Subgroup ($H$) information
| Description: | $A_4:F_5$ |
| Order: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Index: | \(8640\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,5)(2,9)(4,6)(8,10), (2,3)(5,10)(7,9)(12,19,18,14)(13,16,15,20), (1,8) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $\PGL(2,9)\wr C_2:C_2$ |
| Order: | \(2073600\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 5^{2} \) |
| Exponent: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_6^2.D_4$, of order \(4147200\)\(\medspace = 2^{11} \cdot 3^{4} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $F_5\times S_4$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| $W$ | $F_5\times S_4$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $2160$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $\PGL(2,9)\wr C_2:C_2$ |