Subgroup ($H$) information
| Description: | $C_3:S_3$ |
| Order: | \(18\)\(\medspace = 2 \cdot 3^{2} \) |
| Index: | \(324\)\(\medspace = 2^{2} \cdot 3^{4} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$\langle(3,7)(4,6)(5,8)(10,12)(11,14)(13,15), (2,4,6)(5,9,8), (1,3,7)(2,4,6)(10,17,12)(11,18,14)(13,16,15)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, an A-group, and rational.
Ambient group ($G$) information
| Description: | $\He_3^2:C_2^3$ |
| Order: | \(5832\)\(\medspace = 2^{3} \cdot 3^{6} \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), and rational.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4:D_6\wr C_2$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \) |
| $\operatorname{Aut}(H)$ | $C_3^2:\GL(2,3)$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
| $W$ | $C_3:S_3$, of order \(18\)\(\medspace = 2 \cdot 3^{2} \) |
Related subgroups
| Centralizer: | $C_3$ | |||
| Normalizer: | $C_3^2:C_6$ | |||
| Normal closure: | $C_3^3:S_3^2$ | |||
| Core: | $C_1$ | |||
| Minimal over-subgroups: | $C_3^2:C_6$ | $C_3^2:S_3$ | ||
| Maximal under-subgroups: | $C_3^2$ | $S_3$ | $S_3$ | $S_3$ |
Other information
| Number of subgroups in this autjugacy class | $216$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | $0$ |
| Projective image | $\He_3^2:C_2^3$ |