Subgroup ($H$) information
| Description: | $C_2\times A_4:S_6$ |
| Order: | \(17280\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5 \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,2,3,4,5)(7,12,13)(8,11,14)(9,10,15), (13,15), (10,12)(13,15), (3,6)(7,9)(10,15,12,13)(11,14), (7,9)(13,15), (7,12,15)(8,11,14)(9,10,13)\rangle$
|
| Derived length: | $3$ |
The subgroup is maximal, nonabelian, and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:C_2^2.D_6.A_6.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_2\times S_4.A_6.C_2^2$ |
| $W$ | $A_4:S_6$, of order \(8640\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $27$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $(S_3^3\times A_6):S_3$ |