Subgroup ($H$) information
| Description: | $C_{44}$ |
| Order: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Index: | \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| Exponent: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Generators: |
$c^{11}, c^{22}, c^{4}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a semidirect factor, and cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,11$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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: | $C_{10}^2$ |
| Order: | \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $S_3\times \GL(2,5)$, of order \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| Outer Automorphisms: | $S_3\times \GL(2,5)$, of order \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and metacyclic.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{55}.(C_2^4\times C_{20}).C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\operatorname{res}(S)$ | $C_2\times C_{10}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3520\)\(\medspace = 2^{6} \cdot 5 \cdot 11 \) |
| $W$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2\times C_{220}$ | |||
| Normalizer: | $C_{44}:C_{10}^2$ | |||
| Complements: | $C_{10}^2$ | |||
| Minimal over-subgroups: | $C_{220}$ | $C_{11}:C_{20}$ | $C_2\times C_{44}$ | $D_{44}$ |
| Maximal under-subgroups: | $C_{22}$ | $C_4$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $10$ |
| Projective image | $C_{22}:C_{10}^2$ |