Subgroup ($H$) information
| Description: | $S_4\times S_5\wr C_2$ |
| Order: | \(691200\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5^{2} \) |
| Index: | \(1620\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(7,12,11,8)(9,10)(13,16)(15,17), (7,11)(8,12), (7,12)(8,11), (8,12,11), (1,18,6,17,5,15,2,16,4,13,3,14) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, nonsolvable, and rational.
Ambient group ($G$) information
| Description: | $A_6^3.S_4$ |
| Order: | \(1119744000\)\(\medspace = 2^{12} \cdot 3^{7} \cdot 5^{3} \) |
| Exponent: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | Group of order \(4478976000\)\(\medspace = 2^{14} \cdot 3^{7} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $S_4\times S_5\wr C_2$, of order \(691200\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5^{2} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $1620$ |
| Möbius function | not computed |
| Projective image | not computed |