| Name: | $D_4$ |
| Order: | $8$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{7} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0\\0 & 0 & 0 & 0 & \zeta_{12}^{11} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{12}^{5} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $A(1,4)_2$, $J(A(1,2))$${}^{\times 2}$ |
| Minimal supergroups: | $J(A(1,12))$, $J_s(B(1,4;2)_2)$, $J(A(1,8)_1)$${}^{\times 2}$, $J(B(3,2;2))$, $J(E(36))$, $J_s(B(1,4)_2)$, $J(B(1,4;2)_2)$${}^{\times 3}$, $J(B(1,4)_2)$, $J(A(2,4))$${}^{\times 2}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$63$ |
$0$ |
$1830$ |
$0$ |
$57435$ |
$0$ |
$1860138$ |
$0$ |
$61381782$ |
| $a_2$ |
$1$ |
$3$ |
$18$ |
$168$ |
$2022$ |
$26658$ |
$363915$ |
$5054031$ |
$70956810$ |
$1004102358$ |
$14296393653$ |
$204561105741$ |
$2938952791779$ |
| $a_3$ |
$1$ |
$0$ |
$22$ |
$0$ |
$5910$ |
$0$ |
$1932070$ |
$0$ |
$670248782$ |
$0$ |
$239998506552$ |
$0$ |
$87682437663018$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$3$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$18$ |
$7$ |
$28$ |
$63$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$22$ |
$168$ |
$85$ |
$347$ |
$190$ |
$793$ |
$1830$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$261$ |
$2022$ |
$1084$ |
$597$ |
$4563$ |
$2497$ |
$10580$ |
$5775$ |
$24615$ |
$57435$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$3414$ |
$26658$ |
$1866$ |
$14470$ |
$7899$ |
$61801$ |
$33670$ |
$18366$ |
$144366$ |
$78573$ |
$337926$ |
$183722$ |
$792281$ |
$1860138$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$5910$ |
$46075$ |
$363915$ |
$25116$ |
$197651$ |
$107535$ |
$851886$ |
$58534$ |
$462775$ |
$251536$ |
$1999531$ |
$1085353$ |
$589470$ |
$4699405$ |
| $$ |
$2549036$ |
$11057053$ |
$5993526$ |
$26040756$ |
$61381782$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&2&0&0&0&2&8&0&2&0&6&0&0&20\\0&3&0&4&0&18&0&0&32&0&14&0&48&79&0\\2&0&13&0&10&0&40&57&0&39&0&96&0&0&250\\0&4&0&11&0&38&0&0&66&0&28&0&112&169&0\\0&0&10&0&25&0&60&64&0&72&0&147&0&0&360\\0&18&0&38&0&160&0&0&288&0&128&0&468&740&0\\2&0&40&0&60&0&187&216&0&196&0&451&0&0&1136\\8&0&57&0&64&0&216&303&0&237&0&548&0&0&1422\\0&32&0&66&0&288&0&0&527&0&236&0&854&1363&0\\2&0&39&0&72&0&196&237&0&236&0&500&0&0&1264\\0&14&0&28&0&128&0&0&236&0&108&0&382&612&0\\6&0&96&0&147&0&451&548&0&500&0&1134&0&0&2880\\0&48&0&112&0&468&0&0&854&0&382&0&1436&2244&0\\0&79&0&169&0&740&0&0&1363&0&612&0&2244&3573&0\\20&0&250&0&360&0&1136&1422&0&1264&0&2880&0&0&7392\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&13&11&25&160&187&303&527&236&108&1134&1436&3573&7392&3853&3986&9820&8748&2589\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $0$ | $0$ | $0$ | $0$ | $1/2$ |
|---|
