Subgroup ($H$) information
| Description: | $C_3\wr C_3$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$\langle(10,17,13)(12,14,16), (11,15,18)(12,14,16), (1,2,4)(5,7,6)(10,18,16,13,15,12,17,11,14), (5,6,7)(11,15,18)(12,16,14)\rangle$
|
| Nilpotency class: | $3$ |
| Derived length: | $2$ |
The subgroup is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Ambient group ($G$) information
| Description: | $C_6^3:S_3^2$ |
| Order: | \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times C_6^2.C_3^4.C_2^4$ |
| $\operatorname{Aut}(H)$ | $C_3^3:D_6$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| $W$ | $C_3^2:S_3$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $16$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $C_6^3:S_3^2$ |