2} & 0 & 0 \\0 & 0 & \frac{-1+i}{2} & \frac{1-i}{2} & 0 & 0 \\0 & 0 & 0 & 0 & \frac{1-i}{2} & \frac{1-i}{2} \\0 & 0 & 0 & 0 & \frac{-1-i}{2} & \frac{1+i}{2} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{8}^{7} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$L_1(O)$, $L(J(D_3),D_3)$, $L(J(D_4),D_4)$, $L(O_1,T)$, $L(J(T),T)$
Minimal supergroups:$L_2(J(O))$

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $21$ $0$ $330$ $0$ $6440$ $0$ $145152$ $0$ $3642870$
$a_2$ $1$ $2$ $8$ $44$ $329$ $2962$ $29980$ $328757$ $3827507$ $46676330$ $590165678$ $7674277327$ $101983083350$
$a_3$ $1$ $0$ $8$ $0$ $759$ $0$ $131450$ $0$ $31042655$ $0$ $8800156638$ $0$ $2768901182850$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $8$ $3$ $10$ $21$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $8$ $44$ $23$ $75$ $45$ $155$ $330$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $55$ $329$ $186$ $114$ $649$ $389$ $1380$ $820$ $2965$ $6440$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $481$ $2962$ $285$ $1713$ $1013$ $6258$ $3672$ $2170$ $13593$ $7950$ $29750$ $17325$ $65520$ $145152$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $759$ $4641$ $29980$ $2722$ $17272$ $10043$ $65535$ $5856$ $37821$ $21908$ $144947$ $83370$ $48150$ $322360$
$$ $184835$ $720391$ $411768$ $1616790$ $3642870$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&3&0&0&0&1&0&0&2\\0&2&0&1&0&5&0&0&5&0&4&0&5&10&0\\1&0&5&0&2&0&7&10&0&2&0&13&0&0&22\\0&1&0&4&0&7&0&0&7&0&3&0&15&15&0\\0&0&2&0&9&0&9&7&0&10&0&20&0&0&28\\0&5&0&7&0&25&0&0&29&0&17&0&43&62&0\\0&0&7&0&9&0&26&19&0&14&0&43&0&0&80\\3&0&10&0&7&0&19&37&0&16&0&43&0&0&92\\0&5&0&7&0&29&0&0&45&0&20&0&58&90&0\\0&0&2&0&10&0&14&16&0&25&0&33&0&0&72\\0&4&0&3&0&17&0&0&20&0&16&0&27&47&0\\1&0&13&0&20&0&43&43&0&33&0&94&0&0&170\\0&5&0&15&0&43&0&0&58&0&27&0&105&133&0\\0&10&0&15&0&62&0&0&90&0&47&0&133&207&0\\2&0&22&0&28&0&80&92&0&72&0&170&0&0&380\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&4&9&25&26&37&45&25&16&94&105&207&380&187&195&420&350&112\end{bmatrix}$

Event probabilities

$-$$a_2\in\mathbb{Z}$$a_2=-1$$a_2=0$$a_2=1$$a_2=2$$a_2=3$
$-$$1$$1/2$$1/48$$0$$1/8$$1/6$$3/16$
$a_1=0$$1/2$$1/2$$1/48$$0$$1/8$$1/6$$3/16$
$a_3=0$$1/2$$1/2$$1/48$$0$$1/8$$1/6$$3/16$
$a_1=a_3=0$$1/2$$1/2$$1/48$$0$$1/8$$1/6$$3/16$