Subgroup ($H$) information
| Description: | $S_3\times C_3^2:C_{12}$ |
| Order: | \(648\)\(\medspace = 2^{3} \cdot 3^{4} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(2,3,6,4,5,7)(8,9)(10,13,12,11)(14,15), (1,9,8), (14,15,16), (8,9), (2,5,6), (3,4,7), (2,5,6)(3,4,7)(10,12)(11,13)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $(C_3\times S_3^3):S_4$ |
| Order: | \(15552\)\(\medspace = 2^{6} \cdot 3^{5} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^4.C_2^4.D_6^2$ |
| $\operatorname{Aut}(H)$ | $\PSU(3,2).D_6^2.C_2$ |
| $W$ | $C_2\times S_3^3$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \) |
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 | $(C_3\times S_3^3):S_4$ |