Subgroup ($H$) information
| Description: | $C_{107}$ |
| Order: | \(107\) |
| Index: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(107\) |
| Generators: |
$b^{3}$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a direct factor, cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), central, a $107$-Sylow subgroup (hence a Hall subgroup), a $p$-group, and simple.
Ambient group ($G$) information
| Description: | $C_3\times C_{321}$ |
| Order: | \(963\)\(\medspace = 3^{2} \cdot 107 \) |
| Exponent: | \(321\)\(\medspace = 3 \cdot 107 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The ambient group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.
Quotient group ($Q$) structure
| Description: | $C_3^2$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Exponent: | \(3\) |
| Automorphism Group: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Outer Automorphisms: | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The quotient is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{106}\times \GL(2,3)$ |
| $\operatorname{Aut}(H)$ | $C_{106}$, of order \(106\)\(\medspace = 2 \cdot 53 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_{106}$, of order \(106\)\(\medspace = 2 \cdot 53 \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $W$ | $C_1$, of order $1$ |
Related subgroups
| Centralizer: | $C_3\times C_{321}$ | |||
| Normalizer: | $C_3\times C_{321}$ | |||
| Complements: | $C_3^2$ | |||
| Minimal over-subgroups: | $C_{321}$ | $C_{321}$ | $C_{321}$ | $C_{321}$ |
| Maximal under-subgroups: | $C_1$ |
Other information
| Möbius function | $3$ |
| Projective image | $C_3^2$ |