Subgroup ($H$) information
| Description: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Index: | \(119\)\(\medspace = 7 \cdot 17 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$a, b^{238}$
|
| Derived length: | $2$ |
The subgroup is characteristic (hence normal), a direct factor, nonabelian, a Hall subgroup, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, and rational.
Ambient group ($G$) information
| Description: | $S_3\times C_{119}$ |
| Order: | \(714\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 17 \) |
| Exponent: | \(714\)\(\medspace = 2 \cdot 3 \cdot 7 \cdot 17 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 2$.
Quotient group ($Q$) structure
| Description: | $C_{119}$ |
| Order: | \(119\)\(\medspace = 7 \cdot 17 \) |
| Exponent: | \(119\)\(\medspace = 7 \cdot 17 \) |
| Automorphism Group: | $C_2\times C_{48}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Outer Automorphisms: | $C_2\times C_{48}$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary ($p = 7,17$), hyperelementary, metacyclic, metabelian, a Z-group, and an A-group).
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_6\times C_{48}$, of order \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| $\operatorname{Aut}(H)$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| $W$ | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
Related subgroups
| Centralizer: | $C_{119}$ | |
| Normalizer: | $S_3\times C_{119}$ | |
| Complements: | $C_{119}$ | |
| Minimal over-subgroups: | $S_3\times C_{17}$ | $S_3\times C_7$ |
| Maximal under-subgroups: | $C_3$ | $C_2$ |
Other information
| Möbius function | $1$ |
| Projective image | $S_3\times C_{119}$ |