p=3 al=1 be=1 : 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 216: associativity, centrality of (0,s,1), inverses and the projection onto Gamma_p checked on 200000 random triples
|Gamma_p| = 72, conjugacy classes: 6
homomorphisms pi_1(S_2) -> Gamma_p: 1592136
epimorphisms: 1036800
epimorphisms with kappa = 0 (lift): 414720 , of which J intransitive: 25920
epimorphisms with kappa != 0 (do not lift): 622080 , of which J intransitive: 0
   kappa=1: 311040 = 72 * 4320 (remainder 0) ; J intransitive: 0 = 72 * 0
   kappa=2: 311040 = 72 * 4320 (remainder 0) ; J intransitive: 0 = 72 * 0
non-surjective homomorphisms with kappa != 0: 321840
surjectivity test compared with a closure computation on 356481 tuples (all): 0 disagreements
J = <x,[x,u],v> compared with <x,uxu^-1,v,wvw^-1> on 356481 tuples (all): 0 disagreements
RESULT p=3: every epimorphism which does not lift has J transitive
