Subgroup ($H$) information
| Description: | $C_3:S_5$ |
| Order: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Index: | \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(2,3,6), (2,3)(4,7)(5,8)(9,10), (2,3)(4,9,10,8,7,5)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian and nonsolvable.
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 \(3628800\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $S_3\times S_5$, of order \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
| $W$ | $C_3:S_5$, of order \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_1$ | ||||
| Normalizer: | $C_3:S_5$ | ||||
| Normal closure: | $A_{10}$ | ||||
| Core: | $C_1$ | ||||
| Minimal over-subgroups: | $C_3:S_6$ | $A_4:S_5$ | |||
| Maximal under-subgroups: | $\GL(2,4)$ | $S_5$ | $C_3:S_4$ | $C_{15}:C_4$ | $S_3^2$ |
Other information
| Number of subgroups in this conjugacy class | $5040$ |
| Möbius function | $0$ |
| Projective image | $A_{10}$ |