Subgroup ($H$) information
| Description: | $C_3^2$ |
| Order: | \(9\)\(\medspace = 3^{2} \) |
| Index: | \(332\)\(\medspace = 2^{2} \cdot 83 \) |
| Exponent: | \(3\) |
| Generators: |
$a^{664}, b$
|
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
The subgroup is characteristic (hence normal), a semidirect factor, abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $3$-Sylow subgroup (hence a Hall subgroup), a $p$-group (hence elementary and hyperelementary), and metacyclic.
Ambient group ($G$) information
| Description: | $C_3:C_{996}$ |
| Order: | \(2988\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 83 \) |
| Exponent: | \(996\)\(\medspace = 2^{2} \cdot 3 \cdot 83 \) |
| Derived length: | $2$ |
The ambient group is nonabelian, metacyclic (hence solvable, supersolvable, monomial, and metabelian), 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 \(164\)\(\medspace = 2^{2} \cdot 41 \) |
| Nilpotency class: | $1$ |
| 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
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 $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_{246}:C_2^3$, of order \(1968\)\(\medspace = 2^{4} \cdot 3 \cdot 41 \) |
| $\operatorname{Aut}(H)$ | $\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \) |
| $\operatorname{res}(\operatorname{Aut}(G))$ | $C_2^2$, of order \(4\)\(\medspace = 2^{2} \) |
| $\card{\operatorname{ker}(\operatorname{res})}$ | \(492\)\(\medspace = 2^{2} \cdot 3 \cdot 41 \) |
| $W$ | $C_2$, of order \(2\) |
Related subgroups
| Centralizer: | $C_3\times C_{498}$ | ||
| Normalizer: | $C_3:C_{996}$ | ||
| Complements: | $C_{332}$ | ||
| Minimal over-subgroups: | $C_3\times C_{249}$ | $C_3\times C_6$ | |
| Maximal under-subgroups: | $C_3$ | $C_3$ | $C_3$ |
Other information
| Möbius function | $0$ |
| Projective image | $C_3:C_{332}$ |