Name: | $C_3^2:S_4$ |
Order: | $216$ |
Abelian: | no |
Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{6}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{18}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{18}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{9}^{8} & 0 & 0 & 0 \\0 & 0 & 0 &\zeta_{18}^{17} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{18}^{17} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{9}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0& 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$1$ |
$0$ |
$15$ |
$0$ |
$310$ |
$0$ |
$7455$ |
$0$ |
$195426$ |
$0$ |
$5416950$ |
$a_2$ |
$1$ |
$1$ |
$5$ |
$34$ |
$317$ |
$3396$ |
$39094$ |
$469729$ |
$5815989$ |
$73709212$ |
$952347500$ |
$12510595659$ |
$166766654718$ |
$a_3$ |
$1$ |
$0$ |
$6$ |
$0$ |
$822$ |
$0$ |
$184880$ |
$0$ |
$49026950$ |
$0$ |
$14355086256$ |
$0$ |
$4518295755688$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$1$ |
$1$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$5$ |
$2$ |
$7$ |
$15$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$6$ |
$34$ |
$18$ |
$65$ |
$38$ |
$141$ |
$310$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$48$ |
$317$ |
$178$ |
$104$ |
$675$ |
$390$ |
$1495$ |
$860$ |
$3330$ |
$7455$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$500$ |
$3396$ |
$288$ |
$1926$ |
$1104$ |
$7533$ |
$4294$ |
$2456$ |
$16917$ |
$9620$ |
$38130$ |
$21630$ |
$86205$ |
$195426$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$822$ |
$5580$ |
$39094$ |
$3180$ |
$22074$ |
$12516$ |
$88245$ |
$7112$ |
$49818$ |
$28184$ |
$200157$ |
$112780$ |
$63680$ |
$455130$ |
$$ |
$255990$ |
$1037169$ |
$582372$ |
$2368170$ |
$5416950$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&1&2&0&0&0&1&0&0&2\\0&1&0&1&0&4&0&0&5&0&3&0&7&11&0\\0&0&4&0&2&0&6&9&0&5&0&15&0&0&30\\0&1&0&3&0&6&0&0&9&0&5&0&15&21&0\\0&0&2&0&6&0&10&9&0&9&0&21&0&0&40\\0&4&0&6&0&26&0&0&36&0&18&0&56&82&0\\1&0&6&0&10&0&27&27&0&21&0&54&0&0&114\\2&0&9&0&9&0&27&38&0&23&0&61&0&0&136\\0&5&0&9&0&36&0&0&57&0&27&0&87&131&0\\0&0&5&0&9&0&21&23&0&26&0&50&0&0&108\\0&3&0&5&0&18&0&0&27&0&16&0&42&65&0\\1&0&15&0&21&0&54&61&0&50&0&125&0&0&264\\0&7&0&15&0&56&0&0&87&0&42&0&142&207&0\\0&11&0&21&0&82&0&0&131&0&65&0&207&315&0\\2&0&30&0&40&0&114&136&0&108&0&264&0&0&602\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&4&3&6&26&27&38&57&26&16&125&142&315&602&281&293&662&544&151\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $91/108$ | $1/12$ | $55/108$ | $0$ | $1/6$ | $1/12$ |
---|
$a_1=0$ | $91/108$ | $91/108$ | $1/12$ | $55/108$ | $0$ | $1/6$ | $1/12$ |
---|
$a_3=0$ | $1/2$ | $1/2$ | $1/12$ | $1/6$ | $0$ | $1/6$ | $1/12$ |
---|
$a_1=a_3=0$ | $1/2$ | $1/2$ | $1/12$ | $1/6$ | $0$ | $1/6$ | $1/12$ |
---|