Subgroup ($H$) information
| Description: | $S_6:C_2$ |
| Order: | \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| Index: | \(66\)\(\medspace = 2 \cdot 3 \cdot 11 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,2)(4,12)(5,11)(6,10), (1,9,6,5,10,3,2,8,4,7)(11,12)\rangle$
|
| Derived length: | $1$ |
The subgroup is maximal, nonabelian, almost simple, nonsolvable, and rational.
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 \(190080\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $S_6:C_2$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| $W$ | $S_6:C_2$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $66$ |
| Möbius function | $-1$ |
| Projective image | $M_{12}$ |