| Name: | $D_4$ |
| Order: | $8$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}\zeta_8&0&0&0\\0&\zeta_8&0&0\\0&0&\zeta_8^7&0\\0&0&0&\zeta_8^7\end{bmatrix}, \begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$1$ |
$0$ |
$6$ |
$0$ |
$50$ |
$0$ |
$504$ |
$0$ |
$5712$ |
$0$ |
$69696$ |
| $a_2$ |
$1$ |
$1$ |
$3$ |
$7$ |
$26$ |
$96$ |
$422$ |
$1926$ |
$9326$ |
$46402$ |
$236810$ |
$1229394$ |
$6474228$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$1$ |
$1$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$3$ |
$3$ |
$6$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$7$ |
$11$ |
$23$ |
$50$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$26$ |
$45$ |
$98$ |
$220$ |
$504$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$96$ |
$198$ |
$446$ |
$1028$ |
$2408$ |
$5712$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$422$ |
$918$ |
$2124$ |
$5000$ |
$11920$ |
$28704$ |
$69696$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&1&0&0&0&0\\0&1&0&0&1&0&1&0&1&0\\0&0&2&0&0&0&0&2&0&3\\0&0&0&3&0&1&0&2&0&0\\0&1&0&0&3&0&2&0&4&0\\1&0&0&1&0&5&0&1&0&0\\0&1&0&0&2&0&3&0&3&0\\0&0&2&2&0&1&0&8&0&6\\0&1&0&0&4&0&3&0&8&0\\0&0&3&0&0&0&0&6&0&9\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&3&3&5&3&8&8&9\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-2$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ |
|---|
| $-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=0$ | $5/8$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
