Subgroup ($H$) information
| Description: | $S_3\times F_5\times F_7$ |
| Order: | \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Generators: |
$\langle(1,2,3), (1,2), (9,14,12)(10,13,11), (1,2)(4,7,5,8,6), (9,11,12,10,13,15,14) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $S_3\times F_7\times S_5$ |
| Order: | \(30240\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $2$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times F_7\times S_5$, of order \(30240\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $S_3\times F_5\times F_7$, of order \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
| $W$ | $S_3\times F_5\times F_7$, of order \(5040\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $-1$ |
| Projective image | $S_3\times F_7\times S_5$ |