Subgroup ($H$) information
| Description: | $(S_3\times C_6^2):S_4$ |
| Order: | \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Generators: |
$\langle(2,7,4)(8,10,14)(9,16,18)(11,15,17), (20,25)(21,26), (2,7,4)(3,12,13)(11,17,15) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_3^3.S_4^2:C_2^3$ |
| Order: | \(124416\)\(\medspace = 2^{9} \cdot 3^{5} \) |
| Exponent: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $4$ |
The ambient group is nonabelian, solvable, and rational. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^4.C_3^4.D_6.C_2^5$ |
| $\operatorname{Aut}(H)$ | $(C_2^2\times C_6^2).D_6^2$ |
| $\card{W}$ | \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \) |
Related subgroups
| Centralizer: | $C_2$ |
| Normalizer: | $(C_2^2\times C_6^2):D_6^2$ |
| Normal closure: | $(C_2\times C_6^3):S_3^2$ |
| Core: | $C_2\times C_6^3$ |
Other information
| Number of subgroups in this autjugacy class | $12$ |
| Number of conjugacy classes in this autjugacy class | $2$ |
| Möbius function | not computed |
| Projective image | $C_3^3.S_4^2:C_2^3$ |