Subgroup ($H$) information
| Description: | $C_{14}$ |
| Order: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Index: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$\langle(2,4,3,6,8,7,5), (9,11)(10,12)\rangle$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
Ambient group ($G$) information
| Description: | $F_8:C_{12}$ |
| Order: | \(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \) |
| Exponent: | \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| 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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_8:C_6$, of order \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(S)$ | $C_3$, of order \(3\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(14\)\(\medspace = 2 \cdot 7 \) |
| $W$ | $C_3$, of order \(3\) |
Related subgroups
| Centralizer: | $C_{28}$ | ||
| Normalizer: | $C_7:C_{12}$ | ||
| Normal closure: | $C_2\times F_8$ | ||
| Core: | $C_2$ | ||
| Minimal over-subgroups: | $C_2\times F_8$ | $C_7:C_6$ | $C_{28}$ |
| Maximal under-subgroups: | $C_7$ | $C_2$ |
Other information
| Number of subgroups in this conjugacy class | $8$ |
| Möbius function | $-1$ |
| Projective image | $F_8:C_6$ |