Subgroup ($H$) information
| Description: | $A_4$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Index: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,9,7)(3,4,6)(5,8,10)(11,12,13), (1,9)(2,7)(3,5)(6,10), (1,2)(3,5)(4,8)(7,9)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $A_6:S_3$ |
| Order: | \(2160\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_6:D_6$, of order \(8640\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\operatorname{res}(S)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(6\)\(\medspace = 2 \cdot 3 \) |
| $W$ | $A_4$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_3$ | |
| Normalizer: | $C_3\times A_4$ | |
| Normal closure: | $C_3\times A_6$ | |
| Core: | $C_1$ | |
| Minimal over-subgroups: | $C_3\times A_4$ | |
| Maximal under-subgroups: | $C_2^2$ | $C_3$ |
Other information
| Number of subgroups in this conjugacy class | $60$ |
| Möbius function | $0$ |
| Projective image | $A_6:S_3$ |