Subgroup ($H$) information
| Description: | $C_7^3:(C_3\times S_3)$ |
| Order: | \(6174\)\(\medspace = 2 \cdot 3^{2} \cdot 7^{3} \) |
| Index: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$a^{3}d^{21}, f, d^{14}, d^{6}e^{3}f^{2}, c^{2}d^{24}e^{2}, ef^{3}$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and an A-group.
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 \(444528\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7^{3} \) |
| $\operatorname{Aut}(H)$ | $C_7^3:(S_3\times C_6^2)$, of order \(74088\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 7^{3} \) |
| $W$ | $C_7^3:(S_3\times C_3^2)$, of order \(18522\)\(\medspace = 2 \cdot 3^{3} \cdot 7^{3} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $6$ |
| Möbius function | $-2$ |
| Projective image | $C_7^3:(C_3^3:S_4)$ |