Subgroup ($H$) information
| Description: | $C_4$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Index: | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$c$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, and a $p$-group.
Ambient group ($G$) information
| Description: | $C_4^2:F_{11}$ |
| Order: | \(1760\)\(\medspace = 2^{5} \cdot 5 \cdot 11 \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_4\times F_{11}$ |
| Order: | \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Exponent: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Automorphism Group: | $C_2^2\times F_{11}$, of order \(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2^3\times C_{11}:C_5).C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(S)$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3520\)\(\medspace = 2^{6} \cdot 5 \cdot 11 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $C_4\times F_{11}$ |