Subgroup ($H$) information
| Description: | $C_2\times \OD_{16}$ |
| Order: | \(32\)\(\medspace = 2^{5} \) |
| Index: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Generators: |
$ad^{25}, b, d^{50}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_2^4.C_{100}$ |
| Order: | \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| Exponent: | \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The ambient group is nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_{50}$ |
| Order: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Exponent: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Automorphism Group: | $C_{20}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Outer Automorphisms: | $C_{20}$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,5$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5:(C_2^4.C_2^5.C_2^3)$ |
| $\operatorname{Aut}(H)$ | $C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\operatorname{res}(S)$ | $C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| $W$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_2^2\times C_{50}$ |