For more information about dihedral groups see this page.
Cayley Table
The Cayley table has four quadrants shown here in red,green,blue and orange, each of these quadrants is similar to the Cayley table for a cyclic group:
{1,1}  {r,1}  {r²,1}  {1,m}  {r,m}  {r²,m} 
{r,1}  {r²,1}  {1,1}  {r,m}  {r²,m}  {1,m} 
{r²,1}  {1,1}  {r,1}  {r²,m}  {1,m}  {r,m} 
{1,m}  {r,m}  {r²,m}  {1,1}  {r,1}  {r²,1} 
{r,m}  {r²,m}  {1,m}  {r,1}  {r²,1}  {1,1} 
{r²,m}  {1,m}  {r,m}  {r²,1}  {1,1}  {r,1} 
For more information about Cayley table see this page.
Cayley Graph
There are two cycles that go in opposite directions
For more information about Cayley graph see this page.
Cyclic Notation
The two generators are:
 A complete cycle
 A set of 2element cycles which swap opposite points
(1,2,3..n) , ( 1 n ) ( 2 (n1)… )
For more information about cyclic notation see this page.
Group Presentation
Thereare two generators, r (for rotate) is like a cyclic group, and m (for mirror) which flips the shape over.
<r , m  r^{n} =1,r² = 1, m r m= r^{1}>
or equivalently:
<x , y  x² = y² = xy^{n} = 1>
For more information about group presentation see this page.
Group Representation
One generator is a rotation of the identity matrix through 90° (the diagonal from bottomleft to topright), the other is the n^{th} root of the identity matix (such that lesser roots are not identity). See this page for information about taking roots of a matrix. One a matrix which will do this is an n×n matrix on the right:

, 

For more information about group representation see this page.