Subgroup ($H$) information
| Description: | $A_4\times D_6\times F_5$ |
| Order: | \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| Index: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(2,4,3)(10,11)(12,15)(13,14), (1,2)(3,4), (1,3)(2,4), (1,4,2)(6,8,9,7)(10,11) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $A_4\times F_5\times S_6$ |
| Order: | \(172800\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5.A_6.C_2^2\times S_4$ |
| $\operatorname{Aut}(H)$ | $C_2^2\times S_3\times F_5\times S_4$ |
| $W$ | $S_3\times A_4\times F_5$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $120$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $A_4\times F_5\times S_6$ |