Subgroup ($H$) information
| Description: | $S_3\times C_7$ |
| Order: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Index: | \(216\)\(\medspace = 2^{3} \cdot 3^{3} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(11,12), (2,7,6,5,8,9,4), (10,11,12)\rangle$
|
| Derived length: | $2$ |
The subgroup is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $S_3\times {}^2G(2,3)$ |
| Order: | \(9072\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7 \) |
| Exponent: | \(126\)\(\medspace = 2 \cdot 3^{2} \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and nonsolvable.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times {}^2G(2,3)$, of order \(9072\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_6\times S_3$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| $W$ | $C_6\times S_3$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $36$ |
| Möbius function | $-2$ |
| Projective image | $S_3\times {}^2G(2,3)$ |