We say that a sequence {π_{_i}} of permutations is quasirandom if, for each k>1 and each σ∈S_{_k}, the probability that a uniformly chosen k-set of entries of π_{_i} induces σ tends to 1/k! as i tends to infinity. It is known that a much weaker condition already forces π_{_i} to be quasirandom; namely, if the above property holds for all σ∈S₄. We further weaken this condition by exhibiting sets S⊆S₄, such that if randomly chosen four entries of π_{_i} induce an element of S with probability tending to |S|/24, then {π_{_i}} is quasirandom. Moreover, we are able to completely characterise the sets S with this property. In particular, there are exactly ten such sets, the smallest of which has cardinality eight. This is joint work with Timothy Chan, Daniel Král', Jon Noel, Maryam Sharifzadeh and Jan Volec.