Subgroup ($H$) information
| Description: | $C_7^3:C_3^2:D_6$ |
| Order: | \(37044\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7^{3} \) |
| Index: | \(512\)\(\medspace = 2^{9} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(4,5,6,20,24,7,22), (4,22,7,24,20,6,5)(10,18,19,12,16,13,15), (9,17,21) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is maximal, nonabelian, and solvable. Whether it is monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_2^9.C_7^3:C_3^2:D_6$ |
| Order: | \(18966528\)\(\medspace = 2^{11} \cdot 3^{3} \cdot 7^{3} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^{10}.C_7^3:C_3\wr S_3$, of order \(56899584\)\(\medspace = 2^{11} \cdot 3^{4} \cdot 7^{3} \) |
| $\operatorname{Aut}(H)$ | $C_2\times C_7^3.\He_3.Q_8.C_6$ |
| $W$ | $C_7^3:C_3^2:S_3$, of order \(18522\)\(\medspace = 2 \cdot 3^{3} \cdot 7^{3} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_7^3:C_3^2:D_6$ |
| Normal closure: | $C_2^9.C_7^3:C_3^2:D_6$ |
| Core: | $C_2$ |
Other information
| Number of subgroups in this autjugacy class | $512$ |
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_2^9.C_7^3:C_3^2:S_3$ |