Subgroup ($H$) information
| Description: | $C_9$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Index: | \(201600\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$\langle(2,9,4)(3,6,5)(7,8,10), (2,8,3,9,10,6,4,7,5)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group) and a $p$-group.
Ambient group ($G$) information
| Description: | $A_{10}$ |
| Order: | \(1814400\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5^{2} \cdot 7 \) |
| Exponent: | \(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_{10}$, of order \(3628800\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_9$ | |
| Normalizer: | $C_9:C_6$ | |
| Normal closure: | $A_{10}$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $C_9:C_3$ | $D_9$ |
| Maximal under-subgroups: | $C_3$ | |
| Autjugate subgroups: | 1814400.a.201600.d1.b1 |
Other information
| Number of subgroups in this conjugacy class | $33600$ |
| Möbius function | $0$ |
| Projective image | $A_{10}$ |