Subgroup ($H$) information
| Description: | $F_8$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Index: | $1$ |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Generators: |
$\langle(2,6,8,7,4,5,3), (1,4)(2,3)(5,8)(6,7), (1,7)(2,8)(3,5)(4,6), (1,5)(2,6)(3,7)(4,8)\rangle$
|
| Derived length: | $2$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, monomial, metabelian, and an A-group.
Ambient group ($G$) information
| Description: | $F_8$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Exponent: | \(14\)\(\medspace = 2 \cdot 7 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, monomial (hence solvable), metabelian, and an A-group.
Quotient group ($Q$) structure
| Description: | $C_1$ |
| Order: | $1$ |
| Exponent: | $1$ |
| Automorphism Group: | $C_1$, of order $1$ |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $0$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary (for every $p$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group (for every $p$), perfect, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_8:C_3$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $F_8:C_3$, of order \(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \) |
| $W$ | $F_8$, of order \(56\)\(\medspace = 2^{3} \cdot 7 \) |
Related subgroups
| Centralizer: | $C_1$ | |
| Normalizer: | $F_8$ | |
| Complements: | $C_1$ | |
| Maximal under-subgroups: | $C_2^3$ | $C_7$ |
Other information
| Möbius function | $1$ |
| Projective image | $F_8$ |