Subgroup ($H$) information
| Description: | $C_2\times Q_{16}$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$a, b^{5}, c^{11}$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $2$-Sylow subgroup (hence nilpotent, solvable, supersolvable, a Hall subgroup, and monomial), a $p$-group (hence elementary and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_{11}:(C_{10}\times Q_{16})$ |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}.(C_{10}\times D_4).C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_4.D_4^2$, of order \(256\)\(\medspace = 2^{8} \) |
| $\operatorname{res}(S)$ | $D_4^2:C_2$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(10\)\(\medspace = 2 \cdot 5 \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $11$ |
| Möbius function | $1$ |
| Projective image | $D_{22}:C_{10}$ |