Subgroup ($H$) information
| Description: | $A_9^2.C_4$ |
| Order: | \(131681894400\)\(\medspace = 2^{14} \cdot 3^{8} \cdot 5^{2} \cdot 7^{2} \) |
| Index: | \(24310\)\(\medspace = 2 \cdot 5 \cdot 11 \cdot 13 \cdot 17 \) |
| Exponent: | \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Generators: |
$\langle(1,2,3,4,5,6,7,8,9), (1,11,2,10)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18), (10,11,12,13,14,15,16,17,18)\rangle$
|
| Derived length: | $1$ |
The subgroup is maximal, 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 \(263363788800\)\(\medspace = 2^{15} \cdot 3^{8} \cdot 5^{2} \cdot 7^{2} \) |
| $\card{W}$ | 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 | $24310$ |
| Möbius function | not computed |
| Projective image | not computed |