Subgroup ($H$) information
| Description: | $C_{129}$ |
| Order: | \(129\)\(\medspace = 3 \cdot 43 \) |
| Index: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(129\)\(\medspace = 3 \cdot 43 \) |
| Generators: |
$b^{86}, b^{6}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 3,43$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a Hall subgroup.
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: | $C_2^2$ |
| Order: | \(4\)\(\medspace = 2^{2} \) |
| Exponent: | \(2\) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), metacyclic, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{43}:(C_2^2\times C_{42})$ |
| $\operatorname{Aut}(H)$ | $C_2\times C_{42}$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2\times C_{42}$, of order \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(86\)\(\medspace = 2 \cdot 43 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_{258}$ | ||
| Normalizer: | $C_3\times D_{86}$ | ||
| Complements: | $C_2^2$ | ||
| Minimal over-subgroups: | $C_{258}$ | $C_3\times D_{43}$ | $C_3\times D_{43}$ |
| Maximal under-subgroups: | $C_{43}$ | $C_3$ |
Other information
| Möbius function | $2$ |
| Projective image | $D_{86}$ |