For which triples $\{A,B,C\}$ of positive integers does there exist a (finite or compact) group $G$ with unitary irreducible representations of dimensions $A$,$B$, and $ C$ whose tensor product contains the trivial representation?

Is this known in general? If not, what useful necessary and/or sufficient conditions are known?

For example, it is necessary that $A \le BC$ , $B \le AC$, and $C \le AB$. This is clear, since the tensor product of any two of the representations must contain the dual of the third (whose dimension therefore can't be larger than the product of dimensions of the other two).

Given allowed triples $(A_1,B_1,C_1)$ and $(A_2,B_2,C_2)$ corresponding to groups $G_1$ and $G_2$ and representations $(R_1,R_2,R_3)$ and $(R'_1,R'_2,R'_3)$, we have that $(A_1 A_2, B_1 B_2, C_1 C_2)$ is also allowed, since we can take the product group and the combined representations $((R_1,R'_1),(R_2,R'_2),(R_3,R'_3))$. So we could equivalently try to characterize the `elementary' triples that cannot be obtained via this product construction.

**Update:** It is possible to prove that a necessary condition is that

$ABC - A^2 - B^2 - C^2 + gcd(A,B) + gcd(A,C) + gcd(B,C) - gcd(A,B,C) \ge 0$

For $(A,B,C) = (2,B,C)$, this can also be shown to be sufficient. In this case, the allowed cases are $(2,N,N)$ and $(2,NK,(N+1)K)$. See https://arxiv.org/abs/1801.03508. It would be interesting to understand whether or not this condition is sufficient in general.