Subgroup ($H$) information
| Description: | $C_2\times S_4\times \He_3$ |
| Order: | \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
| Index: | \(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,3)(2,6)(10,11,12)(13,14,15), (1,3)(4,5)(7,8,9)(13,14,15), (1,3)(2,6) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:C_2^2.D_6.A_6.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_6^2:(D_6\times \GL(2,3))$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| $W$ | $C_6^2:S_3^2$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | not computed | ||
| Normalizer: | $C_6^3:S_3^2$ | ||
| Normal closure: | $C_3^3:A_4\times S_6$ | ||
| Core: | $C_1$ | ||
| Minimal over-subgroups: | $\He_3\times S_6$ | $C_3^3:C_6\times S_4$ | $C_3^2:D_6\times S_4$ |
Other information
| Number of subgroups in this autjugacy class | $120$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_3^3:S_4\times S_6$ |