Subgroup ($H$) information
| Description: | $C_3^2\times C_6$ |
| Order: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Index: | \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(19,21)(20,22), (10,12,16)(11,13,14), (1,9,6)(2,3,15)(4,7,18)(5,8,17)(10,16,12)(11,13,14), (2,3,15)(10,16,12)(11,13,14)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group) and elementary for $p = 3$ (hence hyperelementary).
Ambient group ($G$) information
| Description: | $C_3^4:D_6\wr C_2$ |
| Order: | \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^2.C_3^4.C_2^6.C_2^3$ |
| $\operatorname{Aut}(H)$ | $\GL(3,3)$, of order \(11232\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 13 \) |
| $\card{W}$ | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | not computed |