Subgroup ($H$) information
| Description: | $D_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Index: | \(38880\)\(\medspace = 2^{5} \cdot 3^{5} \cdot 5 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,5)(7,10)(8,12)(9,11), (1,5)(2,4)(3,6)(7,8)(10,15)(11,13)(12,14), (7,10,14)(8,12,15)(9,11,13)\rangle$
|
| 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: | $(S_3^3\times A_6):S_3$ |
| Order: | \(466560\)\(\medspace = 2^{7} \cdot 3^{6} \cdot 5 \) |
| Exponent: | \(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:C_2^2.D_6.A_6.C_2^2$ |
| $\operatorname{Aut}(H)$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | not computed | ||
| Normalizer: | not computed | ||
| Normal closure: | $(S_3^3\times A_6):S_3$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $C_2\times S_4$ | ||
| Maximal under-subgroups: | $S_3$ | $S_3$ | $C_2^2$ |
Other information
| Number of subgroups in this autjugacy class | $9720$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $(S_3^3\times A_6):S_3$ |