Subgroup ($H$) information
| Description: | $C_3^2$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Index: | \(26244\)\(\medspace = 2^{2} \cdot 3^{8} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(4,5,6)(7,8,9)(16,17,18)(19,20,21)(28,29,30)(31,32,33), (1,3,2)(4,6,5)(7,9,8) \!\cdots\! \rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is normal, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic. Whether it is a direct factor or a semidirect factor has not been computed.
Ambient group ($G$) information
| Description: | $C_3^6.C_3\wr C_2^2$ |
| Order: | \(236196\)\(\medspace = 2^{2} \cdot 3^{10} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Derived length: | $3$ |
The ambient group is nonabelian and supersolvable (hence solvable and monomial).
Quotient group ($Q$) structure
| Order: | \(26244\)\(\medspace = 2^{2} \cdot 3^{8} \) |
| Exponent: | not computed |
| Automorphism Group: | not computed |
| Outer Automorphisms: | not computed |
| Nilpotency class: | not computed |
| Derived length: | not computed |
Properties have not been computed
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^5.C_3^5.C_6^3.C_2$, of order \(25509168\)\(\medspace = 2^{4} \cdot 3^{13} \) |
| $\operatorname{Aut}(H)$ | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_3^6.C_3^4.C_2$ |
| Normalizer: | $C_3^6.C_3\wr C_2^2$ |
Other information
| Number of subgroups in this autjugacy class | $2$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | not computed |