Subgroup ($H$) information
| Description: | $M_{11}$ |
| Order: | \(7920\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Generators: |
$\langle(1,11)(2,4)(5,12)(6,7), (1,3,2,8)(5,11,10,6)\rangle$
|
| Derived length: | $0$ |
The subgroup is maximal, nonabelian, and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
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).
Quotient set structure
Since this subgroup has trivial core, the ambient group $G$ acts faithfully and transitively on the set of cosets of $H$. The resulting permutation representation is isomorphic to 12T295.
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 \(190080\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $M_{11}$, of order \(7920\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| $W$ | $M_{11}$, of order \(7920\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $12$ |
| Möbius function | $-1$ |
| Projective image | $M_{12}$ |