Subgroup ($H$) information
| Description: | $D_5$ |
| Order: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Index: | \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$b^{5}cd^{9}, c^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $C_5\times D_{10}^2$ |
| Order: | \(2000\)\(\medspace = 2^{4} \cdot 5^{3} \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{10}^2.A_4.C_4^3.C_2^2$ |
| $\operatorname{Aut}(H)$ | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\operatorname{res}(S)$ | $F_5$, of order \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| $W$ | $D_5$, of order \(10\)\(\medspace = 2 \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2^2\times C_{10}$ | |||
| Normalizer: | $C_{10}^2:C_2^2$ | |||
| Normal closure: | $C_5:D_5$ | |||
| Core: | $C_5$ | |||
| Minimal over-subgroups: | $C_5:D_5$ | $C_5\times D_5$ | $D_{10}$ | $D_{10}$ |
| Maximal under-subgroups: | $C_5$ | $C_2$ |
Other information
| Number of subgroups in this autjugacy class | $40$ |
| Number of conjugacy classes in this autjugacy class | $8$ |
| Möbius function | $-8$ |
| Projective image | $C_5\times D_{10}^2$ |