| Name: | $D_6$ |
| Order: | $12$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0& 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $J(A(1,2))$, $A(3,2)$, $J(A(3,1))$${}^{\times 2}$ |
| Minimal supergroups: | $J(A(3,4))$${}^{\times 2}$, $J(B(3,2;2))$, $J(A(6,2))$${}^{\times 6}$, $J(A(3,6))$, $J_s(B(3,2))$, $J(B(3,2))$, $J_s(B(T,1))$, $J_s(B(3,2;2))$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$57$ |
$0$ |
$1430$ |
$0$ |
$41265$ |
$0$ |
$1283058$ |
$0$ |
$41551818$ |
| $a_2$ |
$1$ |
$3$ |
$17$ |
$142$ |
$1545$ |
$19148$ |
$252564$ |
$3442869$ |
$47854569$ |
$673553236$ |
$9562469142$ |
$136614883199$ |
$1961147974624$ |
| $a_3$ |
$1$ |
$0$ |
$20$ |
$0$ |
$4340$ |
$0$ |
$1321290$ |
$0$ |
$449725276$ |
$0$ |
$160258222860$ |
$0$ |
$58478646649346$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$17$ |
$7$ |
$26$ |
$57$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$20$ |
$142$ |
$73$ |
$283$ |
$160$ |
$633$ |
$1430$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$211$ |
$1545$ |
$840$ |
$473$ |
$3411$ |
$1901$ |
$7794$ |
$4325$ |
$17895$ |
$41265$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$2546$ |
$19148$ |
$1416$ |
$10498$ |
$5809$ |
$43877$ |
$24154$ |
$13334$ |
$101604$ |
$55803$ |
$236014$ |
$129332$ |
$549661$ |
$1283058$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$4340$ |
$32701$ |
$252564$ |
$18006$ |
$138019$ |
$75667$ |
$587604$ |
$41554$ |
$321033$ |
$175656$ |
$1372401$ |
$748619$ |
$408926$ |
$3211725$ |
| $$ |
$1749482$ |
$7528983$ |
$4095966$ |
$17675784$ |
$41551818$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&2&7&0&1&0&5&0&0&14\\0&3&0&4&0&16&0&0&24&0&12&0&36&57&0\\2&0&12&0&9&0&31&45&0&26&0&71&0&0&174\\0&4&0&9&0&30&0&0&48&0&22&0&80&119&0\\0&0&9&0&22&0&47&45&0&52&0&109&0&0&248\\0&16&0&30&0&122&0&0&204&0&94&0&328&510&0\\2&0&31&0&47&0&136&152&0&137&0&318&0&0&774\\7&0&45&0&45&0&152&217&0&159&0&377&0&0&964\\0&24&0&48&0&204&0&0&363&0&164&0&584&925&0\\1&0&26&0&52&0&137&159&0&163&0&342&0&0&852\\0&12&0&22&0&94&0&0&164&0&78&0&264&418&0\\5&0&71&0&109&0&318&377&0&342&0&781&0&0&1944\\0&36&0&80&0&328&0&0&584&0&264&0&976&1516&0\\0&57&0&119&0&510&0&0&925&0&418&0&1516&2403&0\\14&0&174&0&248&0&774&964&0&852&0&1944&0&0&4962\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&12&9&22&122&136&217&363&163&78&781&976&2403&4962&2581&2675&6558&5850&1731\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $2/3$ | $0$ | $1/6$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=0$ | $2/3$ | $2/3$ | $0$ | $1/6$ | $0$ | $0$ | $1/2$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
