Subgroup ($H$) information
| Description: | $C_5\times D_5^2$ |
| Order: | \(500\)\(\medspace = 2^{2} \cdot 5^{3} \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$acd, e, c^{2}, b^{5}cd^{4}, d^{2}$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $D_5^4$ |
| Order: | \(10000\)\(\medspace = 2^{4} \cdot 5^{4} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4:C_4\wr S_4$, of order \(3840000\)\(\medspace = 2^{11} \cdot 3 \cdot 5^{4} \) |
| $\operatorname{Aut}(H)$ | $C_4\times F_5\wr C_2$, of order \(3200\)\(\medspace = 2^{7} \cdot 5^{2} \) |
| $W$ | $D_5\times D_{10}$, of order \(200\)\(\medspace = 2^{3} \cdot 5^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $120$ |
| Number of conjugacy classes in this autjugacy class | $24$ |
| Möbius function | $-2$ |
| Projective image | $D_5^4$ |