Subgroup ($H$) information
| Description: | $A_4$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Index: | \(240\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,4)(3,5), (1,5)(3,4), (1,3,5)(6,12,8)(7,13,9)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_2^4:\GL(2,4)$ |
| Order: | \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(30\)\(\medspace = 2 \cdot 3 \cdot 5 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, an A-group, 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times S_4\times S_5$, of order \(17280\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\operatorname{res}(S)$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| $W$ | $A_4$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2\times C_6$ | ||
| Normalizer: | $C_2^2:C_6^2$ | ||
| Normal closure: | $A_4\times A_5$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_2^2:A_4$ | $C_3\times A_4$ | $C_2\times A_4$ |
| Maximal under-subgroups: | $C_2^2$ | $C_3$ |
Other information
| Number of subgroups in this autjugacy class | $40$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_2^4:\GL(2,4)$ |