Subgroup ($H$) information
| Description: | not computed |
| Order: | \(3888\)\(\medspace = 2^{4} \cdot 3^{5} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(4,9,7)(10,14,11)(12,15,13)(16,18,17), (2,5)(3,9)(4,6)(7,8)(10,18)(11,16) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $(C_3^3\times C_6^2):S_4$ |
| Order: | \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_2^4.D_6^2$ |
| $\operatorname{Aut}(H)$ | not computed |
| $W$ | $C_2\times C_3^3:D_{12}$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $(C_3^2\times C_6^2):S_4$ |