Subgroup ($H$) information
| Description: | not computed |
| Order: | \(526848\)\(\medspace = 2^{9} \cdot 3 \cdot 7^{3} \) |
| Index: | \(54\)\(\medspace = 2 \cdot 3^{3} \) |
| Exponent: | not computed |
| Generators: |
$\langle(17,22)(18,21)(19,20)(23,24), (17,23)(18,20)(19,21)(22,24), (1,6)(2,5)(3,4) \!\cdots\! \rangle$
|
| Derived length: | not computed |
The subgroup is nonabelian, solvable, and an A-group. Whether it is elementary, hyperelementary, monomial, simple, quasisimple, perfect, almost simple, or rational has not been computed.
Ambient group ($G$) information
| Description: | $C_2^9.C_7^3:C_3\wr S_3$ |
| Order: | \(28449792\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 7^{3} \) |
| Exponent: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Derived length: | $5$ |
The ambient group is nonabelian and solvable. 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^9.C_7^3:C_3\wr S_3$, of order \(28449792\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 7^{3} \) |
| $\operatorname{Aut}(H)$ | not computed |
| $\card{W}$ | not computed |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_2^6.C_7^2.F_8.C_3^3.C_2$ |
| Normal closure: | $C_2^9.C_7^3.C_3^2$ |
| Core: | $C_2^6.C_7^2\times F_8$ |
Other information
| Number of subgroups in this autjugacy class | $3$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^9.C_7^3:C_3\wr S_3$ |