Subgroup ($H$) information
| Description: | $A_6^2.D_4$ |
| Order: | \(1036800\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5^{2} \) |
| Index: | \(270\)\(\medspace = 2 \cdot 3^{3} \cdot 5 \) |
| Exponent: | \(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,4)(2,6,3,5), (7,12)(9,11)(14,16)(15,18), (1,6)(3,5)(7,12)(8,10), (7,12) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian and nonsolvable.
Ambient group ($G$) information
| Description: | $A_6^3.S_3$ |
| Order: | \(279936000\)\(\medspace = 2^{10} \cdot 3^{7} \cdot 5^{3} \) |
| Exponent: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $A_6\wr C_3.C_2^3$, of order \(1119744000\)\(\medspace = 2^{12} \cdot 3^{7} \cdot 5^{3} \) |
| $\operatorname{Aut}(H)$ | $C_2^2\times A_6^2.C_2^3$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $135$ |
| Möbius function | not computed |
| Projective image | not computed |