For symmetric group conjugacy classes and irreducible representation both are parametrized by Young diagramms, so there is a kind of "good" bijection between the two sets. For general finite groups see MO discussion.

**Question:** What are the finite groups where some "good" bijection(s) between conjugacy classes and irreducible representations are known ?

"Good" bijection is an informal "definition", nevertheless I hope example of S_n and other examples listed below, may convince that the question makes sense.

I think that it is far too optimistic to have one unique bijection for general group,
but it seems to me that for certain classes of groups there can be some set of "good" bijections. Let me briefly discuss below some properties which "good" bijection may satisfy, and may be discuss details in the next question.

Some examples:

**1)** symmetric group S_n

**2)** Z/2Z is naturally isomprhic to its dual, as well as $Z/2Z \oplus Z/2Z$ see e.g. MO "fantastic properties of Z/2Z"

**3)** Generally for abelian finite groups: among all set-theoretic bijections $G \to \hat G$, some are distinguished that they are group isomorphisms. So we have non unique, but
a class of "good" bijections.

**4)** For GL(2,F_q) Paul Garret writes: "conjugacy classes match in an ad hoc fashion with specic representations". (See here table at page 11).

**5)** G. Kuperberg describes relation of the McKay correspondence and that kind of bijection for A_5 (or its central extension), see here.

**6)** If I understand correctly here at MO D. Jordan mentions that bijection exists
for Coxeter groups. (I would be thankful for detailed reference).

**7)** Dihedral groups $D_{2n}$ - see answer by Glasby below

**8)** Finite Heisenberg group with $p^{2n+1}$ elements, also known as extraspecial group - see answer by Glasby below

**9)** Quternionic group $Q_{8}$ - this actually can be seen as a particular example of the item above. Or note that it is $Z/2Z$ central extension of $Z/2Z \oplus Z/2Z$,
and $ Z/2Z \oplus Z/2Z$ has natural bijection
as mentioned in item 2 above, and it is easy to extend it to $Q_8$.

**10)**
It seems that for Drinfeld double of a finite group (and probably more generally for "modular categories") there is known some analog of natural bijection.
There is such remark at page 5 of
Drinfeld Doubles for Finite Subgroups
of SU(2) and SU(3) Lie Groups.
R. COQUEREAUX, Jean-Bernard ZUBER:

In other words, there is not only an equal number of classes and irreps in a double, there is also a canonical bijection between them.

There can be several properties which "good" bijection may satisfy, at least for some "good" groups

1) Respect the action of $Out(G)$. Actually the two sets are not isomorphic in general as $Out(G)$-sets (see MO21606), however there are many cases where they are isomorphic see G. Robinson's MO-answer.

2) Reality/Rationality constraints. Again in general there is no correspondence see MO, but there are some cases where corresponding properties of classes and irreps agree - see J. Schmidt's answer on that question.

Two properties below are even more problematic

3) It might be that product on conjugacy classes have something to do with tensor product of representations (at least for abelian groups we may require these two fully agree).

4) If to think about kind of "orbit method" ideology, and think that conjugacy class is in some sense perversed coadjoint orbit, we may hope that structure of conjugacy representation, should somehow respect the "good" bijection. For example for $S_n$ we proved that irrep corresponding to Young diagram "d" lives inside the conjugacy subrepresentation realized as functions on the conjugacy class parametrized exactly by "d". (See MO153561, MO153991 for some discussion of conjugacy (adjoint) representation).

5) For algebraic groups over finite fields conjugacy classes and irreps sometimes naturally divided into families (e.g. conjugacy classes are often parametrized by equations $ F_{t_i}(x_k) = 0 $ - changing "t" we get different conjugacy classes in the same "family"). So we may hope that good bijection respects the families. (It is works fine for Heisenberg group, but for UT(4,p) I have met some problems).