Subgroup ($H$) information
| Description: | $S_3\times A_4$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Index: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,4,12)(2,11,10)(3,8,9)(5,7,6), (1,9)(2,7)(3,4)(5,10)(6,11)(8,12), (1,4)(2,11)(3,9)(6,7), (2,7,9)(3,11,6)(5,8,10), (1,7)(2,9)(3,11)(4,6)(5,12)(8,10)\rangle$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $M_{12}$ |
| Order: | \(95040\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $0$ |
The ambient group is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $M_{12}:C_2$, of order \(190080\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $S_3\times S_4$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $W$ | $S_3\times A_4$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_1$ | |||
| Normalizer: | $S_3\times A_4$ | |||
| Normal closure: | $M_{12}$ | |||
| Core: | $C_1$ | |||
| Minimal over-subgroups: | $M_{12}$ | |||
| Maximal under-subgroups: | $C_3\times A_4$ | $C_2\times A_4$ | $C_2\times D_6$ | $C_3\times S_3$ |
Other information
| Number of subgroups in this conjugacy class | $1320$ |
| Möbius function | $-1$ |
| Projective image | $M_{12}$ |