Let G=Z4 be the cyclic group on 4 elements, generated by S=−1,1, let H=Z2timesZ2 be the Klein four group, generated by T=(0,1),(1,0). Then |S|=|T| and both Cayley graphs are isomorphic to C4, the cycle of length 4.
For n>2 each even cycle C2n is a Cayley graph for the cyclic group Z2n and for the dihedral group Dn of order 2n.
Another well-known example is the graph of a cube Q3 which is a Cayley graph for the abelian group Z4timesZ2 and for the dihedral group D4. In the previous example the dihedral group was generated by two involutions, while in the latter case it is generated by an involution and an element of order 4.
If only generators are counted, without their inverses, the first two examples do not give matching counts.
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