For a finite group G, let AD(G) denote the Fourier norm of the antidiagonal in G × G. The author showed recently that AD(G) coincides with the amenability constant of the Fourier algebra of G and is equal to the normalized sum of the cubes of the character degrees of G. Motivated by a gap result for amenability constants by B. E. Johnson, we determine exactly which numbers in the interval [1,2] arise as values of AD(G). As a by-product, we show that the set of values of AD(G) does not contain all its limit points.

Some other calculations or bounds for AD(G) are given for familiar classes of finite groups. We also indicate a connection between AD(G) and the commuting probability of G, and use this to show that every finite group G satisfying AD(G) < 61/15 must be solvable; here, the value 61/15 is the best possible.