Subgroup ($H$) information
| Description: | $C_{110}$ |
| Order: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Index: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Generators: |
$c^{55}, c^{10}, c^{22}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a semidirect factor, and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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.
Quotient group ($Q$) structure
| Description: | $C_2\times F_{11}$ |
| Order: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(110\)\(\medspace = 2 \cdot 5 \cdot 11 \) |
| Automorphism Group: | $C_2\times F_{11}$, of order \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Outer Automorphisms: | $C_2$, of order \(2\) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{22}^2.C_{15}.C_5.C_{20}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{20}$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | $22$ |
| Projective image | $C_{22}:F_{11}$ |