Name: | $S_3$ |
Order: | $6$ |
Abelian: | no |
Generators: | $\begin{bmatrix}\zeta_6&0&0&0\\0&\zeta_6&0&0\\0&0&\zeta_6^5&0\\0&0&0&\zeta_6^5\end{bmatrix}, \begin{bmatrix}0&0&0&1\\0&0&-1&0\\0&-1&0&0\\1&0&0&0\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$1$ |
$0$ |
$6$ |
$0$ |
$55$ |
$0$ |
$602$ |
$0$ |
$7182$ |
$0$ |
$90156$ |
$a_2$ |
$1$ |
$1$ |
$3$ |
$8$ |
$29$ |
$116$ |
$517$ |
$2437$ |
$11965$ |
$60326$ |
$310265$ |
$1620125$ |
$8564063$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$1$ |
$1$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$3$ |
$3$ |
$6$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$8$ |
$12$ |
$25$ |
$55$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$29$ |
$52$ |
$114$ |
$259$ |
$602$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$116$ |
$239$ |
$543$ |
$1265$ |
$2996$ |
$7182$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$517$ |
$1143$ |
$2666$ |
$6329$ |
$15210$ |
$36894$ |
$90156$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&1&0&0&0&1\\0&1&0&0&1&0&1&0&2&0\\0&0&2&0&0&1&0&2&0&2\\0&0&0&3&0&1&0&3&0&2\\0&1&0&0&4&0&2&0&5&0\\1&0&1&1&0&4&0&3&0&3\\0&1&0&0&2&0&4&0&5&0\\0&0&2&3&0&3&0&9&0&5\\0&2&0&0&5&0&5&0&10&0\\1&0&2&2&0&3&0&5&0&7\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&2&3&4&4&4&9&10&7\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-2$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ |
---|
$-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=0$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|