Fact1: |C_S5(5-cycle)| = 5  all even: True
Fact2: |X| = 12  |N_S5(X)| = 24  all fix 4: True  |N_A5(X)| = 12
Fact2b: #S5-conjugates = 5  #A5-conjugates = 5  equal: True
Fact3: V_4 normal abelian in A_4, A_4 not nilpotent -> F*(A_4) = V_4, A_4/V_4 = C_3
Fact4: number of odd elements of N_S5(X): 12  all invert X/V: True
Fact5: |N_A5(P2)|, |N_A5(P3)|, |N_A5(P5)| = 12 6 10
ALL CHECKS PASSED
