Subgroup ($H$) information
| Description: | $C_2\times C_6$ |
| Order: | \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| Index: | \(107\) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Generators: |
$a, b^{321}, b^{428}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), maximal, a direct factor, central (hence abelian, nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a Hall subgroup, elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_2\times C_{642}$ |
| Order: | \(1284\)\(\medspace = 2^{2} \cdot 3 \cdot 107 \) |
| Exponent: | \(642\)\(\medspace = 2 \cdot 3 \cdot 107 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.
Quotient group ($Q$) structure
| Description: | $C_{107}$ |
| Order: | \(107\) |
| Exponent: | \(107\) |
| Automorphism Group: | $C_{106}$, of order \(106\)\(\medspace = 2 \cdot 53 \) |
| Outer Automorphisms: | $C_{106}$, of order \(106\)\(\medspace = 2 \cdot 53 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
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 \(1272\)\(\medspace = 2^{3} \cdot 3 \cdot 53 \) |
| $\operatorname{Aut}(H)$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $D_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(106\)\(\medspace = 2 \cdot 53 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_2\times C_{642}$ | |||
| Normalizer: | $C_2\times C_{642}$ | |||
| Complements: | $C_{107}$ | |||
| Minimal over-subgroups: | $C_2\times C_{642}$ | |||
| Maximal under-subgroups: | $C_6$ | $C_6$ | $C_6$ | $C_2^2$ |
Other information
| Möbius function | $-1$ |
| Projective image | $C_{107}$ |