Subgroup ($H$) information
| Description: | $C_{22}:C_{10}$ |
| Order: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Generators: |
$a^{10}, a^{4}, c^{2}, c^{11}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and an A-group.
Ambient group ($G$) information
| Description: | $C_2^3.F_{11}$ |
| Order: | \(880\)\(\medspace = 2^{4} \cdot 5 \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2\wr C_2\times F_{11}$, of order \(3520\)\(\medspace = 2^{6} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $S_3\times F_{11}$, of order \(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \) |
| $\operatorname{res}(S)$ | $C_2\times F_{11}$, of order \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_{11}:C_5$, of order \(55\)\(\medspace = 5 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $D_{22}:C_{10}$ |