Subgroup ($H$) information
| Description: | $F_5\times F_8:C_3$ |
| Order: | \(3360\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \) |
| Index: | $1$ |
| Exponent: | \(420\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Generators: |
$\langle(9,13,11,12), (9,13,10,12,11), (1,2)(3,5)(4,7)(6,8), (9,11)(12,13), (1,4)(2,7)(3,6)(5,8), (2,8,7,4,5,6,3), (3,6,4)(5,8,7), (1,3)(2,5)(4,6)(7,8)\rangle$
|
| Derived length: | $3$ |
The subgroup is the radical (hence characteristic, normal, and solvable), a direct factor, nonabelian, a Hall subgroup, monomial, 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.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_5\times F_8:C_3$, of order \(3360\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \) |
| $\operatorname{Aut}(H)$ | $F_5\times F_8:C_3$, of order \(3360\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \) |
| $W$ | $F_5\times F_8:C_3$, of order \(3360\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \cdot 7 \) |
Related subgroups
| Centralizer: | $C_1$ | ||||
| Normalizer: | $F_5\times F_8:C_3$ | ||||
| Complements: | $C_1$ | ||||
| Maximal under-subgroups: | $D_5\times F_8:C_3$ | $F_5\times F_8$ | $F_8:C_{12}$ | $C_2\times A_4\times F_5$ | $C_{35}:C_{12}$ |
Other information
| Möbius function | $1$ |
| Projective image | $F_5\times F_8:C_3$ |