| Name: | $C_3\times S_3$ |
| Order: | $18$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{3}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{2} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0 & 0& 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 &0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 &-1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $J(A(3,1))$, $C(3,1)$, $J_s(A(3,1))$ |
| Minimal supergroups: | $J(C(6,2))$, $J_s(C(6,2))$, $J(C(3,3))$, $J(D(3,1))$${}^{\times 2}$, $J_s(C(3,3))$${}^{\times 3}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$1$ |
$0$ |
$27$ |
$0$ |
$810$ |
$0$ |
$25515$ |
$0$ |
$826686$ |
$0$ |
$27280638$ |
| $a_2$ |
$1$ |
$1$ |
$7$ |
$72$ |
$891$ |
$11826$ |
$161676$ |
$2246049$ |
$31535811$ |
$446266098$ |
$6353947962$ |
$90916032879$ |
$1306201198968$ |
| $a_3$ |
$1$ |
$0$ |
$10$ |
$0$ |
$2622$ |
$0$ |
$858610$ |
$0$ |
$297887030$ |
$0$ |
$106665983640$ |
$0$ |
$38969972011146$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$1$ |
$1$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$7$ |
$3$ |
$12$ |
$27$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$10$ |
$72$ |
$37$ |
$153$ |
$84$ |
$351$ |
$810$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$115$ |
$891$ |
$480$ |
$263$ |
$2025$ |
$1107$ |
$4698$ |
$2565$ |
$10935$ |
$25515$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1514$ |
$11826$ |
$828$ |
$6426$ |
$3507$ |
$27459$ |
$14958$ |
$8154$ |
$64152$ |
$34911$ |
$150174$ |
$81648$ |
$352107$ |
$826686$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2622$ |
$20469$ |
$161676$ |
$11154$ |
$87831$ |
$47781$ |
$378594$ |
$26010$ |
$205659$ |
$111780$ |
$888651$ |
$482355$ |
$261954$ |
$2088585$ |
| $$ |
$1132866$ |
$4914189$ |
$2663766$ |
$11573604$ |
$27280638$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&2&3&0&1&0&3&0&0&8\\0&1&0&2&0&8&0&0&14&0&6&0&22&35&0\\0&0&6&0&5&0&17&25&0&18&0&43&0&0&112\\0&2&0&5&0&16&0&0&30&0&14&0&48&77&0\\0&0&5&0&10&0&27&29&0&30&0&65&0&0&160\\0&8&0&16&0&72&0&0&128&0&56&0&208&328&0\\2&0&17&0&27&0&82&98&0&89&0&200&0&0&504\\3&0&25&0&29&0&98&131&0&105&0&245&0&0&632\\0&14&0&30&0&128&0&0&233&0&104&0&382&605&0\\1&0&18&0&30&0&89&105&0&101&0&222&0&0&560\\0&6&0&14&0&56&0&0&104&0&48&0&172&272&0\\3&0&43&0&65&0&200&245&0&222&0&503&0&0&1280\\0&22&0&48&0&208&0&0&382&0&172&0&632&1000&0\\0&35&0&77&0&328&0&0&605&0&272&0&1000&1587&0\\8&0&112&0&160&0&504&632&0&560&0&1280&0&0&3288\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&6&5&10&72&82&131&233&101&48&503&632&1587&3288&1703&1767&4352&3878&1135\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $17/18$ | $0$ | $7/9$ | $0$ | $0$ | $1/6$ |
|---|
| $a_1=0$ | $17/18$ | $17/18$ | $0$ | $7/9$ | $0$ | $0$ | $1/6$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $0$ | $1/3$ | $0$ | $0$ | $1/6$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $0$ | $1/3$ | $0$ | $0$ | $1/6$ |
|---|
