| Name: | $S_3\times D_6$ |
| Order: | $72$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{9}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{18}^{5} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{18}^{11} & 0 & 0 & 0 \\0 & 0 & 0& \zeta_{9}^{8} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{18}^{13} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{18}^{7} \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 &0 & -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0& 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0& 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0& 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $J(B(3,2))$, $J(A(2,6))$, $B(3,6)$, $J(A(3,6))$, $J_s(A(3,6))$, $J(B(3,3))$${}^{\times 4}$ |
| Minimal supergroups: | $J(B(6,6))$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$24$ |
$0$ |
$480$ |
$0$ |
$12040$ |
$0$ |
$336672$ |
$0$ |
$10001376$ |
| $a_2$ |
$1$ |
$2$ |
$9$ |
$57$ |
$511$ |
$5577$ |
$67107$ |
$849921$ |
$11105703$ |
$148287705$ |
$2012556259$ |
$27670732641$ |
$384512958111$ |
| $a_3$ |
$1$ |
$0$ |
$9$ |
$0$ |
$1311$ |
$0$ |
$330280$ |
$0$ |
$98790223$ |
$0$ |
$32138874414$ |
$0$ |
$11013624565152$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$3$ |
$11$ |
$24$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$9$ |
$57$ |
$28$ |
$100$ |
$57$ |
$216$ |
$480$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$74$ |
$511$ |
$276$ |
$161$ |
$1061$ |
$606$ |
$2365$ |
$1340$ |
$5315$ |
$12040$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$790$ |
$5577$ |
$447$ |
$3096$ |
$1750$ |
$12380$ |
$6958$ |
$3930$ |
$28092$ |
$15750$ |
$64056$ |
$35805$ |
$146608$ |
$336672$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1311$ |
$9200$ |
$67107$ |
$5172$ |
$37279$ |
$20837$ |
$152913$ |
$11664$ |
$85086$ |
$47440$ |
$351066$ |
$194956$ |
$108502$ |
$808365$ |
| $$ |
$448112$ |
$1865892$ |
$1032570$ |
$4315878$ |
$10001376$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&0&0&1&0&0&4\\0&2&0&1&0&6&0&0&8&0&5&0&9&18&0\\1&0&6&0&2&0&10&16&0&5&0&21&0&0&48\\0&1&0&5&0&10&0&0&13&0&5&0&27&32&0\\0&0&2&0&11&0&14&12&0&19&0&35&0&0&66\\0&6&0&10&0&39&0&0&58&0&29&0&91&139&0\\0&0&10&0&14&0&45&39&0&34&0&90&0&0&200\\4&0&16&0&12&0&39&71&0&40&0&98&0&0&242\\0&8&0&13&0&58&0&0&100&0&45&0&147&234&0\\0&0&5&0&19&0&34&40&0&52&0&87&0&0&204\\0&5&0&5&0&29&0&0&45&0&27&0&66&112&0\\1&0&21&0&35&0&90&98&0&87&0&213&0&0&476\\0&9&0&27&0&91&0&0&147&0&66&0&260&369&0\\0&18&0&32&0&139&0&0&234&0&112&0&369&583&0\\4&0&48&0&66&0&200&242&0&204&0&476&0&0&1151\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&5&11&39&45&71&100&52&27&213&260&583&1151&582&610&1411&1210&369\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $19/36$ | $0$ | $7/36$ | $0$ | $0$ | $1/3$ |
|---|
| $a_1=0$ | $19/36$ | $19/36$ | $0$ | $7/36$ | $0$ | $0$ | $1/3$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $0$ | $1/6$ | $0$ | $0$ | $1/3$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $0$ | $1/6$ | $0$ | $0$ | $1/3$ |
|---|
