Subgroup ($H$) information
| Description: | $Q_{64}$ |
| Order: | \(64\)\(\medspace = 2^{6} \) |
| Index: | \(3\) |
| Exponent: | \(32\)\(\medspace = 2^{5} \) |
| Generators: |
$a, b$
|
| Nilpotency class: | $5$ |
| Derived length: | $2$ |
The subgroup is maximal, 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: | $C_3:Q_{64}$ |
| Order: | \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Exponent: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times D_{16}:C_8$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| $\operatorname{Aut}(H)$ | $D_{32}:C_8$, of order \(512\)\(\medspace = 2^{9} \) |
| $\operatorname{res}(S)$ | $D_{16}:C_8$, of order \(256\)\(\medspace = 2^{8} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $D_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $Q_{64}$ | ||
| Normal closure: | $C_3:Q_{64}$ | ||
| Core: | $Q_{32}$ | ||
| Minimal over-subgroups: | $C_3:Q_{64}$ | ||
| Maximal under-subgroups: | $Q_{32}$ | $Q_{32}$ | $C_{32}$ |
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $-1$ |
| Projective image | $C_3:D_{16}$ |