Beautifully written; I greatly appreciate it. For anyone who, like me, was a little "dense" on seeing c as an integer, let me save you the effort. Pick t first, then pick b large enough so that c=tb is an integer.

@Reza: one of the major problems we are having with your question is that you refer to a group, which only has one binary operation, yet you invoke both multiplication (represented by juxtaposition and -1 exponents) and also addition (in the form of summation symbols). What exactly is the algebraic construct in which you are performing these two binary operations? From a modern algebra standpoint, that is (with very high likelihood) a ring.

What we can say is that this answer uses the Schur-Zassenhaus theorem. Note also that a conjugate of the (subgroup of the) acting group is used there.

FWIW, the second of these is even a wreath product: $(C_2\times C_2)\rtimes C_2 = C_2\wr C_2$, though I admit this doesn't compare a wreath product to a wreath product.

If there were a proof, or even a heuristic, that says something like "the ratio $|{\rm Irr}(G)|/|{\rm cd}(G)|$ is bounded above by a subexponential," I'll accept this answer. FWIW, I've now run some (primitive) code that suggests, for large enough $n$ (checked up to $n=55$), the ratio hovers right around the range $2.2$ to $2.4$, so I'm willing to believe (conjecturally) the function is even constant.