| Name: | $C_2^2$ |
| Order: | $4$ |
| Abelian: | yes |
| Generators: | $\begin{bmatrix}i&0&0&0\\0&i&0&0\\0&0&-i&0\\0&0&0&-i\end{bmatrix}, \begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$1$ |
$0$ |
$8$ |
$0$ |
$80$ |
$0$ |
$896$ |
$0$ |
$10752$ |
$0$ |
$135168$ |
| $a_2$ |
$1$ |
$1$ |
$4$ |
$10$ |
$42$ |
$166$ |
$768$ |
$3620$ |
$17902$ |
$90310$ |
$465096$ |
$2429164$ |
$12843988$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$1$ |
$1$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$4$ |
$4$ |
$8$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$10$ |
$17$ |
$36$ |
$80$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$42$ |
$76$ |
$168$ |
$384$ |
$896$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$166$ |
$354$ |
$808$ |
$1888$ |
$4480$ |
$10752$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$768$ |
$1704$ |
$3984$ |
$9472$ |
$22784$ |
$55296$ |
$135168$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&2&0&0&0&0\\0&1&0&0&2&0&1&0&3&0\\0&0&3&0&0&0&0&4&0&6\\0&0&0&4&0&3&0&4&0&1\\0&2&0&0&5&0&4&0&8&0\\2&0&0&3&0&8&0&2&0&0\\0&1&0&0&4&0&5&0&7&0\\0&0&4&4&0&2&0&15&0&12\\0&3&0&0&8&0&7&0&14&0\\0&0&6&1&0&0&0&12&0&16\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&3&4&5&8&5&15&14&16\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-2$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ |
|---|
| $-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
| $a_1=0$ | $3/4$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
|---|
