Subgroup ($H$) information
| Description: | $S_3^2:S_3\times A_6$ |
| Order: | \(77760\)\(\medspace = 2^{6} \cdot 3^{5} \cdot 5 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(10,12,11), (1,6)(2,3)(7,8,9)(10,13,11,15,12,14), (7,8)(13,15), (1,6,2,5) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and nonsolvable.
Ambient group ($G$) information
| Description: | $C_3^3:S_4\times S_6$ |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:C_2^2.D_6.A_6.C_2^2$ |
| $\operatorname{Aut}(H)$ | $(S_3\times S_3\wr C_2).A_6.C_2^2$ |
| $W$ | $S_3^2:S_3.A_6.C_2$, of order \(155520\)\(\medspace = 2^{7} \cdot 3^{5} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^3:S_4\times S_6$ |