Subgroup ($H$) information
| Description: | $Q_{32}$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(46\)\(\medspace = 2 \cdot 23 \) |
| Exponent: | \(16\)\(\medspace = 2^{4} \) |
| Generators: |
$a, b^{690}$
|
| Nilpotency class: | $4$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $C_{23}:Q_{64}$ |
| Order: | \(1472\)\(\medspace = 2^{6} \cdot 23 \) |
| Exponent: | \(736\)\(\medspace = 2^{5} \cdot 23 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{368}.C_{44}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $D_{16}:C_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\operatorname{res}(S)$ | $D_{16}:C_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| $W$ | $D_{16}$, of order \(32\)\(\medspace = 2^{5} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $23$ |
| Möbius function | $1$ |
| Projective image | $D_{368}$ |