Subgroup ($H$) information
| Description: | $C_3.S_{14}$ |
| Order: | \(261534873600\)\(\medspace = 2^{11} \cdot 3^{6} \cdot 5^{2} \cdot 7^{2} \cdot 11 \cdot 13 \) |
| Index: | \(12240\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 17 \) |
| Exponent: | \(360360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \) |
| Generators: |
$\langle(2,3)(5,6), (2,3,4), (2,3)(5,6,7,8,9,10,11,12,13,14,15,16,17,18)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian and nonsolvable.
Ambient group ($G$) information
| Description: | $A_{18}$ |
| Order: | \(3201186852864000\)\(\medspace = 2^{15} \cdot 3^{8} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \) |
| Exponent: | \(12252240\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
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 \(6402373705728000\)\(\medspace = 2^{16} \cdot 3^{8} \cdot 5^{3} \cdot 7^{2} \cdot 11 \cdot 13 \cdot 17 \) |
| $\operatorname{Aut}(H)$ | Group of order \(523069747200\)\(\medspace = 2^{12} \cdot 3^{6} \cdot 5^{2} \cdot 7^{2} \cdot 11 \cdot 13 \) |
| $\card{W}$ | 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 | $12240$ |
| Möbius function | not computed |
| Projective image | not computed |