p=5 al=0 be=2 : Q_8 closed with 8 matrices of determinant 1, non-abelian=1, elements of order 4: 6
elements of Q_8 other than +-1 with an eigenvalue in F_p: 6  (hypothesis (E) FAILS)
extension of order 1000: associativity, centrality of (0,s,1), inverses and the projection onto Gamma_p checked on 200000 random triples
|Gamma_p| = 200, conjugacy classes: 8
homomorphisms pi_1(S_2) -> Gamma_p: 34375000
epimorphisms: 22464000
epimorphisms with kappa = 0 (lift): 5184000 , of which J intransitive: 753600
epimorphisms with kappa != 0 (do not lift): 17280000 , of which J intransitive: 192000
   kappa=1: 4320000 = 200 * 21600 (remainder 0) ; J intransitive: 48000 = 200 * 240
   kappa=2: 4320000 = 200 * 21600 (remainder 0) ; J intransitive: 48000 = 200 * 240
   kappa=3: 4320000 = 200 * 21600 (remainder 0) ; J intransitive: 48000 = 200 * 240
   kappa=4: 4320000 = 200 * 21600 (remainder 0) ; J intransitive: 48000 = 200 * 240
non-surjective homomorphisms with kappa != 0: 9132000
surjectivity test compared with a closure computation on 22567 tuples (random sample): 0 disagreements
J = <x,[x,u],v> compared with <x,uxu^-1,v,wvw^-1> on 22688 tuples (random sample): 0 disagreements
RESULT p=5: there are epimorphisms which do not lift and have J intransitive
