Subgroup ($H$) information
| Description: | $Q_{16}$ |
| Order: | \(16\)\(\medspace = 2^{4} \) |
| Index: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$a^{5}c, b^{11}$
|
| Nilpotency class: | $3$ |
| 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: | $\SD_{16}:F_{11}$ |
| Order: | \(1760\)\(\medspace = 2^{5} \cdot 5 \cdot 11 \) |
| Exponent: | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{22}.(C_2^4\times C_{10})$ |
| $\operatorname{Aut}(H)$ | $D_8:C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| $\operatorname{res}(S)$ | $C_2\times D_4$, of order \(16\)\(\medspace = 2^{4} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_{20}$ | ||
| Normalizer: | $D_8:C_{10}$ | ||
| Normal closure: | $C_{11}:Q_{16}$ | ||
| Core: | $Q_8$ | ||
| Minimal over-subgroups: | $C_{11}:Q_{16}$ | $C_5\times Q_{16}$ | $D_8:C_2$ |
| Maximal under-subgroups: | $Q_8$ | $Q_8$ | $C_8$ |
Other information
| Number of subgroups in this conjugacy class | $11$ |
| Möbius function | $-1$ |
| Projective image | $D_4\times F_{11}$ |