Name: | $C_3^2:C_6$ |
Order: | $54$ |
Abelian: | no |
Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{9}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{9}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{9}^{7} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{9}^{8} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{9}^{8} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{9}^{2} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 1 & 0 & 0 & 0 \\1 & 0 & 0& 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 &0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 &-1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$1$ |
$0$ |
$15$ |
$0$ |
$330$ |
$0$ |
$9135$ |
$0$ |
$282366$ |
$0$ |
$9168390$ |
$a_2$ |
$1$ |
$1$ |
$5$ |
$36$ |
$357$ |
$4266$ |
$55734$ |
$759501$ |
$10576773$ |
$149149026$ |
$2120393700$ |
$30320202771$ |
$435492394110$ |
$a_3$ |
$1$ |
$0$ |
$6$ |
$0$ |
$966$ |
$0$ |
$290620$ |
$0$ |
$99524166$ |
$0$ |
$35567544816$ |
$0$ |
$12990657192744$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$1$ |
$1$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$5$ |
$2$ |
$7$ |
$15$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$6$ |
$36$ |
$18$ |
$67$ |
$38$ |
$147$ |
$330$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$50$ |
$357$ |
$192$ |
$108$ |
$765$ |
$426$ |
$1731$ |
$960$ |
$3960$ |
$9135$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$572$ |
$4266$ |
$318$ |
$2328$ |
$1286$ |
$9675$ |
$5316$ |
$2928$ |
$22329$ |
$12240$ |
$51822$ |
$28350$ |
$120771$ |
$282366$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$966$ |
$7220$ |
$55734$ |
$3966$ |
$30378$ |
$16618$ |
$129285$ |
$9105$ |
$70500$ |
$38493$ |
$301761$ |
$164313$ |
$89574$ |
$706401$ |
$$ |
$384174$ |
$1657341$ |
$900396$ |
$3895290$ |
$9168390$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&0&0&0&1&2&0&0&0&1&0&0&2\\0&1&0&1&0&4&0&0&5&0&3&0&7&13&0\\0&0&4&0&2&0&6&11&0&5&0&15&0&0&38\\0&1&0&3&0&6&0&0&11&0&5&0&17&27&0\\0&0&2&0&6&0&10&9&0&11&0&25&0&0&52\\0&4&0&6&0&28&0&0&44&0&20&0&70&112&0\\1&0&6&0&10&0&31&33&0&31&0&68&0&0&170\\2&0&11&0&9&0&33&50&0&33&0&81&0&0&212\\0&5&0&11&0&44&0&0&81&0&33&0&129&203&0\\0&0&5&0&11&0&31&33&0&38&0&74&0&0&188\\0&3&0&5&0&20&0&0&33&0&20&0&56&93&0\\1&0&15&0&25&0&68&81&0&74&0&173&0&0&424\\0&7&0&17&0&70&0&0&129&0&56&0&214&333&0\\0&13&0&27&0&112&0&0&203&0&93&0&333&533&0\\2&0&38&0&52&0&170&212&0&188&0&424&0&0&1104\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&1&4&3&6&28&31&50&81&38&20&173&214&533&1104&573&605&1456&1302&385\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $47/54$ | $0$ | $19/27$ | $0$ | $0$ | $1/6$ |
---|
$a_1=0$ | $47/54$ | $47/54$ | $0$ | $19/27$ | $0$ | $0$ | $1/6$ |
---|
$a_3=0$ | $1/2$ | $1/2$ | $0$ | $1/3$ | $0$ | $0$ | $1/6$ |
---|
$a_1=a_3=0$ | $1/2$ | $1/2$ | $0$ | $1/3$ | $0$ | $0$ | $1/6$ |
---|