Subgroup ($H$) information
| Description: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Index: | \(332\)\(\medspace = 2^{2} \cdot 83 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$ab^{830}, b^{664}$
|
| Derived length: | $2$ |
The subgroup is normal, a direct factor, nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, and rational.
Ambient group ($G$) information
| Description: | $S_3\times C_{332}$ |
| Order: | \(1992\)\(\medspace = 2^{3} \cdot 3 \cdot 83 \) |
| Exponent: | \(996\)\(\medspace = 2^{2} \cdot 3 \cdot 83 \) |
| 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: | $C_{332}$ |
| Order: | \(332\)\(\medspace = 2^{2} \cdot 83 \) |
| Exponent: | \(332\)\(\medspace = 2^{2} \cdot 83 \) |
| Automorphism Group: | $C_2\times C_{82}$, of order \(164\)\(\medspace = 2^{2} \cdot 41 \) |
| Outer Automorphisms: | $C_2\times C_{82}$, of order \(164\)\(\medspace = 2^{2} \cdot 41 \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 2,83$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = N_G(H) / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{246}:C_2^3$, of order \(1968\)\(\medspace = 2^{4} \cdot 3 \cdot 41 \) |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(S)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(164\)\(\medspace = 2^{2} \cdot 41 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
Other information
| Möbius function | $0$ |
| Projective image | $S_3\times C_{332}$ |