Subgroup ($H$) information
| Description: | $C_{11}^2$ |
| Order: | \(121\)\(\medspace = 11^{2} \) |
| Index: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Exponent: | \(11\) |
| Generators: |
$bd^{2}, cd^{10}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_{11}^3:C_{10}$ |
| Order: | \(13310\)\(\medspace = 2 \cdot 5 \cdot 11^{3} \) |
| 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.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^3.C_5^3.C_2^3$ |
| $\operatorname{Aut}(H)$ | $\GL(2,11)$, of order \(13200\)\(\medspace = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{11}^3$ | |||
| Normalizer: | $C_{11}^3:C_2$ | |||
| Normal closure: | $C_{11}^3$ | |||
| Core: | $C_1$ | |||
| Minimal over-subgroups: | $C_{11}^3$ | $C_{11}:D_{11}$ | ||
| Maximal under-subgroups: | $C_{11}$ | $C_{11}$ | $C_{11}$ | $C_{11}$ |
Other information
| Number of subgroups in this autjugacy class | $100$ |
| Number of conjugacy classes in this autjugacy class | $20$ |
| Möbius function | $0$ |
| Projective image | $C_{11}^3:C_{10}$ |