| Name: | $D_6$ |
| Order: | $12$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0& 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $A(2,2)$, $B(3,1)$${}^{\times 2}$, $A(3,2)$ |
| Minimal supergroups: | $J(B(3,2))$, $J_s(B(3,2))$, $B(3,6)$, $B(6,2)$, $B(3,4)$, $B(O,1)$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$60$ |
$0$ |
$1440$ |
$0$ |
$41300$ |
$0$ |
$1283184$ |
$0$ |
$41552280$ |
| $a_2$ |
$1$ |
$2$ |
$14$ |
$132$ |
$1514$ |
$19052$ |
$252270$ |
$3441972$ |
$47851842$ |
$673544964$ |
$9562444094$ |
$136614807452$ |
$1961147745798$ |
| $a_3$ |
$1$ |
$0$ |
$22$ |
$0$ |
$4362$ |
$0$ |
$1321580$ |
$0$ |
$449729322$ |
$0$ |
$160258281012$ |
$0$ |
$58478647501164$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$4$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$14$ |
$6$ |
$26$ |
$60$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$22$ |
$132$ |
$74$ |
$284$ |
$156$ |
$632$ |
$1440$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$210$ |
$1514$ |
$840$ |
$480$ |
$3412$ |
$1904$ |
$7796$ |
$4310$ |
$17890$ |
$41300$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$2548$ |
$19052$ |
$1404$ |
$10500$ |
$5804$ |
$43880$ |
$24152$ |
$13360$ |
$101604$ |
$55814$ |
$236020$ |
$129276$ |
$549640$ |
$1283184$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$4362$ |
$32702$ |
$252270$ |
$18016$ |
$138022$ |
$75672$ |
$587610$ |
$41508$ |
$321036$ |
$175636$ |
$1372404$ |
$748610$ |
$409024$ |
$3211722$ |
| $$ |
$1749524$ |
$7529004$ |
$4095756$ |
$17675700$ |
$41552280$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&6&0&5&0&6&0&0&14\\0&4&0&2&0&16&0&0&26&0&14&0&30&60&0\\1&0&11&0&10&0&35&38&0&25&0&74&0&0&174\\0&2&0&14&0&30&0&0&42&0&16&0&96&112&0\\2&0&10&0&22&0&38&58&0&58&0&104&0&0&248\\0&16&0&30&0&122&0&0&204&0&94&0&328&510&0\\1&0&35&0&38&0&147&146&0&113&0&318&0&0&774\\6&0&38&0&58&0&146&212&0&188&0&382&0&0&964\\0&26&0&42&0&204&0&0&372&0&172&0&562&934&0\\5&0&25&0&58&0&113&188&0&193&0&332&0&0&852\\0&14&0&16&0&94&0&0&172&0&86&0&244&426&0\\6&0&74&0&104&0&318&382&0&332&0&778&0&0&1944\\0&30&0&96&0&328&0&0&562&0&244&0&1032&1492&0\\0&60&0&112&0&510&0&0&934&0&426&0&1492&2414&0\\14&0&174&0&248&0&774&964&0&852&0&1944&0&0&4962\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&11&14&22&122&147&212&372&193&86&778&1032&2414&4962&2660&2656&6678&5896&1906\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/6$ | $0$ | $1/6$ | $0$ | $0$ | $0$ |
|---|
| $a_1=0$ | $1/6$ | $1/6$ | $0$ | $1/6$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
