Subgroup ($H$) information
| Description: | $C_3\wr C_3$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(144\)\(\medspace = 2^{4} \cdot 3^{2} \) |
| Exponent: | \(9\)\(\medspace = 3^{2} \) |
| Generators: |
$\langle(1,16,14,3,18,15,2,17,13)(4,10,9,5,11,7,6,12,8), (7,8,9)(16,18,17), (4,5,6)(7,8,9)(13,14,15)(16,18,17), (1,3,2)(7,9,8)(10,11,12)(16,17,18)\rangle$
|
| 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: | $C_3^5:(C_2\times S_4)$ |
| Order: | \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3^2\times S_3^3):D_6$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_3^3:D_6$, of order \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| $W$ | $C_3^2:C_6$, of order \(54\)\(\medspace = 2 \cdot 3^{3} \) |
Related subgroups
| Centralizer: | $C_3^2$ | ||
| Normalizer: | $C_3^4:S_3$ | ||
| Normal closure: | $C_3^5:A_4$ | ||
| Core: | $C_3^3$ | ||
| Minimal over-subgroups: | $C_3^3:A_4$ | $C_3^4:C_3$ | $C_3\wr S_3$ |
| Maximal under-subgroups: | $C_3^3$ | $\He_3$ | $C_9:C_3$ |
Other information
| Number of subgroups in this autjugacy class | $24$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3^5:(C_2\times S_4)$ |