Subgroup ($H$) information
| Description: | $C_3^3:F_9:C_2$ |
| Order: | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| Index: | \(162\)\(\medspace = 2 \cdot 3^{4} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(10,12,11)(13,15,14)(16,17,18)(19,20,21)(22,23,24)(25,27,26)(28,29,30)(31,32,33) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_3^8:D_4.D_6$ |
| Order: | \(629856\)\(\medspace = 2^{5} \cdot 3^{9} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^8.C_6^2.C_2.C_2^5$, of order \(15116544\)\(\medspace = 2^{8} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $C_3^4:(S_3\times \SD_{16})$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| $\card{W}$ | 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 | $162$ |
| Möbius function | not computed |
| Projective image | not computed |