Subgroup ($H$) information
| Description: | $C_3^3$ |
| Order: | \(27\)\(\medspace = 3^{3} \) |
| Index: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(1,11,18)(2,7,16)(3,17,8)(5,13,10)(6,14,15)(19,21,23)(20,22,24), (19,23,21)(20,24,22), (20,22,24)\rangle$
|
| 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: | $(C_3^3\times \He_3):D_{12}$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_3.D_6^2.C_2^2$, of order \(1259712\)\(\medspace = 2^{6} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $\GL(3,3)$, of order \(11232\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 13 \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $18$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | $0$ |
| Projective image | $(C_3^3\times \He_3):D_{12}$ |