| Name: | $C_6$ |
| Order: | $6$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}\zeta_{12}&0&0&0\\0&\zeta_{12}^{11}&0&0\\0&0&\zeta_{12}^{11}&0\\0&0&0&\zeta_{12}\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$36$ |
$0$ |
$400$ |
$0$ |
$4900$ |
$0$ |
$63504$ |
$0$ |
$855624$ |
| $a_2$ |
$1$ |
$2$ |
$8$ |
$32$ |
$148$ |
$712$ |
$3586$ |
$18524$ |
$97796$ |
$524744$ |
$2854258$ |
$15701644$ |
$87215618$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$4$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$8$ |
$16$ |
$36$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$32$ |
$72$ |
$168$ |
$400$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$148$ |
$344$ |
$824$ |
$2000$ |
$4900$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$712$ |
$1712$ |
$4176$ |
$10280$ |
$25480$ |
$63504$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$3586$ |
$8772$ |
$21684$ |
$53960$ |
$134988$ |
$339192$ |
$855624$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&2&0&2&0&3&0&2\\0&4&0&0&8&0&8&0&12&0\\1&0&5&6&0&6&0&15&0&10\\2&0&6&12&0&12&0&26&0&16\\0&8&0&0&24&0&24&0&40&0\\2&0&6&12&0&16&0&30&0&20\\0&8&0&0&24&0&28&0&44&0\\3&0&15&26&0&30&0&73&0&50\\0&12&0&0&40&0&44&0&80&0\\2&0&10&16&0&20&0&50&0&42\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&5&12&24&16&28&73&80&42\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-2$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ |
|---|
| $-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=0$ | $1/6$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
