Subgroup ($H$) information
| Description: | $C_5\times A_4$ |
| Order: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Index: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,4)(2,3), (1,3)(2,4), (1,4,2), (5,7,8,6,9)\rangle$
|
| Derived length: | $2$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, monomial, metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_5\times A_4\times A_6$ |
| Order: | \(21600\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
Quotient group ($Q$) structure
| Description: | $A_6$ |
| Order: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Automorphism Group: | $S_6:C_2$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| Outer Automorphisms: | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| Derived length: | $0$ |
The quotient is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_4\times A_6.C_2^2\times S_4$ |
| $\operatorname{Aut}(H)$ | $C_4\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $W$ | $A_4$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $720$ |
| Projective image | $A_4\times A_6$ |