Subgroup ($H$) information
| Description: | $C_3^7.C_3^4:\SL(2,3)$ |
| Order: | \(4251528\)\(\medspace = 2^{3} \cdot 3^{12} \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(4,11,8,6,12,9,5,10,7)(16,22,21,18,23,19,17,24,20)(28,34,33,30,35,31,29,36,32) \!\cdots\! \rangle$
|
| Derived length: | $5$ |
The subgroup is maximal, nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^8.C_3^6:\SL(2,3)$ |
| Order: | \(114791256\)\(\medspace = 2^{3} \cdot 3^{15} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $5$ |
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 \(4132485216\)\(\medspace = 2^{5} \cdot 3^{17} \) |
| $\operatorname{Aut}(H)$ | $C_3\times C_3^7.C_3:S_3.A_4:S_3^2$, of order \(51018336\)\(\medspace = 2^{5} \cdot 3^{13} \) |
| $\card{W}$ | 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 | $27$ |
| Möbius function | not computed |
| Projective image | not computed |