Cayley networks have desirable properties for interconnection networks, but the degree of many well-known Cayley networks grows with the number of nodes. Therefore, fixed-degree Cayley networks have also been introduced. We consider fixed-degree Cayley networks which are also subnetworks of the pancake network P n . The pancake problem concerns the number of prefix reversals or “flips,” required to sort a permutation of length n. This is also the diameter of P n . Restricting the problem to three of the n−1 possible flips, generates a subnetwork of P n . We identify proper subnetworks and spanning subnetworks of P n generated by three flips. We introduce a degree 3 spanning subnetwork of P n , the Triad network, or Triad n . When n is odd and n mod 8≠1, Triad n has n! nodes and diameter Θ(n log n) . Triad n emulates the shuffle-exchange and shuffle-exchange permutation networks with constant slowdown.