Subgroup ($H$) information
| Description: | $D_{16}$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$a, bc^{5}$
|
| Nilpotency class: | $4$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $D_{16}:C_8$ |
| 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: | $C_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism Group: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_4^3).D_4$, of order \(1024\)\(\medspace = 2^{10} \) |
| $\operatorname{Aut}(H)$ | $D_{16}:C_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_{16}:C_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(8\)\(\medspace = 2^{3} \) |
| $W$ | $C_2\times D_8$, of order \(32\)\(\medspace = 2^{5} \) |
Related subgroups
| Centralizer: | $C_8$ | ||
| Normalizer: | $D_{16}:C_8$ | ||
| Complements: | $C_8$ $C_8$ | ||
| Minimal over-subgroups: | $D_{16}:C_2$ | ||
| Maximal under-subgroups: | $D_8$ | $D_8$ | $C_{16}$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_8\times D_8$ |