Subgroup ($H$) information
| Description: | $C_{20}:C_4$ |
| Order: | \(80\)\(\medspace = 2^{4} \cdot 5 \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$abcd^{50}, b^{2}cd^{50}, d^{50}, d^{25}, d^{20}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $(C_2\times C_4).D_{100}$ |
| Order: | \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \) |
| Exponent: | \(100\)\(\medspace = 2^{2} \cdot 5^{2} \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(256000\)\(\medspace = 2^{11} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $C_2^2\wr C_2\times F_5$, of order \(640\)\(\medspace = 2^{7} \cdot 5 \) |
| $\card{W}$ | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $10$ |
| Möbius function | $0$ |
| Projective image | not computed |