Subgroup ($H$) information
| Description: | $C_5$ |
| Order: | \(5\) |
| Index: | \(4000\)\(\medspace = 2^{5} \cdot 5^{3} \) |
| Exponent: | \(5\) |
| Generators: |
$ce$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $(C_5^3\times C_{10}).\OD_{16}$ |
| Order: | \(20000\)\(\medspace = 2^{5} \cdot 5^{4} \) |
| Exponent: | \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4.C_4.C_2^3.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_4$, of order \(4\)\(\medspace = 2^{2} \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $32$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $(C_5^3\times C_{10}).\OD_{16}$ |