Subgroup ($H$) information
| Description: | $\GL(2,3)$ |
| Order: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Index: | \(330\)\(\medspace = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(1,8,9,5)(4,6,11,10), (1,9)(4,11)(5,8)(6,10), (1,8)(2,7)(4,11)(5,9), (1,4,9,11)(5,6,8,10), (1,5,11)(2,7,3)(4,9,8)\rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and solvable.
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 \(7920\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $C_2\times S_4$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $W$ | $S_4$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
Related subgroups
| Centralizer: | $C_2^2$ | ||
| Normalizer: | $C_2\times \GL(2,3)$ | ||
| Normal closure: | $M_{11}$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $M_{11}$ | $C_2\times \GL(2,3)$ | |
| Maximal under-subgroups: | $\SL(2,3)$ | $\SD_{16}$ | $D_6$ |
Other information
| Number of subgroups in this conjugacy class | $165$ |
| Möbius function | $1$ |
| Projective image | $C_2\times M_{11}$ |