| Name: | $C_3$ |
| Order: | $3$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}\zeta_6&0&0&0\\0&\zeta_6^5&0&0\\0&0&\zeta_6^5&0\\0&0&0&\zeta_6\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$36$ |
$0$ |
$440$ |
$0$ |
$6020$ |
$0$ |
$86184$ |
$0$ |
$1262184$ |
| $a_2$ |
$1$ |
$2$ |
$8$ |
$34$ |
$164$ |
$842$ |
$4506$ |
$24726$ |
$137892$ |
$777418$ |
$4417178$ |
$25244606$ |
$144936754$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$4$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$8$ |
$16$ |
$36$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$34$ |
$76$ |
$180$ |
$440$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$164$ |
$388$ |
$952$ |
$2380$ |
$6020$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$842$ |
$2068$ |
$5184$ |
$13144$ |
$33572$ |
$86184$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$4506$ |
$11312$ |
$28740$ |
$73540$ |
$189084$ |
$487872$ |
$1262184$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&2&0&2&0&3&0&4\\0&4&0&0&8&0&8&0&16&0\\1&0&5&6&0&8&0&17&0&14\\2&0&6&12&0&14&0&32&0&26\\0&8&0&0&28&0&28&0&56&0\\2&0&8&14&0&20&0&42&0&34\\0&8&0&0&28&0&36&0&64&0\\3&0&17&32&0&42&0&101&0&78\\0&16&0&0&56&0&64&0&124&0\\4&0&14&26&0&34&0&78&0&66\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&5&12&28&20&36&101&124&66\end{bmatrix}$
$\mathrm{Pr}[a_i=n]=0$ for $i=1,2$ and $n\in\mathbb{Z}$.
