Subgroup ($H$) information
| Description: | $C_3^4$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(1,3,2)(4,6,5)(13,15,14)(16,18,17)(25,27,26)(28,30,29), (1,25,15)(2,26,13) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and a $p$-group (hence elementary and hyperelementary).
Ambient group ($G$) information
| Description: | $C_3^5:F_9:C_2$ |
| Order: | \(34992\)\(\medspace = 2^{4} \cdot 3^{7} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^3.C_3^4.Q_8.C_6.C_2^3$, of order \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \) |
| $\operatorname{Aut}(H)$ | $C_2.\PSL(4,3).C_2$, of order \(24261120\)\(\medspace = 2^{9} \cdot 3^{6} \cdot 5 \cdot 13 \) |
| $W$ | $\SD_{16}$, of order \(16\)\(\medspace = 2^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $36$ |
| Number of conjugacy classes in this autjugacy class | $4$ |
| Möbius function | $0$ |
| Projective image | $C_3^5:F_9:C_2$ |