| Name: | $C_3:C_{12}$ |
| Order: | $36$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{9}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{18}^{5} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{18}^{11} & 0 & 0 & 0 \\0 & 0 & 0& \zeta_{9}^{8} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{18}^{13} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{18}^{7} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & i & 0 & 0 & 0 \\0 & i & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -i \\0 & 0 & 0 & 0 & -i & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $A(3,6)$, $A(1,12)$, $B(3,2;2)$ |
| Minimal supergroups: | $J(B(3,6;2))$, $B(6,6)$, $J_s(B(3,6;2))$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$48$ |
$0$ |
$960$ |
$0$ |
$24080$ |
$0$ |
$673344$ |
$0$ |
$20002752$ |
| $a_2$ |
$1$ |
$2$ |
$12$ |
$96$ |
$968$ |
$10992$ |
$133728$ |
$1698384$ |
$22207032$ |
$296562288$ |
$4025073152$ |
$55341347184$ |
$769025561928$ |
| $a_3$ |
$1$ |
$0$ |
$18$ |
$0$ |
$2622$ |
$0$ |
$660560$ |
$0$ |
$197580446$ |
$0$ |
$64277748828$ |
$0$ |
$22027249130304$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$4$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$12$ |
$6$ |
$22$ |
$48$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$18$ |
$96$ |
$56$ |
$200$ |
$114$ |
$432$ |
$960$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$148$ |
$968$ |
$552$ |
$322$ |
$2122$ |
$1212$ |
$4730$ |
$2680$ |
$10630$ |
$24080$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1580$ |
$10992$ |
$894$ |
$6192$ |
$3500$ |
$24760$ |
$13916$ |
$7860$ |
$56184$ |
$31500$ |
$128112$ |
$71610$ |
$293216$ |
$673344$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2622$ |
$18400$ |
$133728$ |
$10344$ |
$74558$ |
$41674$ |
$305826$ |
$23330$ |
$170172$ |
$94882$ |
$702132$ |
$389914$ |
$217004$ |
$1616732$ |
| $$ |
$896224$ |
$3731784$ |
$2065140$ |
$8631756$ |
$20002752$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&4&0&3&0&4&0&0&8\\0&4&0&2&0&12&0&0&16&0&10&0&18&36&0\\1&0&9&0&8&0&23&24&0&15&0&46&0&0&96\\0&2&0&10&0&20&0&0&26&0&10&0&54&64&0\\2&0&8&0&16&0&24&36&0&30&0&64&0&0&132\\0&12&0&20&0&78&0&0&116&0&58&0&182&278&0\\1&0&23&0&24&0&87&86&0&63&0&176&0&0&400\\4&0&24&0&36&0&86&118&0&96&0&208&0&0&484\\0&16&0&26&0&116&0&0&200&0&90&0&294&468&0\\3&0&15&0&30&0&63&96&0&93&0&166&0&0&408\\0&10&0&10&0&58&0&0&90&0&54&0&132&224&0\\4&0&46&0&64&0&176&208&0&166&0&420&0&0&952\\0&18&0&54&0&182&0&0&294&0&132&0&520&738&0\\0&36&0&64&0&278&0&0&468&0&224&0&738&1166&0\\8&0&96&0&132&0&400&484&0&408&0&952&0&0&2302\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&9&10&16&78&87&118&200&93&54&420&520&1166&2302&1160&1166&2818&2396&722\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/18$ | $0$ | $1/18$ | $0$ | $0$ | $0$ |
|---|
| $a_1=0$ | $1/18$ | $1/18$ | $0$ | $1/18$ | $0$ | $0$ | $0$ |
|---|
| $a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
