Name: | $C_8$ |
Order: | $8$ |
Abelian: | yes |
Generators: | $\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{7} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0\\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{5} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & -i & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -i\\0 & 0 & 0 & 0 & 1 & 0 \\-i & 0 & 0 & 0 & 0 & 0 \\0 & 0 & -i & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\\end{bmatrix}$ |
Maximal subgroups: | $A(1,4)_2$ |
Minimal supergroups: | $J_n(A(2,4))$${}^{\times 2}$, $J_s(A(1,8)_1)$, $J_n(E(36))$, $J_n(A(1,12))$, $J_s(B(1,4)_2)$, $J_s(B(1,4;2)_2)$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$63$ |
$0$ |
$1830$ |
$0$ |
$57435$ |
$0$ |
$1860138$ |
$0$ |
$61381782$ |
$a_2$ |
$1$ |
$2$ |
$14$ |
$155$ |
$1982$ |
$26537$ |
$363551$ |
$5052938$ |
$70953530$ |
$1004092517$ |
$14296364129$ |
$204561017168$ |
$2938952526059$ |
$a_3$ |
$1$ |
$0$ |
$22$ |
$0$ |
$5910$ |
$0$ |
$1932070$ |
$0$ |
$670248782$ |
$0$ |
$239998506552$ |
$0$ |
$87682437663018$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$14$ |
$7$ |
$28$ |
$63$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$22$ |
$155$ |
$85$ |
$347$ |
$190$ |
$793$ |
$1830$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$261$ |
$1982$ |
$1084$ |
$597$ |
$4563$ |
$2497$ |
$10580$ |
$5775$ |
$24615$ |
$57435$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$3414$ |
$26537$ |
$1866$ |
$14470$ |
$7899$ |
$61801$ |
$33670$ |
$18366$ |
$144366$ |
$78573$ |
$337926$ |
$183722$ |
$792281$ |
$1860138$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$5910$ |
$46075$ |
$363551$ |
$25116$ |
$197651$ |
$107535$ |
$851886$ |
$58534$ |
$462775$ |
$251536$ |
$1999531$ |
$1085353$ |
$589470$ |
$4699405$ |
$$ |
$2549036$ |
$11057053$ |
$5993526$ |
$26040756$ |
$61381782$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&3&5&0&4&0&8&0&0&20\\0&3&0&4&0&18&0&0&32&0&14&0&48&79&0\\1&0&11&0&13&0&42&51&0&43&0&99&0&0&250\\0&4&0&11&0&38&0&0&66&0&28&0&112&169&0\\1&0&13&0&21&0&57&73&0&66&0&142&0&0&360\\0&18&0&38&0&160&0&0&288&0&128&0&468&740&0\\3&0&42&0&57&0&185&222&0&192&0&448&0&0&1136\\5&0&51&0&73&0&222&285&0&249&0&557&0&0&1422\\0&32&0&66&0&288&0&0&527&0&236&0&854&1363&0\\4&0&43&0&66&0&192&249&0&228&0&494&0&0&1264\\0&14&0&28&0&128&0&0&236&0&108&0&382&612&0\\8&0&99&0&142&0&448&557&0&494&0&1130&0&0&2880\\0&48&0&112&0&468&0&0&854&0&382&0&1436&2244&0\\0&79&0&169&0&740&0&0&1363&0&612&0&2244&3573&0\\20&0&250&0&360&0&1136&1422&0&1264&0&2880&0&0&7392\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&11&11&21&160&185&285&527&228&108&1130&1436&3573&7392&3853&3946&9820&8730&2589\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
---|
$-$ | $1$ | $1/2$ | $0$ | $0$ | $1/2$ | $0$ | $0$ |
---|
$a_1=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $1/2$ | $0$ | $0$ |
---|
$a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $1/2$ | $0$ | $0$ |
---|
$a_1=a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $1/2$ | $0$ | $0$ |
---|