On Involutions with Many Fixed Points in Gassmann Triples

We show that in a non-trivial Gassmann triple (G,H,H′) of index n there does not exist an involution τ ∈ G such that the value of the permutation character on τ is n− 2. In addition we describe a GAP program designed to search for examples of Gassmann triples and give a brief summary of the results of this search.