Subgroup ($H$) information
| Description: | $D_4$ |
| Order: | \(8\)\(\medspace = 2^{3} \) |
| Index: | \(86\)\(\medspace = 2 \cdot 43 \) |
| Exponent: | \(4\)\(\medspace = 2^{2} \) |
| Generators: |
$a, c^{43}$
|
| Nilpotency class: | $2$ |
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a semidirect factor, nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), metacyclic (hence metabelian), and rational.
Ambient group ($G$) information
| Description: | $D_4:D_{43}$ |
| Order: | \(688\)\(\medspace = 2^{4} \cdot 43 \) |
| Exponent: | \(172\)\(\medspace = 2^{2} \cdot 43 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Quotient group ($Q$) structure
| Description: | $D_{43}$ |
| Order: | \(86\)\(\medspace = 2 \cdot 43 \) |
| Exponent: | \(86\)\(\medspace = 2 \cdot 43 \) |
| Automorphism Group: | $F_{43}$, of order \(1806\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 43 \) |
| Outer Automorphisms: | $C_{21}$, of order \(21\)\(\medspace = 3 \cdot 7 \) |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{86}.C_{42}.C_2^3$ |
| $\operatorname{Aut}(H)$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_4$, of order \(8\)\(\medspace = 2^{3} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3612\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 43 \) |
| $W$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
Related subgroups
| Centralizer: | $C_{43}:C_4$ | ||
| Normalizer: | $D_4:D_{43}$ | ||
| Complements: | $D_{43}$ | ||
| Minimal over-subgroups: | $D_4\times C_{43}$ | $D_4:C_2$ | |
| Maximal under-subgroups: | $C_2^2$ | $C_2^2$ | $C_4$ |
Other information
| Möbius function | $43$ |
| Projective image | $C_2\times D_{86}$ |