| Name: | $C_3\times S_3$ |
| Order: | $18$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{9}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{9}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{9}^{7} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{9}^{8} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{9}^{8} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{9}^{2} \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0& 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $A(1,6)_2$, $B(3,1)$, $A(3,3)$ |
| Minimal supergroups: | $D(3,3)$, $B(3,6)$${}^{\times 2}$, $J(B(3,3))$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$48$ |
$0$ |
$1000$ |
$0$ |
$27440$ |
$0$ |
$847224$ |
$0$ |
$27505632$ |
| $a_2$ |
$1$ |
$2$ |
$12$ |
$98$ |
$1040$ |
$12702$ |
$166908$ |
$2277606$ |
$31727592$ |
$447438806$ |
$6361156052$ |
$90960532566$ |
$1306476953504$ |
| $a_3$ |
$1$ |
$0$ |
$18$ |
$0$ |
$2910$ |
$0$ |
$872040$ |
$0$ |
$298574878$ |
$0$ |
$106702665948$ |
$0$ |
$38971972004416$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$4$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$12$ |
$6$ |
$22$ |
$48$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$18$ |
$98$ |
$56$ |
$204$ |
$114$ |
$444$ |
$1000$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$152$ |
$1040$ |
$580$ |
$330$ |
$2302$ |
$1284$ |
$5202$ |
$2880$ |
$11890$ |
$27440$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1724$ |
$12702$ |
$954$ |
$6996$ |
$3864$ |
$29044$ |
$15960$ |
$8804$ |
$67008$ |
$36740$ |
$155496$ |
$85050$ |
$362348$ |
$847224$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2910$ |
$21680$ |
$166908$ |
$11916$ |
$91166$ |
$49878$ |
$387906$ |
$27314$ |
$211536$ |
$115498$ |
$905340$ |
$492978$ |
$268792$ |
$2119272$ |
| $$ |
$1152592$ |
$4972128$ |
$2701188$ |
$11685996$ |
$27505632$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&4&0&3&0&4&0&0&8\\0&4&0&2&0&12&0&0&16&0&10&0&18&40&0\\1&0&9&0&8&0&23&26&0&17&0&48&0&0&112\\0&2&0&10&0&20&0&0&30&0&10&0&58&76&0\\2&0&8&0&16&0&24&38&0&32&0&70&0&0&156\\0&12&0&20&0&82&0&0&132&0&62&0&210&338&0\\1&0&23&0&24&0&95&100&0&81&0&202&0&0&512\\4&0&26&0&38&0&100&138&0&118&0&250&0&0&636\\0&16&0&30&0&132&0&0&248&0&102&0&378&612&0\\3&0&17&0&32&0&81&118&0&117&0&214&0&0&568\\0&10&0&10&0&62&0&0&102&0&62&0&160&280&0\\4&0&48&0&70&0&202&250&0&214&0&516&0&0&1272\\0&18&0&58&0&210&0&0&378&0&160&0&664&990&0\\0&40&0&76&0&338&0&0&612&0&280&0&990&1602&0\\8&0&112&0&156&0&512&636&0&568&0&1272&0&0&3306\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&9&10&16&82&95&138&248&117&62&516&664&1602&3306&1748&1778&4410&3912&1206\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/9$ | $0$ | $1/9$ | $0$ | $0$ | $0$ |
|---|
| $a_1=0$ | $1/9$ | $1/9$ | $0$ | $1/9$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
