Subgroup ($H$) information
| Description: | $C_3^5:F_9:C_2$ |
| Order: | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| Index: | \(702\)\(\medspace = 2 \cdot 3^{3} \cdot 13 \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Generators: |
$\langle(1,17,34,25)(2,18,36,27)(3,16,35,26)(4,33)(5,31)(6,32)(7,14)(8,15)(9,13) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_3^7.\GL(3,3)$ |
| Order: | \(24564384\)\(\medspace = 2^{5} \cdot 3^{10} \cdot 13 \) |
| Exponent: | \(936\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 13 \) |
| Derived length: | $2$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^7.C_2^2.\SL(3,3)$, of order \(49128768\)\(\medspace = 2^{6} \cdot 3^{10} \cdot 13 \) |
| $\operatorname{Aut}(H)$ | $C_3^3.C_3^4.Q_8.C_6.C_2^3$, of order \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \) |
| $W$ | $C_3^5:F_9:D_6$, of order \(209952\)\(\medspace = 2^{5} \cdot 3^{8} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^6.F_9:D_6$ |
| Normal closure: | $C_3^6:\SL(3,3)$ |
| Core: | $C_1$ |
Other information
| Number of subgroups in this autjugacy class | $39$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^7.\GL(3,3)$ |