Subgroup ($H$) information
| Description: | $C_3^8.C_3^3:\SL(2,3)$ |
| Order: | \(4251528\)\(\medspace = 2^{3} \cdot 3^{12} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(16,17,18)(22,23,24)(28,30,29)(31,33,32), (13,15,14)(19,21,20)(22,23,24) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is maximal, nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^8.C_3^5:\SL(2,3)$ |
| Order: | \(38263752\)\(\medspace = 2^{3} \cdot 3^{14} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 \(12397455648\)\(\medspace = 2^{5} \cdot 3^{18} \) |
| $\operatorname{Aut}(H)$ | Group of order \(459165024\)\(\medspace = 2^{5} \cdot 3^{15} \) |
| $\card{W}$ | 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 | $9$ |
| Möbius function | not computed |
| Projective image | not computed |