Subgroup ($H$) information
| Description: | $A_6$ |
| Order: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Index: | \(44\)\(\medspace = 2^{2} \cdot 11 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(2,9)(3,7)(5,6)(10,11), (2,4,6,11)(7,8,10,9)\rangle$
|
| Derived length: | $0$ |
The subgroup is nonabelian and simple (hence nonsolvable, perfect, quasisimple, and almost simple).
Ambient group ($G$) information
| Description: | $C_2\times M_{11}$ |
| Order: | \(15840\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $1$ |
The ambient group is nonabelian and nonsolvable.
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 44T140.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $M_{11}$, of order \(7920\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $S_6:C_2$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
| $W$ | $A_6.C_2$, of order \(720\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $A_6.C_2^2$ | ||
| Normal closure: | $M_{11}$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_2\times A_6$ | $A_6.C_2$ | $A_6.C_2$ |
| Maximal under-subgroups: | $A_5$ | $C_3^2:C_4$ | $S_4$ |
Other information
| Number of subgroups in this conjugacy class | $11$ |
| Möbius function | $0$ |
| Projective image | $C_2\times M_{11}$ |