Name: | $C_3:C_8$ |
Order: | $24$ |
Abelian: | no |
Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{5} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{7} & 0 \\0 & 0 & 0 & 0 & 0& \zeta_{12}^{7} \\\end{bmatrix}, \begin{bmatrix}i & 0 & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{8}^{1} & 0 & 0 & 0 \\0 & \zeta_{8}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -i& 0 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{8}^{7} \\0 & 0 & 0 & 0 & \zeta_{8}^{7} & 0 \\\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$48$ |
$0$ |
$960$ |
$0$ |
$24500$ |
$0$ |
$708624$ |
$0$ |
$21989352$ |
$a_2$ |
$1$ |
$2$ |
$12$ |
$96$ |
$974$ |
$11252$ |
$140618$ |
$1846224$ |
$25028578$ |
$346593780$ |
$4869399242$ |
$69103707224$ |
$987797537174$ |
$a_3$ |
$1$ |
$0$ |
$18$ |
$0$ |
$2658$ |
$0$ |
$713160$ |
$0$ |
$231278698$ |
$0$ |
$80947591308$ |
$0$ |
$29346093106308$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$4$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$12$ |
$6$ |
$22$ |
$48$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$18$ |
$96$ |
$56$ |
$200$ |
$114$ |
$432$ |
$960$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$148$ |
$974$ |
$554$ |
$322$ |
$2140$ |
$1218$ |
$4782$ |
$2700$ |
$10780$ |
$24500$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1596$ |
$11252$ |
$900$ |
$6304$ |
$3546$ |
$25464$ |
$14224$ |
$7988$ |
$58074$ |
$32340$ |
$133156$ |
$73892$ |
$306600$ |
$708624$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2658$ |
$18962$ |
$140618$ |
$10592$ |
$77796$ |
$43166$ |
$323696$ |
$24006$ |
$178642$ |
$98832$ |
$748304$ |
$411992$ |
$227392$ |
$1735624$ |
$$ |
$953540$ |
$4036844$ |
$2213400$ |
$9411696$ |
$21989352$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&2&0&1&4&0&3&0&4&0&0&8\\0&4&0&2&0&12&0&0&16&0&10&0&18&36&0\\1&0&9&0&8&0&23&24&0&15&0&46&0&0&98\\0&2&0&10&0&20&0&0&26&0&10&0&54&66&0\\2&0&8&0&16&0&24&36&0&30&0&64&0&0&136\\0&12&0&20&0&78&0&0&118&0&58&0&186&286&0\\1&0&23&0&24&0&87&88&0&67&0&180&0&0&420\\4&0&24&0&36&0&88&120&0&100&0&214&0&0&516\\0&16&0&26&0&118&0&0&208&0&92&0&310&498&0\\3&0&15&0&30&0&67&100&0&95&0&176&0&0&448\\0&10&0&10&0&58&0&0&92&0&54&0&138&232&0\\4&0&46&0&64&0&180&214&0&176&0&434&0&0&1024\\0&18&0&54&0&186&0&0&310&0&138&0&544&796&0\\0&36&0&66&0&286&0&0&498&0&232&0&796&1270&0\\8&0&98&0&136&0&420&516&0&448&0&1024&0&0&2574\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&9&10&16&78&87&120&208&95&54&434&544&1270&2574&1340&1360&3362&2972&894\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $1/12$ | $0$ | $1/12$ | $0$ | $0$ | $0$ |
---|
$a_1=0$ | $1/12$ | $1/12$ | $0$ | $1/12$ | $0$ | $0$ | $0$ |
---|
$a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|