Subgroup ($H$) information
| Description: | not computed |
| Order: | \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \) |
| Index: | \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Exponent: | not computed |
| Generators: |
$\langle(13,18,17), (1,2,8,7)(3,4,6,5)(13,18)(19,20), (1,8)(2,7)(3,6)(4,5)(13,17,18) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is nonabelian, supersolvable (hence solvable and monomial), and metabelian. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_6^4:(C_2\times S_4)$ |
| Order: | \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \) |
| Exponent: | \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_5^4:D_4$, of order \(497664\)\(\medspace = 2^{11} \cdot 3^{5} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | not computed |
Related subgroups
Other information
| Number of subgroups in this autjugacy class | $6$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_6^3:(S_3\times S_4)$ |