This paper investigates the structure of groups with non-trivial center which can be written as a product of two simple subgroups. Examples must be covering groups of simple groups. For example \(2\cdot PSU(4,3)=AB\) where \(A\simeq PSp(4,3)\) and \(B\simeq PSL(4,3)\) and \(3\cdot PSU(4,3)=AB\) where \(A\simeq PSp(4,3)\) and \(B\simeq PSU(3,3).\) The main results of the paper include the fact that no covering group of an alternating group can be written as a product of two simple subgroups. The fact that if \(G=AB\) where A and B are simple subgroups with G/N a sporadic simple group for some normal subgroup N of G, then either \(G\simeq A\times B\), \(N=1\) and G is a simple group, (all these factorizations are known), or \(| N| =3\) and G/N\(\simeq Suz\), \(A\simeq PSU(5,2)\) and \(B\simeq G_ 2(4)\). A similar result is proven when G/N is a group of Lie type 1 or 2 and N is the center of G and has prime order. In this case there are several possibilities.