Subgroup ($H$) information
| Description: | $C_{32}$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(32\)\(\medspace = 2^{5} \) |
| Generators: |
$ac$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, 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: | $C_2\times Q_{64}$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(32\)\(\medspace = 2^{5} \) |
| Nilpotency class: | $5$ |
| 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: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_8^2.C_8:C_2^3$, of order \(4096\)\(\medspace = 2^{12} \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_8$, of order \(16\)\(\medspace = 2^{4} \) |
| $\operatorname{res}(S)$ | $C_2\times C_8$, of order \(16\)\(\medspace = 2^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(128\)\(\medspace = 2^{7} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_2\times C_{32}$ | ||
| Normalizer: | $C_2\times Q_{64}$ | ||
| Minimal over-subgroups: | $C_2\times C_{32}$ | $Q_{64}$ | $Q_{64}$ |
| Maximal under-subgroups: | $C_{16}$ | ||
| Autjugate subgroups: | 128.993.4.c1.a1 |
Other information
| Möbius function | $2$ |
| Projective image | $C_2\times D_{16}$ |