Subgroup ($H$) information
| Description: | $C_4\wr C_2$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$a, d$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is normal, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $D_8.D_8$ |
| Order: | \(256\)\(\medspace = 2^{8} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $4$ |
| Derived length: | $2$ |
The ambient group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $D_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Automorphism Group: | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_4\times C_8).C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| $\operatorname{res}(S)$ | $C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(64\)\(\medspace = 2^{6} \) |
| $W$ | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
| Centralizer: | $C_{16}$ | ||
| Normalizer: | $D_8.D_8$ | ||
| Minimal over-subgroups: | $D_8:C_4$ | $D_4.D_4$ | |
| Maximal under-subgroups: | $D_4:C_2$ | $C_4^2$ | $\OD_{16}$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $D_4\times D_8$ |