Subgroup ($H$) information
| Description: | $\SD_{16}$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(2187\)\(\medspace = 3^{7} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\langle(1,4,27,17)(2,6,25,16)(3,5,26,18)(7,10,21,22)(8,11,19,23)(9,12,20,24)(13,28) \!\cdots\! \rangle$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and metacyclic (hence metabelian).
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 \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $\SD_{16}$ | ||
| Normal closure: | $C_3^5:F_9:C_2$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_3^2:\SD_{16}$ | $F_9:C_2$ | $C_3:\SD_{16}$ |
| Maximal under-subgroups: | $D_4$ | $Q_8$ | $C_8$ |
Other information
| Number of subgroups in this autjugacy class | $2187$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3^5:F_9:C_2$ |