Rotation
Cyclic groups can represent finite rotations of symmetrical shapes in a plane.
c2
c3
Modulo Addition
Cyclic groups are equivalent to (isomorphic to) modulo n addition (denoted Z_{n}). If we take the group operation to be + then we would tend to refer to Z_{n} but if we take the group operation to be * then we would tend to refer to C_{n}.
Generators
If the group operation is multiplication then:
<r  r ^{n} =1>
If the group operation is addition then:
<z  n z = 0 >
Properties
Cyclic groups are Abelian which means that the group operation is commutative.
Cyclic Groups
A group whose elements can be written as e, a, a²… a^{n1}
shifting rows
One possibility would be to start with a row containing all the elements in order, this is the 'identity' row:
0  1  2  3 
Then shift the row to the right (modulo n).
1  2  3  0 
Repeat this until we have done a complete cycle, then put all the rows above each other, the completed table is:
0  1  2  3 
1  2  3  0 
2  3  0  1 
3  0  1  2 
Generating a Cyclic Group using a Program
We can use a computer program to generate these groups, here I have used Axiom/FriCAS which is described here.
c1 := cyclicGroup(1) <1> Type: PermutationGroup(Integer) toTable()$toFiniteGroup(c1,1)
Type: List(Matrix(Integer)) c2 := cyclicGroup(2)
<
(
1 2
)
>
Type: List(Matrix(Integer)) c3 := cyclicGroup(3)
<
(
1 2 3
)
>
Type: List(Matrix(Integer)) c4 := cyclicGroup(4)
<
(
1 2 3 4
)
>
permutationRepresentation(c4,4)
Type: List(Matrix(Integer)) c5 := cyclicGroup(5)
<
(
1 2 3 4 5
)
>
permutationRepresentation(c5,5)

where:
 The points of the permutation are numbered 1..n
 The elements of the group are named: "i" for the identity, single letters "a","b"... for the generators, and products of these.
 numbers in brackets are points of permutations represented in cyclic notation.
 cyclicGroup[1] is not really valid and the results for this case are nonsense.
 The Axiom/FriCAS program can't work in terms of the Cayley table, so I have added my own code to do this.