Subgroup ($H$) information
| Description: | $C_3^6.(C_3^6.C_2^6:S_4)$ |
| Order: | \(816293376\)\(\medspace = 2^{9} \cdot 3^{13} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(1,10,3,12,2,11)(4,6,5)(7,18,9,17,8,16)(13,15,14)(19,29,21,28,20,30)(22,23,24) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^{12}.C_2^6.D_6.(C_2\times D_4)$ |
| Order: | \(6530347008\)\(\medspace = 2^{12} \cdot 3^{13} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian, solvable, and rational. 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 \(26121388032\)\(\medspace = 2^{14} \cdot 3^{13} \) |
| $\operatorname{Aut}(H)$ | Group of order \(6530347008\)\(\medspace = 2^{12} \cdot 3^{13} \) |
| $\card{W}$ | 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 | $2$ |
| Möbius function | not computed |
| Projective image | not computed |