Subgroup ($H$) information
| Description: | $C_3\wr C_4$ |
| Order: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(1,31,26,9)(2,33,27,8)(3,32,25,7)(4,36,30,10)(5,34,28,11)(6,35,29,12)(13,20) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3^5:\PSU(3,2)$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_3^4.(Q_8.A_4).D_6$, of order \(2519424\)\(\medspace = 2^{7} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_2\times F_9:D_6$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \) |
| $W$ | $C_3^3:C_4$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_3$ | |||
| Normalizer: | $C_3\wr C_4$ | |||
| Normal closure: | $C_3^5.C_3^2.C_4$ | |||
| Core: | $C_3$ | |||
| Minimal over-subgroups: | $C_3^4:C_{12}$ | |||
| Maximal under-subgroups: | $C_3^2\wr C_2$ | $C_3^2:C_{12}$ | $C_3^3:C_4$ | $C_3:C_{12}$ |
Other information
| Number of subgroups in this autjugacy class | $648$ |
| Number of conjugacy classes in this autjugacy class | $12$ |
| Möbius function | $0$ |
| Projective image | $C_3^5:\PSU(3,2)$ |