Subgroup ($H$) information
| Description: | $C_3^6.(C_3^6.C_2^7:S_4)$ |
| Order: | \(1632586752\)\(\medspace = 2^{10} \cdot 3^{13} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(5,6)(8,9)(23,24)(26,27), (25,26,27), (28,29,30)(31,32,33), (1,16,2,18,3,17) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6.(C_3^6.(C_2^{10}.S_4))$ |
| Order: | \(13060694016\)\(\medspace = 2^{13} \cdot 3^{13} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Derived length: | $5$ |
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 \(104485552128\)\(\medspace = 2^{16} \cdot 3^{13} \) |
| $\operatorname{Aut}(H)$ | 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 | $4$ |
| Möbius function | not computed |
| Projective image | not computed |