Why isn't the composition equal to the product? Let $G_1, G_2, H$ be three isomorph graphs. We are given the permutation $\phi:G_1\rightarrow G_2$ with $\phi=(1,2,4,3)$ and the permutation $\psi:G_2\rightarrow H$ with $\psi =(1,2)$. In my book there is the following: $$\psi \circ\phi =\phi \cdot \psi :G_1\rightarrow H$$ in order to compute a permuation of $G_1$ to get $H$. Isn't the composition the same as the product of permutations? Why are the permutations revesed at the product above?

There are two different conventions: the product of permutations can be defined either as composition or as composition in the reverse order. With the second convention, $(1\ 2)\cdot (1\ 2\ 4\ 3)$ means the permuation that is obtained by first doing $(1\ 2)$ and then doing $(1\ 2\ 4\ 3)$ (note that this is the opposite order of usual function composition!). This is presumably the convention your book uses.