p=7 al=2 be=3 : 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: 0  (hypothesis (E) holds)
extension of order 2744: associativity, centrality of (0,s,1), inverses and the projection onto Gamma_p checked on 200000 random triples
|Gamma_p| = 392, conjugacy classes: 11
homomorphisms pi_1(S_2) -> Gamma_p: 261651376
epimorphisms: 169344000
epimorphisms with kappa = 0 (lift): 27095040 , of which J intransitive: 1072512
epimorphisms with kappa != 0 (do not lift): 142248960 , of which J intransitive: 0
   kappa=1: 23708160 = 392 * 60480 (remainder 0) ; J intransitive: 0 = 392 * 0
   kappa=2: 23708160 = 392 * 60480 (remainder 0) ; J intransitive: 0 = 392 * 0
   kappa=3: 23708160 = 392 * 60480 (remainder 0) ; J intransitive: 0 = 392 * 0
   kappa=4: 23708160 = 392 * 60480 (remainder 0) ; J intransitive: 0 = 392 * 0
   kappa=5: 23708160 = 392 * 60480 (remainder 0) ; J intransitive: 0 = 392 * 0
   kappa=6: 23708160 = 392 * 60480 (remainder 0) ; J intransitive: 0 = 392 * 0
non-surjective homomorphisms with kappa != 0: 77545440
surjectivity test compared with a closure computation on 164452 tuples (random sample): 0 disagreements
J = <x,[x,u],v> compared with <x,uxu^-1,v,wvw^-1> on 163328 tuples (random sample): 0 disagreements
RESULT p=7: every epimorphism which does not lift has J transitive
