Subgroup ($H$) information
| Description: | $C_7^3:C_3^2:S_3$ |
| Order: | \(18522\)\(\medspace = 2 \cdot 3^{3} \cdot 7^{3} \) |
| Index: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$a^{3}c^{3}d^{21}, ef, d^{14}, c^{2}d^{24}e^{2}, bc^{5}d^{3}e^{4}f^{4}, f, d^{6}e^{6}f^{5}$
|
| Derived length: | $4$ |
The subgroup is nonabelian and monomial (hence solvable).
Ambient group ($G$) information
| Description: | $C_7^3:(C_3^3:S_4)$ |
| Order: | \(222264\)\(\medspace = 2^{3} \cdot 3^{4} \cdot 7^{3} \) |
| Exponent: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_7^3.C_6^2.(C_6\times S_3)$, of order \(444528\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7^{3} \) |
| $\operatorname{Aut}(H)$ | $C_7^3.\He_3.Q_8.C_6$ |
| $W$ | $C_7^3:C_3\wr S_3$, of order \(55566\)\(\medspace = 2 \cdot 3^{4} \cdot 7^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $4$ |
| Möbius function | $1$ |
| Projective image | $C_7^3:(C_3^3:S_4)$ |