Subgroup ($H$) information
| Description: | $A_4^2:D_{10}$ |
| 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: |
$\langle(1,2)(3,4), (2,3,4)(5,7)(6,8)(10,12,14)(11,13,15), (2,4,3)(10,11)(12,15) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5.A_6.C_2^2\times S_4$ |
| $\operatorname{Aut}(H)$ | $F_5\times S_4^2$, of order \(11520\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \) |
| $W$ | $A_4\times F_5\times S_4$, of order \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $30$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $A_4\times F_5\times S_6$ |