For instance, when p=2 this implies that every connected 2-compact group can be written BG = BH × BDI(4), where BH is the 2-completion of the classifying space of a connected compact Lie group, and BDI(4) denotes s copies of the " Dwyer -Wilkerson 2-compact group" BDI(4) of rank 3, constructed in Dwyer & Wilkerson (1993) with Weyl group corresponding to group number 24 in the Shepard - Todd enumeration of complex reflection groups. For p=3 a similar statement holds but the new exotic 3-compact group is now group number 12 on the Shepard-Todd list, of rank 2. For primes greater than 3, family 2 on the Shepard-Todd list will contain infinitely many exotic p-compact groups.