Subgroup ($H$) information
| Description: | $C_8:C_4$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$ab^{5}, c^{25}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is normal, a direct factor, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $C_4.C_{20}^2$ |
| Order: | \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, nilpotent (hence solvable, supersolvable, and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_5\times C_{10}$ |
| Order: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $\GL(2,5)$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Outer Automorphisms: | $\GL(2,5)$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 5$ (hence hyperelementary), 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^5.C_2^6.C_2^2.S_5$ |
| $\operatorname{Aut}(H)$ | $C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\operatorname{res}(S)$ | $C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_2\times C_{10}\times C_{20}$ | |
| Normalizer: | $C_4.C_{20}^2$ | |
| Complements: | $C_5\times C_{10}$ | |
| Minimal over-subgroups: | $C_8:C_{20}$ | $C_4.C_4^2$ |
| Maximal under-subgroups: | $C_4^2$ | $C_2\times C_8$ |
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $-5$ |
| Projective image | $C_2\times C_{10}^2$ |