Subgroup ($H$) information
| Description: | $C_2\times C_{50}$ |
| Order: | \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Generators: |
$a, d^{2}, d^{10}, d^{25}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_{10}^2.D_{10}$ |
| Order: | \(2000\)\(\medspace = 2^{4} \cdot 5^{3} \) |
| Exponent: | \(50\)\(\medspace = 2 \cdot 5^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2\times C_{10}$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_4\times S_3$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Outer Automorphisms: | $C_4\times S_3$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_{25}.C_{10}\times C_2^3.\PSL(2,7)$ |
| $\operatorname{Aut}(H)$ | $S_3\times C_{20}$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(S)$ | $S_3\times C_{20}$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
| $W$ | $C_{10}$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $7$ |
| Number of conjugacy classes in this autjugacy class | $7$ |
| Möbius function | $-2$ |
| Projective image | $C_{50}:C_{10}$ |