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Let  V  be a general three-dimensional quartic in the complex projective space ℙ 4  such that the only singularity of  V  is a double point of rank 3. We prove that  V  is a birationally rigid variety. Its group of birational self-maps is, up to the finite subgroup of biregular automorphisms, a free product of 25 cyclic groups of order 2. It follows that the complement to the set of birationally rigid factorial quartics with terminal singularities is of codimension at least 3 in the natural parameter space.

Birational rigidity of quartic three-folds with a double point of rank 3