Subgroup ($H$) information
| Description: | $C_3^3:C_6^2$ |
| Order: | \(972\)\(\medspace = 2^{2} \cdot 3^{5} \) |
| Index: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,7,5)(2,8,3)(4,9,6), (10,13,14)(11,12,15), (1,7,5)(2,9)(3,4)(6,8)(10,14) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3^6:(C_4\times S_3)$ |
| Order: | \(17496\)\(\medspace = 2^{3} \cdot 3^{7} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $3$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3\times C_3.C_3^5.C_4.C_2^5$ |
| $\operatorname{Aut}(H)$ | $C_3^2.Q_8^2.S_3^3$ |
| $W$ | $C_3:S_3^2$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | not computed |