Subgroup ($H$) information
| Description: | $D_{20}$ |
| Order: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Index: | \(28\)\(\medspace = 2^{2} \cdot 7 \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Generators: |
$abc^{61}, c^{70}, c^{84}, b^{2}c^{70}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_{28}.D_{20}$ |
| Order: | \(1120\)\(\medspace = 2^{5} \cdot 5 \cdot 7 \) |
| Exponent: | \(280\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times C_{70}).C_6.C_2^6$ |
| $\operatorname{Aut}(H)$ | $D_4\times F_5$, of order \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| $\operatorname{res}(S)$ | $D_4\times F_5$, of order \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $W$ | $D_{20}$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_4$ | |||
| Normalizer: | $C_4.D_{20}$ | |||
| Normal closure: | $D_{140}$ | |||
| Core: | $C_{20}$ | |||
| Minimal over-subgroups: | $D_{140}$ | $D_{20}:C_2$ | $C_{40}:C_2$ | $C_{40}:C_2$ |
| Maximal under-subgroups: | $C_{20}$ | $D_{10}$ | $D_4$ |
Other information
| Number of subgroups in this conjugacy class | $7$ |
| Möbius function | $-2$ |
| Projective image | $C_{14}:D_{20}$ |