Subgroup ($H$) information
| Description: | $S_3^2\times S_5$ |
| Order: | \(4320\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,7,2)(3,8,6)(4,5,9), (1,4)(2,5)(3,8)(7,9)(10,11), (2,7)(3,9)(4,6)(5,8) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, nonsolvable, and rational.
Ambient group ($G$) information
| Description: | $S_5\times C_3^2:\GL(2,3)$ |
| Order: | \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Derived length: | $5$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_5\times C_3^2:\GL(2,3)$, of order \(51840\)\(\medspace = 2^{7} \cdot 3^{4} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $S_5\times \SOPlus(4,2)$, of order \(8640\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \) |
| $W$ | $S_5\times \SOPlus(4,2)$, of order \(8640\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $2$ |
| Projective image | $S_5\times C_3^2:\GL(2,3)$ |