Subgroup ($H$) information
| Description: | $F_{11}$ |
| Order: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Index: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Generators: |
$a^{5}b^{5}c^{61}, c^{6}, a^{2}b^{6}$
|
| Derived length: | $2$ |
The subgroup is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).
Ambient group ($G$) information
| Description: | $C_{66}:C_{10}^2$ |
| Order: | \(6600\)\(\medspace = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| Exponent: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^2.C_{165}.C_{60}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $F_{11}$, of order \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| $W$ | $F_{11}$, of order \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $60$ |
| Number of conjugacy classes in this autjugacy class | $20$ |
| Möbius function | $2$ |
| Projective image | $C_{66}:C_{10}^2$ |