Subgroup ($H$) information
| Description: | $C_{11}:D_{11}$ |
| Order: | \(242\)\(\medspace = 2 \cdot 11^{2} \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(22\)\(\medspace = 2 \cdot 11 \) |
| Generators: |
$a^{5}, c^{2}, b$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_{22}:F_{11}$ |
| Order: | \(2420\)\(\medspace = 2^{2} \cdot 5 \cdot 11^{2} \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{10}$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Outer Automorphisms: | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{11}^2:C_2^2$, of order \(48400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 11^{2} \) |
| $\operatorname{Aut}(H)$ | $C_{11}^2.\GL(2,11)$, of order \(1597200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{3} \) |
| $\operatorname{res}(S)$ | $F_{11}\wr C_2$, of order \(24200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | $1$ |
| $W$ | $C_{11}:F_{11}$, of order \(1210\)\(\medspace = 2 \cdot 5 \cdot 11^{2} \) |
Related subgroups
Other information
| Möbius function | $1$ |
| Projective image | $C_{22}:F_{11}$ |