Subgroup ($H$) information
| Description: | $C_3^4$ |
| Order: | \(81\)\(\medspace = 3^{4} \) |
| Index: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(1,3,6)(2,5,9)(7,11,12)(13,17,15)(14,19,18)(16,20,21), (2,9,5)(4,8,10)(7,12,11) \!\cdots\! \rangle$
|
| 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: | $C_3^6.\SL(2,3)$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_6.C_6^2.D_6$, of order \(1889568\)\(\medspace = 2^{5} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $C_2.\PSL(4,3).C_2$, of order \(24261120\)\(\medspace = 2^{9} \cdot 3^{6} \cdot 5 \cdot 13 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $324$ |
| Number of conjugacy classes in this autjugacy class | $27$ |
| Möbius function | $0$ |
| Projective image | $C_3^6.\SL(2,3)$ |