Subgroup ($H$) information
| Description: | $C_{214}$ |
| Order: | \(214\)\(\medspace = 2 \cdot 107 \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(214\)\(\medspace = 2 \cdot 107 \) |
| Generators: |
$b^{321}, b^{6}$
|
| 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 ($p = 2,107$), hyperelementary, metacyclic, and a Z-group).
Ambient group ($G$) information
| Description: | $S_3\times C_{214}$ |
| Order: | \(1284\)\(\medspace = 2^{2} \cdot 3 \cdot 107 \) |
| Exponent: | \(642\)\(\medspace = 2 \cdot 3 \cdot 107 \) |
| 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: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Nilpotency class: | $-1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, 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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $D_6\times C_{106}$, of order \(1272\)\(\medspace = 2^{3} \cdot 3 \cdot 53 \) |
| $\operatorname{Aut}(H)$ | $C_{106}$, of order \(106\)\(\medspace = 2 \cdot 53 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{106}$, of order \(106\)\(\medspace = 2 \cdot 53 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $S_3\times C_{214}$ | |
| Normalizer: | $S_3\times C_{214}$ | |
| Complements: | $S_3$ $S_3$ | |
| Minimal over-subgroups: | $C_{642}$ | $C_2\times C_{214}$ |
| Maximal under-subgroups: | $C_{107}$ | $C_2$ |
Other information
| Möbius function | $3$ |
| Projective image | $S_3$ |