| Name: | $C_3:S_3$ |
| Order: | $18$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{9}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{9}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{9}^{7} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{9}^{8} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{9}^{8} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{9}^{2} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0& 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $J(A(1,3))$${}^{\times 3}$, $J(A(3,1))$, $A(3,3)$ |
| Minimal supergroups: | $J(A(3,6))$${}^{\times 2}$, $J(C(3,3))$, $J(B(3,3))$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$45$ |
$0$ |
$990$ |
$0$ |
$27405$ |
$0$ |
$847098$ |
$0$ |
$27505170$ |
| $a_2$ |
$1$ |
$3$ |
$15$ |
$108$ |
$1071$ |
$12798$ |
$167202$ |
$2278503$ |
$31730319$ |
$447447078$ |
$6361181100$ |
$90960608313$ |
$1306477182330$ |
| $a_3$ |
$1$ |
$0$ |
$16$ |
$0$ |
$2892$ |
$0$ |
$871840$ |
$0$ |
$298572428$ |
$0$ |
$106702634196$ |
$0$ |
$38971971577308$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$15$ |
$6$ |
$21$ |
$45$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$16$ |
$108$ |
$54$ |
$201$ |
$114$ |
$441$ |
$990$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$150$ |
$1071$ |
$576$ |
$324$ |
$2295$ |
$1278$ |
$5193$ |
$2880$ |
$11880$ |
$27405$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1716$ |
$12798$ |
$954$ |
$6984$ |
$3858$ |
$29025$ |
$15948$ |
$8784$ |
$66987$ |
$36720$ |
$155466$ |
$85050$ |
$362313$ |
$847098$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2892$ |
$21660$ |
$167202$ |
$11898$ |
$91134$ |
$49854$ |
$387855$ |
$27315$ |
$211500$ |
$115479$ |
$905283$ |
$492939$ |
$268722$ |
$2119203$ |
| $$ |
$1152522$ |
$4972023$ |
$2701188$ |
$11685870$ |
$27505170$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&1&6&0&0&0&3&0&0&8\\0&3&0&3&0&12&0&0&15&0&9&0&21&39&0\\2&0&10&0&6&0&20&33&0&15&0&45&0&0&112\\0&3&0&7&0&20&0&0&33&0&13&0&51&79&0\\0&0&6&0&18&0&30&27&0&33&0&75&0&0&156\\0&12&0&20&0&82&0&0&132&0&62&0&210&338&0\\1&0&20&0&30&0&91&99&0&93&0&204&0&0&512\\6&0&33&0&27&0&99&150&0&99&0&243&0&0&636\\0&15&0&33&0&132&0&0&243&0&99&0&387&609&0\\0&0&15&0&33&0&93&99&0&110&0&222&0&0&568\\0&9&0&13&0&62&0&0&99&0&58&0&168&277&0\\3&0&45&0&75&0&204&243&0&222&0&519&0&0&1272\\0&21&0&51&0&210&0&0&387&0&168&0&642&999&0\\0&39&0&79&0&338&0&0&609&0&277&0&999&1597&0\\8&0&112&0&156&0&512&636&0&568&0&1272&0&0&3306\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&10&7&18&82&91&150&243&110&58&519&642&1597&3306&1719&1809&4366&3906&1145\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $11/18$ | $0$ | $1/9$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=0$ | $11/18$ | $11/18$ | $0$ | $1/9$ | $0$ | $0$ | $1/2$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
