Subgroup ($H$) information
| Description: | $F_8:C_6$ |
| Order: | \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| Index: | \(16\)\(\medspace = 2^{4} \) |
| Exponent: | \(42\)\(\medspace = 2 \cdot 3 \cdot 7 \) |
| Generators: |
$\langle(1,3,5)(2,4,7)(8,11,10)(9,12,14)(15,17)(16,18), (2,11)(3,12)(4,10)(5,14) \!\cdots\! \rangle$
|
| Derived length: | $3$ |
The subgroup is nonabelian, monomial (hence solvable), and an A-group.
Ambient group ($G$) information
| Description: | $C_4\times C_2^3.\GL(3,2)$ |
| Order: | \(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \) |
| Exponent: | \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| Derived length: | $1$ |
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_2^5.\GL(3,2)$, of order \(5376\)\(\medspace = 2^{8} \cdot 3 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $F_8:C_3$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $W$ | $F_8:C_3$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
Related subgroups
Other information
| Number of subgroups in this conjugacy class | $8$ |
| Möbius function | $1$ |
| Projective image | $C_2^4.\PSL(2,7)$ |