Subgroup ($H$) information
| Description: | $Q_8$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a^{5}, b^{22}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is normal, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), and rational.
Ambient group ($G$) information
| Description: | $C_{88}.C_{10}$ |
| Order: | \(880\)\(\medspace = 2^{4} \cdot 5 \cdot 11 \) |
| Exponent: | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and metacyclic (hence solvable, supersolvable, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_8:C_2\times F_{11}$, of order \(3520\)\(\medspace = 2^{6} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\operatorname{res}(S)$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| $W$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
| Centralizer: | $C_{11}:C_{10}$ | ||
| Normalizer: | $C_{88}.C_{10}$ | ||
| Minimal over-subgroups: | $Q_8\times C_{11}$ | $C_5\times Q_8$ | $Q_{16}$ |
| Maximal under-subgroups: | $C_4$ | $C_4$ | |
| Autjugate subgroups: | 880.30.110.a1.b1 |
Other information
| Möbius function | $-11$ |
| Projective image | $C_{44}:C_{10}$ |