Subgroup ($H$) information
| Description: | $C_3^4:C_6$ |
| Order: | \(486\)\(\medspace = 2 \cdot 3^{5} \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Generators: |
$\langle(2,4,7)(3,6)(5,8,11,9,10,12)(14,16,18,20,19,21)(15,17), (2,9,5)(4,8,10) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), 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 \(1889568\)\(\medspace = 2^{5} \cdot 3^{10} \) |
| $\operatorname{Aut}(H)$ | $C_3^4.C_3^4.C_2^3$, of order \(52488\)\(\medspace = 2^{3} \cdot 3^{8} \) |
| $W$ | $C_3^4:C_6$, of order \(486\)\(\medspace = 2 \cdot 3^{5} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $108$ |
| Number of conjugacy classes in this autjugacy class | $3$ |
| Möbius function | $1$ |
| Projective image | $C_3^6.\SL(2,3)$ |