Subgroup ($H$) information
| Description: | $A_6.S_5$ |
| Order: | \(43200\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5^{2} \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(3,8,6,7)(4,10,5,9)(14,18)(15,17)(19,20), (1,7)(2,6)(3,5)(8,10)(11,16,13,20,18)(12,19,17,15,14)\rangle$
|
| Derived length: | $1$ |
The subgroup is nonabelian and nonsolvable.
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 \(4147200\)\(\medspace = 2^{11} \cdot 3^{4} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $S_5\times S_6:C_2$, of order \(172800\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5^{2} \) |
| $W$ | $\PGL(2,9):S_5$, of order \(86400\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $24$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $\PGL(2,9)\wr C_2:C_2$ |