Subgroup ($H$) information
| Description: | $D_{113}$ |
| Order: | \(226\)\(\medspace = 2 \cdot 113 \) |
| Index: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Exponent: | \(226\)\(\medspace = 2 \cdot 113 \) |
| Generators: |
$a^{56}, b$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Ambient group ($G$) information
| Description: | $F_{113}$ |
| Order: | \(12656\)\(\medspace = 2^{4} \cdot 7 \cdot 113 \) |
| Exponent: | \(12656\)\(\medspace = 2^{4} \cdot 7 \cdot 113 \) |
| Derived length: | $2$ |
The ambient group is nonabelian and a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group).
Quotient group ($Q$) structure
| Description: | $C_{56}$ |
| Order: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Exponent: | \(56\)\(\medspace = 2^{3} \cdot 7 \) |
| Automorphism Group: | $C_2^2\times C_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Outer Automorphisms: | $C_2^2\times C_6$, of order \(24\)\(\medspace = 2^{3} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,7$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $F_{113}$, of order \(12656\)\(\medspace = 2^{4} \cdot 7 \cdot 113 \) |
| $\operatorname{Aut}(H)$ | $F_{113}$, of order \(12656\)\(\medspace = 2^{4} \cdot 7 \cdot 113 \) |
| $W$ | $F_{113}$, of order \(12656\)\(\medspace = 2^{4} \cdot 7 \cdot 113 \) |
Related subgroups
| Centralizer: | $C_1$ | |
| Normalizer: | $F_{113}$ | |
| Minimal over-subgroups: | $C_{113}:C_{14}$ | $C_{113}:C_4$ |
| Maximal under-subgroups: | $C_{113}$ | $C_2$ |
Other information
| Möbius function | $0$ |
| Projective image | $F_{113}$ |