Subgroup ($H$) information
| Description: | $S_3\times S_5$ |
| Order: | \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
| Index: | \(160\)\(\medspace = 2^{5} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(2,10)(5,8)(11,12)(13,14), (1,8,6)(2,10)(3,9,4)(5,7)(11,12)(13,14), (3,9,4), (2,10)(3,9)(11,12)(13,14)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, nonsolvable, and rational.
Ambient group ($G$) information
| Description: | $S_5^2:D_4$ |
| Order: | \(115200\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, nonsolvable, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_2\times A_5^2).D_4^2$, of order \(460800\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $S_3\times S_5$, of order \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
| $W$ | $S_3\times S_5$, of order \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $80$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | not computed |
| Projective image | $S_5^2:D_4$ |