Subgroup ($H$) information
| Description: | $A_5$ |
| Order: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Index: | \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,2)(3,4), (2,5,4)\rangle$
|
| Derived length: | $0$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and an A-group.
Ambient group ($G$) information
| Description: | $D_5^2.C_2^2\times S_5$ |
| Order: | \(48000\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $D_5^2.C_2^3$ |
| Order: | \(800\)\(\medspace = 2^{5} \cdot 5^{2} \) |
| Exponent: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Automorphism Group: | $D_5^2.C_2^3.S_4$, of order \(19200\)\(\medspace = 2^{8} \cdot 3 \cdot 5^{2} \) |
| Outer Automorphisms: | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Derived length: | $3$ |
The quotient is nonabelian and solvable. Whether it is monomial has not been computed.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5:D_5.A_4.C_4.S_5$ |
| $\operatorname{Aut}(H)$ | $S_5$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $W$ | $S_5$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $D_5^2.C_2^2\times S_5$ |