Subgroup ($H$) information
| Description: | $C_3^6:\SL(3,3)$ |
| Order: | \(4094064\)\(\medspace = 2^{4} \cdot 3^{9} \cdot 13 \) |
| Index: | \(729\)\(\medspace = 3^{6} \) |
| Exponent: | \(936\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 13 \) |
| Generators: |
$\langle(10,11,12)(19,20,21)(25,27,26)(28,30,29)(34,36,35)(37,38,39), (1,36,20) \!\cdots\! \rangle$
|
| Derived length: | $0$ |
The subgroup is maximal, nonabelian, and perfect (hence nonsolvable).
Ambient group ($G$) information
| Description: | $C_3^{12}.\SL(3,3)$ |
| Order: | \(2984572656\)\(\medspace = 2^{4} \cdot 3^{15} \cdot 13 \) |
| Exponent: | \(936\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 13 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and perfect (hence nonsolvable).
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(5969145312\)\(\medspace = 2^{5} \cdot 3^{15} \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_3^6.\GL(3,3)$, of order \(8188128\)\(\medspace = 2^{5} \cdot 3^{9} \cdot 13 \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $729$ |
| Möbius function | not computed |
| Projective image | not computed |