Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(2\) |
| Generators: |
$a$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $p$-group, simple, and rational.
Ambient group ($G$) information
| Description: | $C_2\times C_{64}$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(64\)\(\medspace = 2^{6} \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
Quotient group ($Q$) structure
| Description: | $C_{64}$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(64\)\(\medspace = 2^{6} \) |
| Automorphism Group: | $C_2\times C_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
| Outer Automorphisms: | $C_2\times C_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
| 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
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $\OD_{32}:C_2^2$, of order \(128\)\(\medspace = 2^{7} \) |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $\operatorname{res}(S)$ | $C_1$, of order $1$ |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(64\)\(\medspace = 2^{6} \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2\times C_{64}$ |
| Normalizer: | $C_2\times C_{64}$ |
| Complements: | $C_{64}$ $C_{64}$ |
| Minimal over-subgroups: | $C_2^2$ |
| Maximal under-subgroups: | $C_1$ |
| Autjugate subgroups: | 128.159.64.a1.b1 |
Other information
| Möbius function | $0$ |
| Projective image | $C_{64}$ |