Subgroup ($H$) information
| Description: | $C_{11}$ |
| Order: | \(11\) |
| Index: | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Exponent: | \(11\) |
| Generators: |
$c$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{11}^2:C_{40}$ |
| Order: | \(4840\)\(\medspace = 2^{3} \cdot 5 \cdot 11^{2} \) |
| Exponent: | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{121}:C_2$, of order \(29040\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \) |
| $\operatorname{Aut}(H)$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| $\operatorname{res}(S)$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(242\)\(\medspace = 2 \cdot 11^{2} \) |
| $W$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $C_{11}^2:C_{40}$ |