I think that, rather than starting with the genes, you might consider starting with the alleles, or better yet, an explanation of the only two colors (pigments/melanin) found in chickens - black (eumelanin) and red (pheomelanin).
With the exception of the 3 pattern genes, basically all genes either intensify, extend, dilute, remove, or have no effect, on red, black, or both. How these genes all appear and interact with one another depends generally on which allele they they are found.
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The four groups come from Van Dort's book. My understanding of what I read is that these groups are based upon the loosely similar way in which they act. For example, in the second group all of the genes (Quote)"Change the pigment uniformly over the total chciken". Even though there are several different genes in the group, the general net affect of each is that the color change is across the total chicken. I will be elaborate through each of these groups over several posts so that it isn't too overwhelming.