From this lecture note, it is known that the automorphism group of the $n$-cycle $C_n$ is the dihedral group $D_n$ with $2 n$ elements.

I use the following code to create a cycle graph of $6$ nodes.

g = CycleGraph[6] 

It gives me the following output.

Now I use the following code to compute its automorphism group.

GraphAutomorphismGroup[g] 

It gives me the following output.

PermutationGroup[{Cycles[{{2, 6}, {3, 5}}], Cycles[{{1, 2, 3, 4, 5, 6}}]}] 

On the other hand if I directly create a dihedral group it looks as follows.

In[15]:= DihedralGroup[6] // GroupGenerators Out[15]= {Cycles[{{1, 6}, {2, 5}, {3, 4}}], Cycles[{{1, 2, 3, 4, 5, 6}}]} 

So, why isn't the dihedral group same as the automorphism group of the cycle graph?