Subgroup ($H$) information
| Description: | $Q_8\times F_{11}$ |
| Order: | \(880\)\(\medspace = 2^{4} \cdot 5 \cdot 11 \) |
| Index: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Generators: |
$a^{5}c^{80}, b^{2}c^{198}, c^{55}, a^{2}c^{12}, b^{11}c^{24}, c^{110}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Ambient group ($G$) information
| Description: | $C_{55}:(Q_8\times F_{11})$ |
| Order: | \(48400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 11^{2} \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_{10}^2.C_{10}.C_2^6$ |
| $\operatorname{Aut}(H)$ | $C_2\times S_4\times F_{11}$, of order \(5280\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 11 \) |
| $W$ | $C_2^2\times F_{11}$, of order \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $110$ |
| Number of conjugacy classes in this autjugacy class | $10$ |
| Möbius function | $1$ |
| Projective image | $C_5\times D_{22}:F_{11}$ |