Subgroup ($H$) information
| Description: | $A_5:\SD_{16}$ |
| Order: | \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
| Index: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(6,15)(7,8)(10,13)(11,14), (6,15)(9,12)(10,11)(13,14), (1,5)(6,11,8,13,15,14,7,10) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian and nonsolvable.
Ambient group ($G$) information
| Description: | $S_6:S_5$ |
| Order: | \(86400\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_5\times S_6:C_2$, of order \(172800\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5^{2} \) |
| $\operatorname{Aut}(H)$ | $C_2\times D_4\times S_5$, of order \(1920\)\(\medspace = 2^{7} \cdot 3 \cdot 5 \) |
| $W$ | $C_2^3:S_5$, of order \(960\)\(\medspace = 2^{6} \cdot 3 \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $45$ |
| Möbius function | $1$ |
| Projective image | $S_6:S_5$ |