Subgroup ($H$) information
| Description: | $F_9$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Index: | \(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(1,10)(2,9)(3,5)(8,11), (1,2,11)(3,5,7)(8,9,10), (1,5,10,3)(2,11,9,8), (1,8,3,9,10,11,5,2)(4,6), (1,9,3)(2,10,5)(7,11,8)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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.
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 \(7920\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $W$ | $F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_2$ | ||
| Normalizer: | $F_9:C_2^2$ | ||
| Normal closure: | $M_{11}$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_2\times F_9$ | $F_9:C_2$ | $F_9:C_2$ |
| Maximal under-subgroups: | $C_3^2:C_4$ | $C_8$ |
Other information
| Number of subgroups in this conjugacy class | $55$ |
| Möbius function | $0$ |
| Projective image | $C_2\times M_{11}$ |