Subgroup ($H$) information
| Description: | $C_5\times A_5$ |
| Order: | \(300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \) |
| Index: | \(6048\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 7 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(2,4,3), (1,3)(2,5)(6,7,9,8,10), (6,8,7,10,9)\rangle$
|
| Derived length: | $1$ |
The subgroup is nonabelian, an A-group, 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 \(3628800\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_4\times S_5$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| $W$ | $A_5:C_4$, of order \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
Related subgroups
| Centralizer: | $C_5$ | |||
| Normalizer: | $A_5:F_5$ | |||
| Normal closure: | $A_{10}$ | |||
| Core: | $C_1$ | |||
| Minimal over-subgroups: | $D_5\times A_5$ | |||
| Maximal under-subgroups: | $C_5\times A_4$ | $A_5$ | $C_5\times D_5$ | $C_5\times S_3$ |
Other information
| Number of subgroups in this conjugacy class | $1512$ |
| Möbius function | $0$ |
| Projective image | $A_{10}$ |