Subgroup ($H$) information
| Description: | $D_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Index: | \(3960\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,6,7)(2,5,8)(4,9,11)(12,14), (12,14), (1,6)(3,10)(5,8)(9,11)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, an A-group, and rational.
Ambient group ($G$) information
| Description: | $S_3\times M_{11}$ |
| Order: | \(47520\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| Exponent: | \(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \) |
| Derived length: | $2$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times M_{11}$, of order \(47520\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \cdot 11 \) |
| $\operatorname{Aut}(H)$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $660$ |
| Möbius function | $6$ |
| Projective image | $S_3\times M_{11}$ |