Subgroup ($H$) information
| Description: | $C_{334}$ |
| Order: | \(334\)\(\medspace = 2 \cdot 167 \) |
| Index: | \(3\) |
| Exponent: | \(334\)\(\medspace = 2 \cdot 167 \) |
| Generators: |
$a, b^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is maximal, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,167$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), and a Hall subgroup.
Ambient group ($G$) information
| Description: | $S_3\times C_{167}$ |
| Order: | \(1002\)\(\medspace = 2 \cdot 3 \cdot 167 \) |
| Exponent: | \(1002\)\(\medspace = 2 \cdot 3 \cdot 167 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Automorphism information
While the subgroup $H$ is not characteristic, the stabilizer $S$ of $H$ in the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : S \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphisms $\operatorname{Inn}(G) \cap S$ is the Weyl group $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $S_3\times C_{166}$, of order \(996\)\(\medspace = 2^{2} \cdot 3 \cdot 83 \) |
| $\operatorname{Aut}(H)$ | $C_{166}$, of order \(166\)\(\medspace = 2 \cdot 83 \) |
| $\operatorname{res}(S)$ | $C_{166}$, of order \(166\)\(\medspace = 2 \cdot 83 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(2\) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_{334}$ | |
| Normalizer: | $C_{334}$ | |
| Normal closure: | $S_3\times C_{167}$ | |
| Core: | $C_{167}$ | |
| Minimal over-subgroups: | $S_3\times C_{167}$ | |
| Maximal under-subgroups: | $C_{167}$ | $C_2$ |
Other information
| Number of subgroups in this conjugacy class | $3$ |
| Möbius function | $-1$ |
| Projective image | $S_3$ |