Subgroup ($H$) information
| Description: | $C_3^3:A_4\times A_6$ |
| Order: | \(116640\)\(\medspace = 2^{5} \cdot 3^{6} \cdot 5 \) |
| Index: | \(1120\)\(\medspace = 2^{5} \cdot 5 \cdot 7 \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Generators: |
$\langle(1,5)(2,6,3,4)(7,15,12,9,14,11,8,13,10), (7,14,11)(8,13,10)(9,15,12), (8,9) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and nonsolvable.
Ambient group ($G$) information
| Description: | $S_6\times A_9$ |
| Order: | \(130636800\)\(\medspace = 2^{10} \cdot 3^{6} \cdot 5^{2} \cdot 7 \) |
| Exponent: | \(1260\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Derived length: | $1$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_9.A_6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $C_3^3:C_2^2.D_6.A_6.C_2^2$ |
| $\card{W}$ | 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 | $280$ |
| Möbius function | not computed |
| Projective image | not computed |