Subgroup ($H$) information
| Description: | $Q_{64}$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(3\) |
| Exponent: | \(32\)\(\medspace = 2^{5} \) |
| Generators: |
$a, b^{3}$
|
| Nilpotency class: | $5$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $C_3\times Q_{64}$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Nilpotency class: | $5$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
Quotient group ($Q$) structure
| Description: | $C_3$ |
| Order: | \(3\) |
| Exponent: | \(3\) |
| Automorphism Group: | $C_2$, of order \(2\) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| 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), a $p$-group, and simple.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times D_{32}:C_8$, of order \(1024\)\(\medspace = 2^{10} \) |
| $\operatorname{Aut}(H)$ | $D_{32}:C_8$, of order \(512\)\(\medspace = 2^{9} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_{32}:C_8$, of order \(512\)\(\medspace = 2^{9} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
Related subgroups
| Centralizer: | $C_6$ | ||
| Normalizer: | $C_3\times Q_{64}$ | ||
| Complements: | $C_3$ | ||
| Minimal over-subgroups: | $C_3\times Q_{64}$ | ||
| Maximal under-subgroups: | $Q_{32}$ | $Q_{32}$ | $C_{32}$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_3\times D_{16}$ |