Subgroup ($H$) information
| Description: | $D_4\times C_5^2$ |
| Order: | \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$ac^{5}, c^{2}, d^{4}, d^{5}, d^{10}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, nonabelian, nilpotent (hence solvable, supersolvable, and monomial), and metacyclic (hence metabelian).
Ambient group ($G$) information
| Description: | $C_{10}^2:C_2^3$ |
| Order: | \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.
Quotient group ($Q$) structure
| Description: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5:(C_2^7.C_2^5)$ |
| $\operatorname{Aut}(H)$ | $D_4\times \GL(2,5)$, of order \(3840\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \) |
| $\operatorname{res}(S)$ | $D_4\times C_4^2$, of order \(128\)\(\medspace = 2^{7} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| $W$ | $C_2^3$, of order \(8\)\(\medspace = 2^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $2$ |
| Projective image | $C_2^2\times D_{10}$ |