Subgroup ($H$) information
| Description: | $C_6\times S_6$ |
| Order: | \(4320\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Generators: |
$\langle(1,6)(2,5)(3,4)(8,9)(10,15)(11,14)(12,13), (10,11,12)(13,15,14), (1,2,3,4,5)(8,9)(10,15)(11,14)(12,13), (8,9)(10,15)(11,14)(12,13)\rangle$
|
| Derived length: | $1$ |
The subgroup is nonabelian and nonsolvable.
Ambient group ($G$) information
| Description: | $C_3^2:D_6\times S_6$ |
| Order: | \(77760\)\(\medspace = 2^{6} \cdot 3^{5} \cdot 5 \) |
| Exponent: | \(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, nonsolvable, and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3.S_3\wr C_2.A_6.C_2^2$ |
| $\operatorname{Aut}(H)$ | $S_6:C_2^3$, of order \(5760\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \) |
| $W$ | $C_2\times S_6$, of order \(1440\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $18$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_3^2:D_6\times S_6$ |