Subgroup ($H$) information
| Description: | $C_6$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Index: | \(86\)\(\medspace = 2 \cdot 43 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$b^{129}, b^{86}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is the center (hence characteristic, normal, abelian, central, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a direct factor, and cyclic (hence elementary ($p = 2,3$), hyperelementary, metacyclic, and a Z-group).
Ambient group ($G$) information
| Description: | $C_3\times D_{86}$ |
| Order: | \(516\)\(\medspace = 2^{2} \cdot 3 \cdot 43 \) |
| Exponent: | \(258\)\(\medspace = 2 \cdot 3 \cdot 43 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), hyperelementary for $p = 2$, and an A-group.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{43}:(C_2^2\times C_{42})$ |
| $\operatorname{Aut}(H)$ | $C_2$, of order \(2\) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2$, of order \(2\) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(3612\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 43 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_3\times D_{86}$ | |
| Normalizer: | $C_3\times D_{86}$ | |
| Complements: | $D_{43}$ $D_{43}$ | |
| Minimal over-subgroups: | $C_{258}$ | $C_2\times C_6$ |
| Maximal under-subgroups: | $C_3$ | $C_2$ |
Other information
| Möbius function | $43$ |
| Projective image | $D_{43}$ |