Subgroup ($H$) information
| Description: | $F_9$ |
| Order: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| Index: | \(27\)\(\medspace = 3^{3} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(11,14)(12,17)(13,18)(15,16), (10,16,15)(11,17,18)(12,14,13), (1,2)(3,4,5,6) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3^3:F_9$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5.C_4^2.C_2^4$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | $F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\operatorname{res}(S)$ | $F_9:C_2$, of order \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(16\)\(\medspace = 2^{4} \) |
| $W$ | $F_9$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_1$ | |
| Normalizer: | $F_9$ | |
| Normal closure: | $C_3^3:F_9$ | |
| Core: | $C_3:S_3$ | |
| Minimal over-subgroups: | $C_3^2:F_9$ | $C_3:F_9$ |
| Maximal under-subgroups: | $C_3^2:C_4$ | $C_8$ |
Other information
| Number of subgroups in this autjugacy class | $27$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $1$ |
| Projective image | $C_3^3:F_9$ |