Name: | $C_6$ |
Order: | $6$ |
Abelian: | yes |
Generators: | $\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}$ |
Maximal subgroups: | $A(1,2)$, $A(3,1)$ |
Minimal supergroups: | $J(A(3,2))$, $B(3,2;2)$, $A(6,2)$${}^{\times 3}$, $B(T,1)$, $A(3,4)$, $J_n(A(3,2))$, $B(3,2)$, $A(3,6)$, $J_s(A(3,2))$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$6$ |
$0$ |
$114$ |
$0$ |
$2860$ |
$0$ |
$82530$ |
$0$ |
$2566116$ |
$0$ |
$83103636$ |
$a_2$ |
$1$ |
$3$ |
$25$ |
$257$ |
$3009$ |
$38053$ |
$504399$ |
$6883551$ |
$95702577$ |
$1347086789$ |
$19124879235$ |
$273229589251$ |
$3922295417807$ |
$a_3$ |
$1$ |
$0$ |
$40$ |
$0$ |
$8680$ |
$0$ |
$2642580$ |
$0$ |
$899450552$ |
$0$ |
$320516445720$ |
$0$ |
$116957293298692$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$6$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$25$ |
$14$ |
$52$ |
$114$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$40$ |
$257$ |
$146$ |
$566$ |
$320$ |
$1266$ |
$2860$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$422$ |
$3009$ |
$1680$ |
$946$ |
$6822$ |
$3802$ |
$15588$ |
$8650$ |
$35790$ |
$82530$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$5092$ |
$38053$ |
$2832$ |
$20996$ |
$11618$ |
$87754$ |
$48308$ |
$26668$ |
$203208$ |
$111606$ |
$472028$ |
$258664$ |
$1099322$ |
$2566116$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$8680$ |
$65402$ |
$504399$ |
$36012$ |
$276038$ |
$151334$ |
$1175208$ |
$83108$ |
$642066$ |
$351312$ |
$2744802$ |
$1497238$ |
$817852$ |
$6423450$ |
$$ |
$3498964$ |
$15057966$ |
$8191932$ |
$35351568$ |
$83103636$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&3&0&6&8&0&6&0&13&0&0&28\\0&6&0&8&0&32&0&0&48&0&24&0&72&114&0\\2&0&20&0&24&0&66&78&0&60&0&148&0&0&348\\0&8&0&18&0&60&0&0&96&0&44&0&160&238&0\\3&0&24&0&35&0&88&108&0&92&0&209&0&0&496\\0&32&0&60&0&244&0&0&408&0&188&0&656&1020&0\\6&0&66&0&88&0&268&316&0&266&0&630&0&0&1548\\8&0&78&0&108&0&316&398&0&342&0&772&0&0&1928\\0&48&0&96&0&408&0&0&726&0&328&0&1168&1850&0\\6&0&60&0&92&0&266&342&0&310&0&672&0&0&1704\\0&24&0&44&0&188&0&0&328&0&156&0&528&836&0\\13&0&148&0&209&0&630&772&0&672&0&1553&0&0&3888\\0&72&0&160&0&656&0&0&1168&0&528&0&1952&3032&0\\0&114&0&238&0&1020&0&0&1850&0&836&0&3032&4806&0\\28&0&348&0&496&0&1548&1928&0&1704&0&3888&0&0&9924\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&6&20&18&35&244&268&398&726&310&156&1553&1952&4806&9924&5162&5269&13116&11664&3462\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $1/3$ | $0$ | $1/3$ | $0$ | $0$ | $0$ |
---|
$a_1=0$ | $1/3$ | $1/3$ | $0$ | $1/3$ | $0$ | $0$ | $0$ |
---|
$a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=a_3=0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|