Subgroup ($H$) information
| Description: | $A_4\times C_3^3:D_6$ |
| Order: | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| Index: | \(20\)\(\medspace = 2^{2} \cdot 5 \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(1,2,4,7,6,9)(3,5,8)(11,12,14), (1,8,2,4,3,7)(5,9,6)(11,14)(12,13), (1,8,4,3,6,5) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $A_5\times S_3\wr S_3$ |
| Order: | \(77760\)\(\medspace = 2^{6} \cdot 3^{5} \cdot 5 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\wr S_3\times S_5$, of order \(155520\)\(\medspace = 2^{7} \cdot 3^{5} \cdot 5 \) |
| $\operatorname{Aut}(H)$ | $C_6^2.C_3^4.C_2^3$ |
| $W$ | $A_4\times C_3^3:D_6$, of order \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $20$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $A_5\times S_3\wr S_3$ |