Subgroup ($H$) information
| Description: | not computed |
| Order: | \(102036672\)\(\medspace = 2^{6} \cdot 3^{13} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(10,13,12,15,11,14)(16,17,18)(22,27,23,25,24,26)(28,29,30)(31,32,33)(34,36,35) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is nonabelian and solvable. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^8.C_2^5:(\He_3^2:C_4)$ |
| Order: | \(612220032\)\(\medspace = 2^{7} \cdot 3^{14} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| 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 \(132239526912\)\(\medspace = 2^{10} \cdot 3^{17} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | 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 | $6$ |
| Möbius function | not computed |
| Projective image | not computed |