Subgroup ($H$) information
| Description: | $C_5\times D_4$ |
| Order: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$ac^{5}, d^{4}, d^{10}, d^{5}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is normal, a direct factor, nonabelian, elementary for $p = 2$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), 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: | $D_{10}$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5:(C_2^7.C_2^5)$ |
| $\operatorname{Aut}(H)$ | $C_4\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| $\operatorname{res}(S)$ | $C_4\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $-10$ |
| Projective image | $C_2^2\times D_{10}$ |