Subgroup ($H$) information
| Description: | $C_2^2:C_8$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(32\)\(\medspace = 2^{5} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$\langle(5,6)(7,8)(9,10)(11,12), (1,2)(3,4)(13,14)(15,16), (1,16,2,15)(3,13)(4,14) \!\cdots\! \rangle$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $(D_4\times C_2^3).Q_{16}$ |
| Order: | \(1024\)\(\medspace = 2^{10} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^5:C_4.D_4^2$, of order \(8192\)\(\medspace = 2^{13} \) |
| $\operatorname{Aut}(H)$ | $C_2^3:D_4$, of order \(64\)\(\medspace = 2^{6} \) |
| $\operatorname{res}(S)$ | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_2\times C_4$ | |
| Normalizer: | $C_2^3:C_8$ | |
| Normal closure: | $C_2^5.\OD_{16}$ | |
| Core: | $C_2^2$ | |
| Minimal over-subgroups: | $C_2^3:C_8$ | |
| Maximal under-subgroups: | $C_2^2\times C_4$ | $C_2\times C_8$ |
Other information
| Number of subgroups in this conjugacy class | $16$ |
| Möbius function | $0$ |
| Projective image | $(C_2^4:C_4) . (C_2\times C_4)$ |