| Name: | $C_3$ |
| Order: | $3$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}\zeta_6&0&0&0\\0&\zeta_6&0&0\\0&0&\zeta_6^5&0\\0&0&0&\zeta_6^5\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$12$ |
$0$ |
$110$ |
$0$ |
$1204$ |
$0$ |
$14364$ |
$0$ |
$180312$ |
| $a_2$ |
$1$ |
$1$ |
$4$ |
$13$ |
$52$ |
$222$ |
$1014$ |
$4839$ |
$23860$ |
$120526$ |
$620278$ |
$3239788$ |
$17127202$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$1$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$4$ |
$6$ |
$12$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$13$ |
$24$ |
$50$ |
$110$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$52$ |
$104$ |
$228$ |
$518$ |
$1204$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$222$ |
$478$ |
$1086$ |
$2530$ |
$5992$ |
$14364$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$1014$ |
$2286$ |
$5332$ |
$12658$ |
$30420$ |
$73788$ |
$180312$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&1&0&1&0&0&0&2\\0&2&0&0&2&0&2&0&4&0\\0&0&3&1&0&2&0&4&0&3\\1&0&1&4&0&3&0&6&0&5\\0&2&0&0&8&0&4&0&10&0\\1&0&2&3&0&6&0&7&0&6\\0&2&0&0&4&0&8&0&10&0\\0&0&4&6&0&7&0&17&0&10\\0&4&0&0&10&0&10&0&20&0\\2&0&3&5&0&6&0&10&0&12\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&3&4&8&6&8&17&20&12\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2$ and $n\in\mathbb{Z}$.
