Subgroup ($H$) information
| Description: | $C_{20}$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Index: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$a^{5}b^{5}c^{37}, c^{22}, b^{2}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), 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.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{110}.C_{10}.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\operatorname{res}(S)$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(320\)\(\medspace = 2^{6} \cdot 5 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $55$ |
| Number of conjugacy classes in this autjugacy class | $5$ |
| Möbius function | $2$ |
| Projective image | $C_{22}:C_{10}^2$ |