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In this article we study the bicanonical map $\phi_2$ of quadruple Galoiscanonical covers X of surfaces of minimal degree. We show that $\phi_2$ hasdiverse behavior and exhibit most of the complexities that are possible for abicanonical map of surfaces of general type, depending on the type of X. Thereare cases in which $\phi_2$ is an embedding, and if so happens, $\phi_2$ embeds$X$ as a projectively normal variety, and cases in which $\phi_2$ is not anembedding. If the latter, $\phi_2$ is finite of degree 1, 2 or 4. We also studythe canonical ring of X, proving that it is generated in degree less than orequal to 3 and finding the number of generators in each degree. For generatorsof degree 2 we find a nice general formula which holds for canonical covers ofarbitrary degrees. We show that this formula depends only on the geometric andthe arithmetic genus of X.

On the bicanonical morphism of quadruple Galois canonical covers