Subgroup ($H$) information
| Description: | $D_4\times C_3^3:S_4$ |
| Order: | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| Index: | \(90\)\(\medspace = 2 \cdot 3^{2} \cdot 5 \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(1,5)(2,6)(3,4)(7,8)(10,15,12,14)(11,13), (1,3)(2,6)(7,8)(11,12)(13,14,15) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3:C_2^2.D_6.A_6.C_2^2$ |
| $\operatorname{Aut}(H)$ | $C_6^3.(C_2^2\times S_4)$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \) |
| $W$ | $C_2^2\times C_3^3:S_4$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $45$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^3:S_4\times S_6$ |