C-groups of PSL(2,q) and PGL(2,q)

We classify the C-group representations of the groups PSL(2,q) and PGL(2,q). We obtain that the C-rank of PSL(2,q) and PGL(2,q) is 3 except when q∈{7,9,11,19,31} for PSL(2,q) and when q=5 for PGL(2,q), in which case it is 4. We provide all representations of rank four.