Subgroup ($H$) information
| Description: | $C_{11}$ |
| Order: | \(11\) |
| Index: | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| Exponent: | \(11\) |
| Generators: |
$c^{4}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $11$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_{44}:C_{10}^2$ |
| Order: | \(4400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 11 \) |
| 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: | $C_4:C_{10}^2$ |
| Order: | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Automorphism Group: | $C_2^5.C_2^3.S_5$ |
| Outer Automorphisms: | $C_2\times D_4\times \GL(2,5)$, of order \(7680\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The quotient is nonabelian, nilpotent (hence solvable, supersolvable, and monomial), and metabelian.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{55}.(C_2^4\times C_{20}).C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(14080\)\(\medspace = 2^{8} \cdot 5 \cdot 11 \) |
| $W$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2\times C_{220}$ | ||||
| Normalizer: | $C_{44}:C_{10}^2$ | ||||
| Complements: | $C_4:C_{10}^2$ | ||||
| Minimal over-subgroups: | $C_{55}$ | $C_{11}:C_5$ | $C_{22}$ | $C_{22}$ | $D_{11}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_{44}:C_{10}^2$ |