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: |
$ab^{5}, c^{11}$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is normal, a direct factor, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence 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.
Quotient group ($Q$) structure
| Description: | $C_{11}:C_{10}$ |
| Order: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Automorphism Group: | $F_{11}$, of order \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 5$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_{44}).C_{10}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $D_8:C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| $\operatorname{res}(S)$ | $D_8:C_2$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Möbius function | $-11$ |
| Projective image | $C_2\times C_{44}:C_{10}$ |