Subgroup ($H$) information
| Description: | $C_{16}$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$\langle(1,3,6,8,2,4,5,7)(9,11,13,16,10,12,14,15), (1,9,3,11,6,13,8,16,2,10,4,12,5,14,7,15) \!\cdots\! \rangle$
|
| 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: | $C_2^6.\OD_{16}$ |
| Order: | \(1024\)\(\medspace = 2^{10} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $7$ |
| Derived length: | $3$ |
The ambient group is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^7.C_2\wr D_4$, of order \(16384\)\(\medspace = 2^{14} \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\operatorname{res}(S)$ | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(32\)\(\medspace = 2^{5} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{16}$ |
| Normalizer: | $\OD_{32}$ |
| Normal closure: | $C_2^6.C_8$ |
| Core: | $C_2$ |
| Minimal over-subgroups: | $\OD_{32}$ |
| Maximal under-subgroups: | $C_8$ |
Other information
| Number of subgroups in this autjugacy class | $64$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $(C_2^3\times C_4) . (C_2\times C_8)$ |