That leaves the elements , each of which is central and therefore forms a singleton conjugacy class. The correponding unirreps must (I guess) have dimension p. Then we have one-dimensional unirreps, and p-dimensional unirreps, and the squares of their dimensions add up to the order of the group, as they should: .
Edit. The analysis for group 2) looks very similar. Here, the centre is the cyclic subgroup Z generated by . There are again one-dimensional unirreps corresponding to representations of the quotient group G/Z, and p-dimensional unirreps corresponding to the elements of Z (other than the identity).