Subgroup ($H$) information
| Description: | $C_7^3:C_3^2:D_6$ |
| Order: | \(37044\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7^{3} \) |
| Index: | \(1536\)\(\medspace = 2^{9} \cdot 3 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(1,19,12)(2,5,8,13,11,17)(3,7,20)(4,6,10,15,9,16)(18,21)(22,23), (1,19,12) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is nonabelian and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2\times \PSL(2,7)\wr S_3$ |
| Order: | \(56899584\)\(\medspace = 2^{11} \cdot 3^{4} \cdot 7^{3} \) |
| Exponent: | \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2\times \PSL(2,7)^3.D_6$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_7^3.\He_3.Q_8.C_6$ |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | not computed |
| Normal closure: | not computed |
| Core: | not computed |
| Autjugate subgroups: | Subgroups are not computed up to automorphism. |
Other information
| Number of subgroups in this conjugacy class | $512$ |
| Möbius function | not computed |
| Projective image | not computed |