Subgroup ($H$) information
| Description: | $C_3^5:C_3$ |
| Order: | \(729\)\(\medspace = 3^{6} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(3\) |
| Generators: |
$\langle(4,10,8)(7,12,11)(13,20,14)(15,16,18)(17,21,19), (2,9,5)(4,8,10)(7,12,11) \!\cdots\! \rangle$
|
| Nilpotency class: | $2$ |
| 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: | $C_3^6.\SL(2,3)$ |
| 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 \(1889568\)\(\medspace = 2^{5} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $C_3^9.C_3^4.Q_8.D_6.C_2$, of order \(306110016\)\(\medspace = 2^{6} \cdot 3^{14} \) |
| $W$ | $C_3^3:C_6$, of order \(162\)\(\medspace = 2 \cdot 3^{4} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $4$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3^6.\SL(2,3)$ |