Subgroup ($H$) information
| Description: | $C_2\times A_5$ |
| Order: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Index: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(2,3)(4,5)(6,8)(7,9), (1,12,14)(2,3)(4,5)(6,8)(7,9), (2,3)(4,5)(6,8)(7,9)(11,12)(13,14)\rangle$
|
| Derived length: | $1$ |
The subgroup is nonabelian, an A-group, and nonsolvable.
Ambient group ($G$) information
| Description: | $A_4\times A_5^2$ |
| Order: | \(43200\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, an A-group, and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_4\times S_5\wr C_2$, of order \(691200\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $S_5$, of order \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| $W$ | $A_5$, of order \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $90$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $A_4\times A_5^2$ |