Subgroup ($H$) information
| Description: | $C_2$ |
| Order: | \(2\) |
| Index: | \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
| Exponent: | \(2\) |
| Generators: |
$a$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $p$-group, simple, and rational.
Ambient group ($G$) information
| Description: | $C_{10}^2:C_2^2$ |
| Order: | \(400\)\(\medspace = 2^{4} \cdot 5^{2} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_{10}\times D_{10}$ |
| Order: | \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_4\times F_5\times S_4$, of order \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| Outer Automorphisms: | $C_2^4.D_6$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, supersolvable (hence solvable and monomial), metabelian, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times C_2^3.\PSL(2,7)\times F_5$ |
| $\operatorname{Aut}(H)$ | $C_1$, of order $1$ |
| $\operatorname{res}(S)$ | $C_1$, of order $1$ |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(15360\)\(\medspace = 2^{10} \cdot 3 \cdot 5 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{10}^2:C_2^2$ | ||||
| Normalizer: | $C_{10}^2:C_2^2$ | ||||
| Complements: | $C_{10}\times D_{10}$ | ||||
| Minimal over-subgroups: | $C_{10}$ | $C_{10}$ | $C_{10}$ | $C_2^2$ | $C_2^2$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Number of subgroups in this autjugacy class | $7$ |
| Number of conjugacy classes in this autjugacy class | $7$ |
| Möbius function | $-40$ |
| Projective image | $C_{10}\times D_{10}$ |