Find the number of non isomorphic directed graphs that after making all edges undirected are isomorphic with $K_{3,3}$.
I need to use Burnside's Lemma - which says that number of orbits (non isomorphic graphs) is equal to the average number of points fixed by an element in group. But I am not sure how the group of isomorphisms looks here. I read that group of permutations of $K_n$ is $S_n$ but I don't know how to find groups for other graphs. Also, in this case, do I count vertices or edges as fixed points? This is difficult, because until now I only used Burnside's Lemma to count some polyhedrons or necklaces
For the isomorphism group of the $K_{3,3}$ graph, think of its six vertices are partitioned into two subsets: three red vertices and three blue vertices.
Intuitively, every permutation of those six vertices that preserves the "red-blue" partition is an isomorphism of the $K_{3,3}$ graph: you can permute the red vertices amongst themselves and the blue vertices amongst them selves, and you can also turn the whole graph upside down, swapping the red vertex set and the blue vertex set. This intuitive idea can be used to derive an algebraic formula for the automorphism group, like this.
First, the product group $S_3 \times S_3$ acts like this: number the red vertices $R_1$, $R_2$, $R_3$ and the blue vertices $B_1$, $B_2$, $B_3$, choose any $(\sigma,\tau) \in S_3 \times S_3$, each expressed as a bijection of the set $\{1,2,3\}$, and let $$(\sigma,\tau)(R_i) = (R_{\sigma(i)}), \quad (\sigma,\tau)(B_j) = B_{\tau(j)} $$ Next, there is a "red-blue swap" isomorphism $$\rho(R_i)=B_i, \quad \rho(B_i)=R_i, \quad i \in \{1,2,3\} $$ and this is an order 2 permutation.
The order 2 permutation $\rho$ does not commute with the elements of the $S_3 \times S_3$ subgroup, and the isomorphism group is not isomorphic to the direct product $(S_3 \times S_3) \times S_2$. But the next best thing is true, namely you get a semidirect product structure $(S_3 \times S_3) \rtimes S_2$.
This might be enough for you to proceed with the problem: you might not need to know the details of the semidirect product structure in order to proceed with solving your problem. So I think I'll finish my answer off here, but let me know if you need additional details.