Subgroup ($H$) information
| Description: | $C_3^2:C_6$ |
| Order: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Index: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(1,2,3)(7,9,8)(10,11,12)(16,18,17), (2,3)(4,9,5,8,6,7)(10,12)(13,16,15,18,14,17) \!\cdots\! \rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $C_3^5:(C_2\times S_4)$ |
| Order: | \(11664\)\(\medspace = 2^{4} \cdot 3^{6} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $(C_3^2\times S_3^3):D_6$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_3^2:\GL(2,3)$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \) |
| $W$ | $S_3^2$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $36$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | $0$ |
| Projective image | $C_3^5:(C_2\times S_4)$ |