Subgroup ($H$) information
| Description: | $C_2\times C_{10}^2$ |
| Order: | \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
| Index: | \(121\)\(\medspace = 11^{2} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$a^{5}, c^{55}, a^{2}, b^{11}, c^{22}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a Hall subgroup.
Ambient group ($G$) information
| Description: | $C_2\times C_{110}:F_{11}$ |
| Order: | \(24200\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 11^{2} \) |
| 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.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{22}^2.C_5.C_{30}.C_{10}.C_2^4$ |
| $\operatorname{Aut}(H)$ | $\GL(2,5)\times \GL(3,2)$, of order \(80640\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $121$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_{11}:F_{11}$ |