Subgroup ($H$) information
| Description: | $C_{11}:C_5$ |
| Order: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Index: | \(2662\)\(\medspace = 2 \cdot 11^{3} \) |
| Exponent: | \(55\)\(\medspace = 5 \cdot 11 \) |
| Generators: |
$a^{22}, bd^{10}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 5$.
Ambient group ($G$) information
| Description: | $C_{11}^3:C_{110}$ |
| Order: | \(146410\)\(\medspace = 2 \cdot 5 \cdot 11^{4} \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{11}^3.C_{11}^2.C_{10}^2$ |
| $\operatorname{Aut}(H)$ | $F_{11}$, of order \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| $W$ | $F_{11}$, of order \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
Related subgroups
| Centralizer: | $C_1$ | ||
| Normalizer: | $F_{11}$ | ||
| Normal closure: | $C_{11}^3:C_5$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_{11}^2:C_5$ | $C_{11}^2:C_5$ | $F_{11}$ |
| Maximal under-subgroups: | $C_{11}$ | $C_5$ |
Other information
| Number of subgroups in this autjugacy class | $14641$ |
| Number of conjugacy classes in this autjugacy class | $11$ |
| Möbius function | $0$ |
| Projective image | $C_{11}^3:C_{110}$ |