  We prove that the group of reversible cellular automata (RCA), on any
alphabet $A$, contains a subgroup generated by three involutions which contains
an isomorphic copy of every finitely generated group of RCA on any alphabet
$B$. This result follows from a case study of groups of RCA generated by symbol
permutations and partial shifts (equivalently, partitioned cellular automata)
with respect to a fixed Cartesian product decomposition of the alphabet. For
prime alphabets, we show that this group is virtually cyclic, and that for
composite alphabets it is non-amenable. For alphabet size four, it is a linear
group. For non-prime non-four alphabets, it contains copies of all finitely
generated groups of RCA. We also prove this property for the group generated by
RCA of biradius one on any full shift with large enough alphabet, and also for
some perfect finitely generated groups of RCA.
