Subgroup ($H$) information
| Description: | $C_4:C_{20}$ |
| Order: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$b^{5}, b^{4}, b^{10}, b^{10}c^{15}, c^{10}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is normal, nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $C_{20}^2.C_2^2$ |
| Order: | \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_5:C_4$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Automorphism Group: | $C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{10}^2.C_2^4.C_2^4.S_5$ |
| $\operatorname{Aut}(H)$ | $C_2^3.C_2^4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\operatorname{res}(S)$ | $C_2^3.C_2^4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2000\)\(\medspace = 2^{4} \cdot 5^{3} \) |
| $W$ | $C_2^2:C_4$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
| Centralizer: | $C_{10}^2$ | ||
| Normalizer: | $C_{20}^2.C_2^2$ | ||
| Minimal over-subgroups: | $C_{20}:C_{20}$ | $C_{20}:Q_8$ | |
| Maximal under-subgroups: | $C_2\times C_{20}$ | $C_2\times C_{20}$ | $C_4:C_4$ |
Other information
| Number of subgroups in this autjugacy class | $12$ |
| Number of conjugacy classes in this autjugacy class | $12$ |
| Möbius function | $0$ |
| Projective image | $C_{10}^2:C_4$ |