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Properties

Label 1.6.L.48.51a
  
Name \(L_2(J(D_6))\)
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 \(C_2^2\times D_6\)

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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:$C_2^2\times D_6$
Order:$48$
Abelian:no
Generators:$\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 &0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{12}^{11} & 0 & 0 \\0 &0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{1} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 1 & 0 & 0 & 0 & 0 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\\end{bmatrix}$

Subgroups and supergroups

Maximal subgroups:$L_2(J(D_3))$${}^{\times 2}$, $L(J(D_6),D_{6,1})$${}^{\times 2}$, $L_2(D_6)$, $L_2(D_{6,2})$, $L_2(J(C_6))$, $L_2(D_{6,1})$${}^{\times 2}$, $L_1(J(D_6))$, $L(J(D_6),D_6)$, $L(J(D_6),J(C_6))$, $L_2(J(D_2))$, $L(J(D_6),D_{6,2})$, $L(J(D_6),J(D_3))$${}^{\times 2}$
Minimal supergroups:

Moment sequences

$x$ $\mathrm{E}[x^{0}]$ $\mathrm{E}[x^{1}]$ $\mathrm{E}[x^{2}]$ $\mathrm{E}[x^{3}]$ $\mathrm{E}[x^{4}]$ $\mathrm{E}[x^{5}]$ $\mathrm{E}[x^{6}]$ $\mathrm{E}[x^{7}]$ $\mathrm{E}[x^{8}]$ $\mathrm{E}[x^{9}]$ $\mathrm{E}[x^{10}]$ $\mathrm{E}[x^{11}]$ $\mathrm{E}[x^{12}]$
$a_1$ $1$ $0$ $2$ $0$ $18$ $0$ $290$ $0$ $6230$ $0$ $154602$ $0$ $4169550$
$a_2$ $1$ $2$ $8$ $41$ $302$ $2847$ $30876$ $360971$ $4410720$ $55490405$ $713074433$ $9315788946$ $123354482653$
$a_3$ $1$ $0$ $8$ $0$ $708$ $0$ $142110$ $0$ $36859480$ $0$ $10667229678$ $0$ $3296616909894$

Moment simplex

$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ $2$ $2$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ $8$ $3$ $9$ $18$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ $8$ $41$ $21$ $66$ $39$ $135$ $290$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ $50$ $302$ $167$ $102$ $593$ $351$ $1282$ $750$ $2810$ $6230$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ $442$ $2847$ $258$ $1617$ $942$ $6128$ $3539$ $2062$ $13617$ $7840$ $30470$ $17465$ $68495$ $154602$
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ $708$ $4528$ $30876$ $2616$ $17509$ $10024$ $68966$ $5754$ $39219$ $22378$ $155709$ $88320$ $50280$ $352840$
$$ $199675$ $801829$ $452718$ $1826496$ $4169550$

Moment matrix

$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&3&0&0&0&0&0&0&2\\0&2&0&1&0&4&0&0&5&0&3&0&4&9&0\\1&0&5&0&1&0&6&9&0&2&0&11&0&0&22\\0&1&0&4&0&6&0&0&6&0&2&0&15&14&0\\0&0&1&0&8&0&8&6&0&11&0&19&0&0&30\\0&4&0&6&0&22&0&0&28&0&16&0&44&64&0\\0&0&6&0&8&0&25&17&0&13&0&44&0&0&86\\3&0&9&0&6&0&17&38&0&17&0&44&0&0&102\\0&5&0&6&0&28&0&0&46&0&22&0&61&100&0\\0&0&2&0&11&0&13&17&0&29&0&37&0&0&82\\0&3&0&2&0&16&0&0&22&0&16&0&30&51&0\\0&0&11&0&19&0&44&44&0&37&0&102&0&0&198\\0&4&0&15&0&44&0&0&61&0&30&0&120&152&0\\0&9&0&14&0&64&0&0&100&0&51&0&152&242&0\\2&0&22&0&30&0&86&102&0&82&0&198&0&0&450\end{bmatrix}$

$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&5&4&8&22&25&38&46&29&16&102&120&242&450&228&236&519&431&140\end{bmatrix}$

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$$1/24$$1/12$$0$$1/12$$7/24$
$a_1=0$$19/48$$1/4$$1/48$$1/24$$0$$1/24$$7/48$
$a_3=0$$19/48$$1/4$$1/48$$1/24$$0$$1/24$$7/48$
$a_1=a_3=0$$19/48$$1/4$$1/48$$1/24$$0$$1/24$$7/48$