Subgroup ($H$) information
| Description: | $C_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(4374\)\(\medspace = 2 \cdot 3^{7} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\langle(1,36,30,19,13,11,17,7)(2,34,29,20,14,12,16,8)(3,35,28,21,15,10,18,9)(4,31,25,22) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
Ambient group ($G$) information
| Description: | $C_3^5:F_9:C_2$ |
| Order: | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_3^4.Q_8.C_6.C_2^3$, of order \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_8$ | |||
| Normalizer: | $\SD_{16}$ | |||
| Normal closure: | $C_3^5:F_9$ | |||
| Core: | $C_1$ | |||
| Minimal over-subgroups: | $C_3^2:C_8$ | $F_9$ | $C_3:C_8$ | $\SD_{16}$ |
| Maximal under-subgroups: | $C_4$ |
Other information
| Number of subgroups in this autjugacy class | $2187$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3^5:F_9:C_2$ |