| Name: | $C_6$ |
| Order: | $6$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}\zeta_{12}&0&0&0\\0&\zeta_{12}&0&0\\0&0&\zeta_{12}^{11}&0\\0&0&0&\zeta_{12}^{11}\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$12$ |
$0$ |
$100$ |
$0$ |
$980$ |
$0$ |
$10584$ |
$0$ |
$122232$ |
| $a_2$ |
$1$ |
$1$ |
$4$ |
$11$ |
$44$ |
$172$ |
$754$ |
$3397$ |
$16020$ |
$77516$ |
$384578$ |
$1944626$ |
$9997970$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$1$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$4$ |
$6$ |
$12$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$11$ |
$22$ |
$46$ |
$100$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$44$ |
$88$ |
$192$ |
$430$ |
$980$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$172$ |
$376$ |
$846$ |
$1940$ |
$4508$ |
$10584$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$754$ |
$1686$ |
$3884$ |
$9074$ |
$21420$ |
$50988$ |
$122232$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&1&0&1&0&0&0&0\\0&2&0&0&2&0&2&0&2&0\\0&0&3&1&0&0&0&4&0&3\\1&0&1&4&0&3&0&4&0&1\\0&2&0&0&6&0&4&0&6&0\\1&0&0&3&0&6&0&3&0&0\\0&2&0&0&4&0&6&0&6&0\\0&0&4&4&0&3&0&13&0&8\\0&2&0&0&6&0&6&0&12&0\\0&0&3&1&0&0&0&8&0&12\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&3&4&6&6&6&13&12&12\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ |
|---|
| $-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=0$ | $1/6$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
