Bounding the order of a group with a large character degree

Let d be the degree of some irreducible character of a finite group G. We can write |G|=d(d+e) for some nonnegative integer e, and we show that if e>1, then |G|⩽Be6 for some universal constant B. This result, which improves a non-polynomial bound of Snyder, relies on recent work of Larsen, Malle and Tiep on character degrees of simple groups, and so it uses the simple group classification.