| Name: | $D_4:C_2$ |
| Order: | $16$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0& 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{5} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{7} & 0 \\0 & 0 & 0 & 0 & 0& \zeta_{12}^{7} \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & -1 & 0 & 0 \\0 & 0& 0 & 0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $A(2,4)$, $J_n(A(1,4)_1)$, $J_s(A(2,2))$${}^{\times 2}$, $J(A(1,4)_1)$, $J_s(A(1,4)_2)$${}^{\times 2}$ |
| Minimal supergroups: | $J(B(2,4))$${}^{\times 3}$, $J_s(A(4,4))$, $J_s(B(3,4))$, $J(B(2,4;4))$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$3$ |
$0$ |
$51$ |
$0$ |
$1230$ |
$0$ |
$34195$ |
$0$ |
$1028538$ |
$0$ |
$32495694$ |
| $a_2$ |
$1$ |
$2$ |
$14$ |
$119$ |
$1298$ |
$15757$ |
$202991$ |
$2711830$ |
$37118778$ |
$516721493$ |
$7280181149$ |
$103473714280$ |
$1480305914879$ |
| $a_3$ |
$1$ |
$0$ |
$18$ |
$0$ |
$3622$ |
$0$ |
$1043490$ |
$0$ |
$344715406$ |
$0$ |
$121226070888$ |
$0$ |
$44001348061594$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$3$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$14$ |
$6$ |
$23$ |
$51$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$18$ |
$119$ |
$64$ |
$245$ |
$138$ |
$545$ |
$1230$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$182$ |
$1298$ |
$716$ |
$407$ |
$2873$ |
$1611$ |
$6526$ |
$3640$ |
$14900$ |
$34195$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$2140$ |
$15757$ |
$1194$ |
$8708$ |
$4845$ |
$35945$ |
$19902$ |
$11058$ |
$82728$ |
$45706$ |
$191092$ |
$105322$ |
$442729$ |
$1028538$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$3622$ |
$26748$ |
$202991$ |
$14812$ |
$111550$ |
$61481$ |
$469757$ |
$33936$ |
$257925$ |
$141854$ |
$1091395$ |
$598204$ |
$328450$ |
$2541914$ |
| $$ |
$1391096$ |
$5932913$ |
$3242148$ |
$13873482$ |
$32495694$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&1&0&2&6&0&1&0&4&0&0&12\\0&3&0&3&0&14&0&0&21&0&11&0&29&49&0\\1&0&11&0&8&0&27&35&0&24&0&63&0&0&142\\0&3&0&9&0&26&0&0&39&0&17&0&70&96&0\\1&0&8&0&19&0&38&43&0&42&0&90&0&0&200\\0&14&0&26&0&104&0&0&168&0&80&0&268&416&0\\2&0&27&0&38&0&117&126&0&102&0&258&0&0&616\\6&0&35&0&43&0&126&177&0&132&0&307&0&0&762\\0&21&0&39&0&168&0&0&295&0&134&0&459&734&0\\1&0&24&0&42&0&102&132&0&140&0&268&0&0&664\\0&11&0&17&0&80&0&0&134&0&68&0&208&338&0\\4&0&63&0&90&0&258&307&0&268&0&634&0&0&1520\\0&29&0&70&0&268&0&0&459&0&208&0&784&1186&0\\0&49&0&96&0&416&0&0&734&0&338&0&1186&1885&0\\12&0&142&0&200&0&616&762&0&664&0&1520&0&0&3832\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&3&11&9&19&104&117&177&295&140&68&634&784&1885&3832&1995&2048&5024&4456&1351\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_1=0$ | $1/2$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $1/4$ | $0$ | $0$ | $0$ | $1/4$ |
|---|
