| Name: | $D_4$ |
| Order: | $8$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0& 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & -1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $A(2,2)$, $J_n(A(1,2))$, $J(A(1,2))$ |
| Minimal supergroups: | $J_s(B(3,2))$, $J_s(A(2,4))$${}^{\times 2}$, $J_s(A(2,6))$, $J(B(1,4)_2)$${}^{\times 2}$, $J_s(A(6,2))$, $J_s(C(2,2))$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$63$ |
$0$ |
$1830$ |
$0$ |
$57435$ |
$0$ |
$1860138$ |
$0$ |
$61381782$ |
| $a_2$ |
$1$ |
$2$ |
$16$ |
$161$ |
$2002$ |
$26597$ |
$363733$ |
$5053484$ |
$70955170$ |
$1004097437$ |
$14296378891$ |
$204561061454$ |
$2938952658919$ |
| $a_3$ |
$1$ |
$0$ |
$22$ |
$0$ |
$5910$ |
$0$ |
$1932070$ |
$0$ |
$670248782$ |
$0$ |
$239998506552$ |
$0$ |
$87682437663018$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$16$ |
$6$ |
$27$ |
$63$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$22$ |
$161$ |
$84$ |
$345$ |
$186$ |
$789$ |
$1830$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$258$ |
$2002$ |
$1080$ |
$597$ |
$4557$ |
$2493$ |
$10572$ |
$5760$ |
$24600$ |
$57435$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$3408$ |
$26597$ |
$1854$ |
$14460$ |
$7887$ |
$61785$ |
$33654$ |
$18366$ |
$144342$ |
$78558$ |
$337896$ |
$183666$ |
$792225$ |
$1860138$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$5910$ |
$46056$ |
$363733$ |
$25104$ |
$197622$ |
$107511$ |
$851841$ |
$58488$ |
$462735$ |
$251490$ |
$1999467$ |
$1085292$ |
$589470$ |
$4699314$ |
| $$ |
$2548980$ |
$11056941$ |
$5993316$ |
$26040546$ |
$61381782$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&2&8&0&3&0&6&0&0&20\\0&3&0&3&0&18&0&0&33&0&15&0&45&81&0\\1&0&13&0&10&0&41&53&0&40&0&99&0&0&250\\0&3&0&13&0&38&0&0&63&0&25&0&120&166&0\\1&0&10&0&25&0&56&71&0&74&0&144&0&0&360\\0&18&0&38&0&160&0&0&288&0&128&0&468&740&0\\2&0&41&0&56&0&193&214&0&182&0&450&0&0&1136\\8&0&53&0&71&0&214&301&0&250&0&549&0&0&1422\\0&33&0&63&0&288&0&0&531&0&240&0&843&1368&0\\3&0&40&0&74&0&182&250&0&254&0&498&0&0&1264\\0&15&0&25&0&128&0&0&240&0&112&0&372&616&0\\6&0&99&0&144&0&450&549&0&498&0&1134&0&0&2880\\0&45&0&120&0&468&0&0&843&0&372&0&1464&2232&0\\0&81&0&166&0&740&0&0&1368&0&616&0&2232&3577&0\\20&0&250&0&360&0&1136&1422&0&1264&0&2880&0&0&7392\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&13&13&25&160&193&301&531&254&112&1134&1464&3577&7392&3891&3978&9880&8778&2675\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_1=0$ | $1/2$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
