| Name: | $D_{12}:C_2$ |
| Order: | $48$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{3}^{1} & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{3}^{2} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{6}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{12}^{5} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{12}^{5} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{6}^{5} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{12}^{7} & 0 \\0 & 0 & 0 & 0 & 0& \zeta_{12}^{7} \\\end{bmatrix}, \begin{bmatrix}-1 & 0 & 0 & 0 & 0 & 0 \\0 & 0& -1 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & -1 & 0 & 0 \\0 & 0 & 0 &0 & 0 & -1 \\0 & 0 & 0 & 0 & -1 & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 &-1 & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & -1 \\1 & 0 & 0 & 0 & 0 & 0 \\0 & -1 & 0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $J_s(B(3,2;2))$${}^{\times 2}$, $J_s(B(3,2))$${}^{\times 2}$, $J_s(A(2,4))$, $J(A(3,4))$, $J_n(A(3,4))$, $B(3,4)$ |
| Minimal supergroups: | |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$24$ |
$0$ |
$480$ |
$0$ |
$12250$ |
$0$ |
$354312$ |
$0$ |
$10994676$ |
| $a_2$ |
$1$ |
$2$ |
$9$ |
$56$ |
$510$ |
$5692$ |
$70502$ |
$923680$ |
$12515972$ |
$173301896$ |
$2434714554$ |
$34551898240$ |
$493898902130$ |
| $a_3$ |
$1$ |
$0$ |
$9$ |
$0$ |
$1333$ |
$0$ |
$356670$ |
$0$ |
$115640917$ |
$0$ |
$40473820854$ |
$0$ |
$14673046945414$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$9$ |
$3$ |
$11$ |
$24$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$9$ |
$56$ |
$28$ |
$100$ |
$57$ |
$216$ |
$480$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$74$ |
$510$ |
$277$ |
$162$ |
$1070$ |
$610$ |
$2392$ |
$1350$ |
$5390$ |
$12250$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$798$ |
$5692$ |
$450$ |
$3152$ |
$1774$ |
$12732$ |
$7114$ |
$4000$ |
$29040$ |
$16176$ |
$66584$ |
$36946$ |
$153300$ |
$354312$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$1333$ |
$9481$ |
$70502$ |
$5300$ |
$38898$ |
$21588$ |
$161848$ |
$12003$ |
$89328$ |
$49422$ |
$374162$ |
$206008$ |
$113724$ |
$867830$ |
| $$ |
$476798$ |
$2018450$ |
$1106700$ |
$4705848$ |
$10994676$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&0&4&0&0&0&1&0&0&4\\0&2&0&1&0&6&0&0&8&0&5&0&9&18&0\\1&0&6&0&2&0&10&15&0&6&0&22&0&0&49\\0&1&0&5&0&10&0&0&13&0&5&0&28&32&0\\0&0&2&0&11&0&14&13&0&19&0&34&0&0&68\\0&6&0&10&0&39&0&0&59&0&29&0&93&143&0\\0&0&10&0&14&0&46&40&0&33&0&91&0&0&210\\4&0&15&0&13&0&40&71&0&44&0&102&0&0&258\\0&8&0&13&0&59&0&0&104&0&47&0&153&250&0\\0&0&6&0&19&0&33&44&0&58&0&92&0&0&224\\0&5&0&5&0&29&0&0&47&0&27&0&67&117&0\\1&0&22&0&34&0&91&102&0&92&0&220&0&0&512\\0&9&0&28&0&93&0&0&153&0&67&0&278&395&0\\0&18&0&32&0&143&0&0&250&0&117&0&395&637&0\\4&0&49&0&68&0&210&258&0&224&0&512&0&0&1287\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&6&5&11&39&46&71&104&58&27&220&278&637&1287&684&703&1699&1508&483\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $13/24$ | $1/12$ | $1/24$ | $0$ | $1/6$ | $1/4$ |
|---|
| $a_1=0$ | $13/24$ | $13/24$ | $1/12$ | $1/24$ | $0$ | $1/6$ | $1/4$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $1/12$ | $0$ | $0$ | $1/6$ | $1/4$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $1/12$ | $0$ | $0$ | $1/6$ | $1/4$ |
|---|
