Name: | $S_3$ |
Order: | $6$ |
Abelian: | no |
Generators: | $\begin{bmatrix}\zeta_6&0&0&0\\0&\zeta_6^5&0&0\\0&0&\zeta_6^5&0\\0&0&0&\zeta_6\end{bmatrix}, \begin{bmatrix}0&0&-1&0\\0&0&0&-1\\1&0&0&0\\0&1&0&0\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$18$ |
$0$ |
$220$ |
$0$ |
$3010$ |
$0$ |
$43092$ |
$0$ |
$631092$ |
$a_2$ |
$1$ |
$2$ |
$6$ |
$21$ |
$90$ |
$437$ |
$2285$ |
$12427$ |
$69074$ |
$388965$ |
$2209101$ |
$12623327$ |
$72470425$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$6$ |
$8$ |
$18$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$21$ |
$38$ |
$90$ |
$220$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$90$ |
$194$ |
$476$ |
$1190$ |
$3010$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$437$ |
$1034$ |
$2592$ |
$6572$ |
$16786$ |
$43092$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$2285$ |
$5656$ |
$14370$ |
$36770$ |
$94542$ |
$243936$ |
$631092$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&2&0&1&0&3\\0&2&0&0&4&0&4&0&8&0\\1&0&3&2&0&5&0&8&0&8\\0&0&2&8&0&5&0&17&0&11\\0&4&0&0&14&0&14&0&28&0\\2&0&5&5&0&12&0&20&0&19\\0&4&0&0&14&0&18&0&32&0\\1&0&8&17&0&20&0&51&0&38\\0&8&0&0&28&0&32&0&62&0\\3&0&8&11&0&19&0&38&0&35\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&3&8&14&12&18&51&62&35\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-2$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ |
---|
$-$ | $1$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
---|
$a_1=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
---|