Name: | $S_4$ |
Order: | $24$ |
Abelian: | no |
Generators: | $\begin{bmatrix}0&1&0&0\\-1&0&0&0\\0&0&0&1\\0&0&-1&0\end{bmatrix}, \begin{bmatrix}\frac{i+1}{2}&\frac{i+1}{2}&0&0\\\frac{i-1}{2}&\frac{-i+1}{2}&0&0\\0&0&\frac{-i+1}{2}&\frac{-i+1}{2}\\0&0&\frac{-i-1}{2}&\frac{i+1}{2}\end{bmatrix}, \begin{bmatrix}0&0&0&\zeta_8\\0&0&-\zeta_8^7&0\\0&-\zeta_8^7&0&0\\\zeta_8&0&0&0\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$1$ |
$0$ |
$6$ |
$0$ |
$60$ |
$0$ |
$770$ |
$0$ |
$10836$ |
$0$ |
$158004$ |
$a_2$ |
$1$ |
$1$ |
$3$ |
$8$ |
$30$ |
$126$ |
$617$ |
$3221$ |
$17586$ |
$98090$ |
$554673$ |
$3162501$ |
$18136681$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$1$ |
$1$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$3$ |
$3$ |
$6$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$8$ |
$12$ |
$26$ |
$60$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$30$ |
$55$ |
$128$ |
$310$ |
$770$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$126$ |
$276$ |
$672$ |
$1676$ |
$4242$ |
$10836$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$617$ |
$1462$ |
$3658$ |
$9282$ |
$23758$ |
$61152$ |
$158004$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&1&0&0&0&1\\0&1&0&0&1&0&1&0&2&0\\0&0&2&0&0&1&0&2&0&3\\0&0&0&3&0&1&0&4&0&2\\0&1&0&0&4&0&3&0&7&0\\1&0&1&1&0&5&0&4&0&4\\0&1&0&0&3&0&5&0&8&0\\0&0&2&4&0&4&0&14&0&10\\0&2&0&0&7&0&8&0&16&0\\1&0&3&2&0&4&0&10&0&12\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&3&4&5&5&14&16&12\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-2$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ |
---|
$-$ | $1$ | $1/2$ | $0$ | $0$ | $1/4$ | $0$ | $1/4$ |
---|
$a_1=0$ | $5/8$ | $1/2$ | $0$ | $0$ | $1/4$ | $0$ | $1/4$ |
---|