Subgroup ($H$) information
| Description: | $C_{11}:C_{20}$ |
| Order: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Index: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Generators: |
$a^{5}c^{11}, c^{22}, b, a^{2}c^{22}$
|
| Derived length: | $2$ |
The subgroup is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).
Ambient group ($G$) information
| Description: | $C_{44}.F_{11}$ |
| Order: | \(4840\)\(\medspace = 2^{3} \cdot 5 \cdot 11^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^2.C_{10}^2.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times F_{11}$, of order \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| $\operatorname{res}(S)$ | $C_2\times F_{11}$, of order \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $W$ | $C_2\times F_{11}$, of order \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $11$ |
| Möbius function | $1$ |
| Projective image | $C_{22}:F_{11}$ |