Name: | $A_4$ |
Order: | $12$ |
Abelian: | no |
Generators: | $\begin{bmatrix}\frac{i+1}{2}&\frac{i+1}{2}&0&0\\\frac{i-1}{2}&\frac{-i+1}{2}&0&0\\0&0&\frac{-i+1}{2}&\frac{-i+1}{2}\\0&0&\frac{-i-1}{2}&\frac{i+1}{2}\end{bmatrix}, \begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$12$ |
$0$ |
$120$ |
$0$ |
$1540$ |
$0$ |
$21672$ |
$0$ |
$316008$ |
$a_2$ |
$1$ |
$1$ |
$4$ |
$12$ |
$52$ |
$236$ |
$1202$ |
$6378$ |
$35044$ |
$195924$ |
$1108834$ |
$6323978$ |
$36271314$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$1$ |
$2$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$4$ |
$6$ |
$12$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$12$ |
$24$ |
$52$ |
$120$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$52$ |
$110$ |
$256$ |
$620$ |
$1540$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$236$ |
$552$ |
$1344$ |
$3352$ |
$8484$ |
$21672$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$1202$ |
$2924$ |
$7316$ |
$18564$ |
$47516$ |
$122304$ |
$316008$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&1&0&1&0&0&0&1\\0&2&0&0&2&0&2&0&4&0\\0&0&3&1&0&1&0&5&0&5\\1&0&1&4&0&4&0&7&0&6\\0&2&0&0&8&0&6&0&14&0\\1&0&1&4&0&8&0&9&0&6\\0&2&0&0&6&0&10&0&16&0\\0&0&5&7&0&9&0&27&0&21\\0&4&0&0&14&0&16&0&32&0\\1&0&5&6&0&6&0&21&0&22\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&3&4&8&8&10&27&32&22\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-2$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ |
---|
$-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=0$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|