Subgroup ($H$) information
| Description: | $F_8:C_6$ |
| Order: | \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| Index: | \(10\)\(\medspace = 2 \cdot 5 \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(1,2)(3,5)(4,7)(6,8), (9,11)(12,13), (1,4)(2,7)(3,6)(5,8), (2,8,7,4,5,6,3), (3,6,4)(5,8,7), (1,3)(2,5)(4,6)(7,8)\rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and an A-group.
Ambient group ($G$) information
| Description: | $F_5\times F_8:C_3$ |
| Order: | \(3360\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \) |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Derived length: | $3$ |
The ambient group is nonabelian, monomial (hence solvable), and an A-group.
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5\times F_8:C_3$, of order \(3360\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $F_8:C_3$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(S)$ | $F_8:C_3$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(4\)\(\medspace = 2^{2} \) |
| $W$ | $F_8:C_3$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $5$ |
| Möbius function | $1$ |
| Projective image | $F_5\times F_8:C_3$ |