In Part I we use the two smallest non-abelian finite simple groups, namely the alternating group A5 and the general linear group L3(2) to define larger permutation groups of degrees 12 and 24, respectively. Specifically, we shall obtain highly symmetric sets of generators for each of the new groups and use these generating sets to deduce the groups' main properties. The first group will turn out to be the Mathieu group M12 of order 12 × 11 × 10 × 9 × 8 = 95 040 [70] and the second the Mathieu group M24 of order 24 × 23 × 22 × 21 × 20 × 16 × 3 = 244 823 040 [71]; they will be shown to be quintuply transitive on 12 and 24 letters, respectively. These constructions were first described in refs. [31] and [32].