# Finite groups of small orders

From Groupprops

This page lists, with small notes, finite groups of all small orders, starting from 1.

## 1

- Trivial group: This is the one and only one group with one element

## 2

- Cyclic group:Z2: This is the unique group of two elements. It is the cyclic group of order two. It's also the sign group and it's also the symmetric group on 2 elements.

## 3

- Cyclic group:Z3: This is the unique group of three elements. It is the cyclic group of order three. It's also the alternating group on three elements.

## 4

- Cyclic group:Z4: This is a cyclic group of four elements. It is the smallest nontrivial non-simple group.
- Klein four-group: This is the direct product of the cyclic group of order 2, with itself. It also occurs as the group of double transpositions in the symmetric group of order 4. It is the smallest non-cyclic elementary Abelian group.

## 5

- Cyclic group:Z5: This is a cyclic group on 5 elements.

## 6

- Cyclic group:Z6: This is the cyclic group on 6 elements. It is also the direct product of the cyclic group on 3 elements and the cyclic group on 2 elements.
- Symmetric group:S3: This is the symmetric group on 3 elements. It is also the same as the dihedral group on 3 elements, and is the semidirect product of the cyclic group of order 3 with the cyclic group of order 2. It is also the holomorph of the cyclic group of order 3, or equivalently, the affine group of order 1 of the finite field of 3 elements. It is the smallest

## 7

## 8

- Cyclic group:Z8
- Abelian group:Z4XZ2
- Elementary Abelian group:E8
- Quaternion group: This is smallest nilpotent non-Abelian group. It is also the smallest non-Abelian group tht is semidirectly indecomposable.
- Dihedral group:D8: This is also a smallest nilpotent non-Abelian group