Is there somewhere a database of incidence matrices of generalized quadrangles that one can download?
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You can compute them with the FinInG package in the computer algebra system GAP. FinInG has available all the classical generalised quadrangles, elation generalised quadrangles (with a $q$clan, flock, or Kantor family as input), and it shouldn't be difficult to compute the known Paynederived quadrangles from the symplectic generalised quadrangles. Here is an example:


