Subgroup ($H$) information
| Description: | $C_7^3:C_3^2:S_3$ |
| Order: | \(18522\)\(\medspace = 2 \cdot 3^{3} \cdot 7^{3} \) |
| Index: | \(8\)\(\medspace = 2^{3} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$ab^{3}c^{2}d^{36}f^{6}, ef^{6}, d^{14}e^{5}f, c^{2}f^{4}, b^{2}cd^{30}e, f, d^{6}e^{3}f^{6}$
|
| Derived length: | $4$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_7^3:(C_6^2:D_6)$ |
| Order: | \(148176\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 7^{3} \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| Derived length: | $4$ |
The ambient group is nonabelian and monomial (hence solvable).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2.F_7\wr S_3$, of order \(889056\)\(\medspace = 2^{5} \cdot 3^{4} \cdot 7^{3} \) |
| $\operatorname{Aut}(H)$ | $C_7^3.\He_3.Q_8.C_6$ |
| $W$ | $C_7^3:C_3^2:S_3$, of order \(18522\)\(\medspace = 2 \cdot 3^{3} \cdot 7^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $1$ |
| Projective image | $C_7^3:(C_6^2:D_6)$ |