Subgroup ($H$) information
| Description: | $C_7$ |
| Order: | \(7\) |
| Index: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(7\) |
| Generators: |
$b^{32}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $7$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_7:Q_{64}$ |
| Order: | \(448\)\(\medspace = 2^{6} \cdot 7 \) |
| Exponent: | \(224\)\(\medspace = 2^{5} \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $Q_{64}$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Exponent: | \(32\)\(\medspace = 2^{5} \) |
| Automorphism Group: | $D_{32}:C_8$, of order \(512\)\(\medspace = 2^{9} \) |
| Outer Automorphisms: | $C_2\times C_8$, of order \(16\)\(\medspace = 2^{4} \) |
| Nilpotency class: | $5$ |
| Derived length: | $2$ |
The quotient is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence metabelian).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{112}.C_{12}.C_2^4$, of order \(21504\)\(\medspace = 2^{10} \cdot 3 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3584\)\(\medspace = 2^{9} \cdot 7 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{224}$ |
| Normalizer: | $C_7:Q_{64}$ |
| Complements: | $Q_{64}$ |
| Minimal over-subgroups: | $C_{14}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_7:Q_{64}$ |