Subgroup ($H$) information
| Description: | not computed |
| Order: | \(1259712\)\(\medspace = 2^{6} \cdot 3^{9} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(2,11,9,6,8,17)(3,16,15,4,13,14)(7,12,18)(21,24)(22,25,27)(23,26), (7,12,18) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is nonabelian and solvable. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_3^9.C_2^6.D_6$ |
| Order: | \(15116544\)\(\medspace = 2^{8} \cdot 3^{10} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3:S_3)^3.D_6\wr S_3$, of order \(60466176\)\(\medspace = 2^{10} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^9.C_2^5.C_2^2$ |
| Normal closure: | $C_3^9.C_2^6.S_3$ |
| Core: | $C_3^8.D_6$ |
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^9.C_2^6.D_6$ |