| Name: | $C_3:D_4$ |
| Order: | $24$ |
| Abelian: | no |
| Generators: | $\begin{bmatrix}\zeta_{3}^{1} & 0 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 & 0 \\0 &0 & \zeta_{3}^{2} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{2} & 0 & 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{3}^{1} \\\end{bmatrix}, \begin{bmatrix}\zeta_{3}^{2} & 0 & 0 & 0 & 0 & 0 \\0 & \zeta_{6}^{1} & 0 & 0 & 0 & 0 \\0 & 0 & \zeta_{6}^{1} & 0 & 0 & 0 \\0 & 0 & 0 & \zeta_{3}^{1} & 0 & 0 \\0 & 0 & 0 & 0 & \zeta_{6}^{5} & 0 \\0 & 0 & 0 & 0 & 0 & \zeta_{6}^{5} \\\end{bmatrix}, \begin{bmatrix}1 & 0 & 0 & 0 & 0 & 0 \\0 & 0 & i& 0 & 0 & 0 \\0 & i & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 & 0 & -i \\0 & 0 & 0 & 0 & -i & 0 \\\end{bmatrix}, \begin{bmatrix}0 & 0 & 0 & 1 & 0& 0 \\0 & 0 & 0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 & 0 & 1 \\-1 & 0 & 0 & 0 & 0 & 0 \\0& -1 & 0 & 0 & 0 & 0 \\0 & 0 & -1 & 0 & 0 & 0 \\\end{bmatrix}$ |
| Maximal subgroups: | $J(A(3,2))$, $J_s(A(3,2))$, $J(A(1,4)_2)$, $B(3,2;2)$ |
| Minimal supergroups: | $J(B(3,6;2))$, $J(B(6,2))$, $J(B(3,4))$${}^{\times 2}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$30$ |
$0$ |
$720$ |
$0$ |
$20650$ |
$0$ |
$641592$ |
$0$ |
$20776140$ |
| $a_2$ |
$1$ |
$2$ |
$10$ |
$75$ |
$784$ |
$9607$ |
$126378$ |
$1721715$ |
$23928108$ |
$336779043$ |
$4781241730$ |
$68307462775$ |
$980574050046$ |
| $a_3$ |
$1$ |
$0$ |
$11$ |
$0$ |
$2181$ |
$0$ |
$660790$ |
$0$ |
$224864661$ |
$0$ |
$80129140506$ |
$0$ |
$29239323750582$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$2$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=4\right)\colon$ |
$10$ |
$4$ |
$14$ |
$30$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=6\right)\colon$ |
$11$ |
$75$ |
$38$ |
$144$ |
$82$ |
$320$ |
$720$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=8\right)\colon$ |
$108$ |
$784$ |
$424$ |
$240$ |
$1712$ |
$956$ |
$3906$ |
$2170$ |
$8960$ |
$20650$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=10\right)\colon$ |
$1280$ |
$9607$ |
$714$ |
$5260$ |
$2914$ |
$21956$ |
$12092$ |
$6680$ |
$50826$ |
$27922$ |
$118040$ |
$64694$ |
$274876$ |
$641592$ |
| $\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}a_3^{e_3}\right]:\sum ie_i=12\right)\colon$ |
$2181$ |
$16370$ |
$126378$ |
$9020$ |
$69040$ |
$37860$ |
$293850$ |
$20800$ |
$160558$ |
$87864$ |
$686266$ |
$374366$ |
$204512$ |
$1605952$ |
| $$ |
$874818$ |
$3764614$ |
$2048088$ |
$8838060$ |
$20776140$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&0&0&0&1&4&0&0&0&2&0&0&7\\0&2&0&2&0&8&0&0&12&0&6&0&18&28&0\\1&0&7&0&4&0&15&23&0&13&0&35&0&0&87\\0&2&0&5&0&15&0&0&24&0&11&0&40&59&0\\0&0&4&0&12&0&24&21&0&27&0&55&0&0&124\\0&8&0&15&0&61&0&0&102&0&47&0&164&255&0\\1&0&15&0&24&0&69&75&0&69&0&159&0&0&387\\4&0&23&0&21&0&75&112&0&77&0&187&0&0&482\\0&12&0&24&0&102&0&0&182&0&82&0&292&462&0\\0&0&13&0&27&0&69&77&0&84&0&172&0&0&426\\0&6&0&11&0&47&0&0&82&0&39&0&132&209&0\\2&0&35&0&55&0&159&187&0&172&0&392&0&0&972\\0&18&0&40&0&164&0&0&292&0&132&0&488&758&0\\0&28&0&59&0&255&0&0&462&0&209&0&758&1203&0\\7&0&87&0&124&0&387&482&0&426&0&972&0&0&2481\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&7&5&12&61&69&112&182&84&39&392&488&1203&2481&1292&1345&3279&2930&867\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ | $a_2=3$ |
|---|
| $-$ | $1$ | $7/12$ | $0$ | $1/4$ | $0$ | $0$ | $1/3$ |
|---|
| $a_1=0$ | $7/12$ | $7/12$ | $0$ | $1/4$ | $0$ | $0$ | $1/3$ |
|---|
| $a_3=0$ | $1/2$ | $1/2$ | $0$ | $1/6$ | $0$ | $0$ | $1/3$ |
|---|
| $a_1=a_3=0$ | $1/2$ | $1/2$ | $0$ | $1/6$ | $0$ | $0$ | $1/3$ |
|---|
