Subgroup ($H$) information
| Description: | $C_5:D_5^2$ |
| Order: | \(500\)\(\medspace = 2^{2} \cdot 5^{3} \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Generators: |
$ad^{9}, b^{2}e, c^{2}, b^{5}cd^{9}, e$
|
| Derived length: | $2$ |
The subgroup is normal, a semidirect factor, 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.
Quotient group ($Q$) structure
| Description: | $D_{10}$ |
| Order: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Automorphism Group: | $C_2\times F_5$, of order \(40\)\(\medspace = 2^{3} \cdot 5 \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $2$ |
The quotient is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, 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 \(3840000\)\(\medspace = 2^{11} \cdot 3 \cdot 5^{4} \) |
| $\operatorname{Aut}(H)$ | $F_5\wr S_3$, of order \(48000\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{3} \) |
| $W$ | $D_5^3$, of order \(1000\)\(\medspace = 2^{3} \cdot 5^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $-10$ |
| Projective image | $D_5^4$ |