p=13 al=0 be=5 : 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 17576: associativity, centrality of (0,s,1), inverses and the projection onto Gamma_p checked on 200000 random triples
|Gamma_p| = 1352, conjugacy classes: 26
homomorphisms pi_1(S_2) -> Gamma_p: 11314040296
epimorphisms: 6950361600
epimorphisms with kappa = 0 (lift): 572382720 , of which J intransitive: 41371200
epimorphisms with kappa != 0 (do not lift): 6377978880 , of which J intransitive: 30371328
   kappa=1: 531498240 = 1352 * 393120 (remainder 0) ; J intransitive: 2530944 = 1352 * 1872
   kappa=2: 531498240 = 1352 * 393120 (remainder 0) ; J intransitive: 2530944 = 1352 * 1872
   kappa=3: 531498240 = 1352 * 393120 (remainder 0) ; J intransitive: 2530944 = 1352 * 1872
   kappa=4: 531498240 = 1352 * 393120 (remainder 0) ; J intransitive: 2530944 = 1352 * 1872
   kappa=5: 531498240 = 1352 * 393120 (remainder 0) ; J intransitive: 2530944 = 1352 * 1872
   kappa=6: 531498240 = 1352 * 393120 (remainder 0) ; J intransitive: 2530944 = 1352 * 1872
   kappa=7: 531498240 = 1352 * 393120 (remainder 0) ; J intransitive: 2530944 = 1352 * 1872
   kappa=8: 531498240 = 1352 * 393120 (remainder 0) ; J intransitive: 2530944 = 1352 * 1872
   kappa=9: 531498240 = 1352 * 393120 (remainder 0) ; J intransitive: 2530944 = 1352 * 1872
   kappa=10: 531498240 = 1352 * 393120 (remainder 0) ; J intransitive: 2530944 = 1352 * 1872
   kappa=11: 531498240 = 1352 * 393120 (remainder 0) ; J intransitive: 2530944 = 1352 * 1872
   kappa=12: 531498240 = 1352 * 393120 (remainder 0) ; J intransitive: 2530944 = 1352 * 1872
non-surjective homomorphisms with kappa != 0: 4008382560
surjectivity test compared with a closure computation on 132628 tuples (random sample): 0 disagreements
J = <x,[x,u],v> compared with <x,uxu^-1,v,wvw^-1> on 131479 tuples (random sample): 0 disagreements
RESULT p=13: there are epimorphisms which do not lift and have J intransitive
