Subgroup ($H$) information
| Description: | $C_3^3:F_9$ |
| Order: | \(1944\)\(\medspace = 2^{3} \cdot 3^{5} \) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(1,4)(2,6)(3,8,12,7)(5,11,10,9)(15,18)(16,17), (1,2,4,6)(3,5,8,11,12,10,7,9) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is maximal, nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3^7:C_8$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metabelian (hence solvable), and an A-group. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^7.S_3^2$, of order \(8957952\)\(\medspace = 2^{12} \cdot 3^{7} \) |
| $\operatorname{Aut}(H)$ | $C_3^5.C_4^2.C_2^4$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $9$ |
| Möbius function | not computed |
| Projective image | not computed |