====================================================================================================
PART A. Commutator tables derived from the presentations
G1: [c,b], [d,b], [e,c], [e,d], [d,c] = [b,a], [e,b] = [c,a], [e,a] = [d,a]^w
   number of relations 7, rank 7, det(relations + W-basis) = -1
[OK]   G1: the 7 relation vectors are independent
[OK]   G1: the W-basis ['[b,a]', '[c,a]', '[d,a]'] spans a complement; determinant is a unit for every p
   nonzero B(e_i,e_j), i>j: {(2, 1): (1, 0, 0), (3, 1): (0, 1, 0), (4, 1): (0, 0, 1), (4, 3): (1, 0, 0), (5, 1): (0, 0, omega), (5, 2): (0, 1, 0)}
[OK]   G1: commutator table equals the table of the note
G2: [d,a], [e,a], [c,b], [d,b], [e,c], [d,c] = [b,a], [e,b] = [c,a]
   number of relations 7, rank 7, det(relations + W-basis) = 1
[OK]   G2: the 7 relation vectors are independent
[OK]   G2: the W-basis ['[b,a]', '[c,a]', '[e,d]'] spans a complement; determinant is a unit for every p
   nonzero B(e_i,e_j), i>j: {(2, 1): (1, 0, 0), (3, 1): (0, 1, 0), (4, 3): (1, 0, 0), (5, 2): (0, 1, 0), (5, 4): (0, 0, 1)}
[OK]   G2: commutator table equals the table of the note
====================================================================================================
PART B. Symbolic identities (sympy)
-- G1
[OK]   G1: M(y) is the matrix of Section 3
   Pfaffians Pf_1..Pf_5 = [y1*y2, omega*y1*y3, -y2*y3, -y2**2, y1**2]
[OK]   G1: v(y) = (Pf_1,-Pf_2,Pf_3,-Pf_4,Pf_5) is the vector of Section 3
[OK]   G1: M(y) v(y) = 0 identically
   exponent vectors of the Pfaffians: ['[(0, 1, 1)]', '[(0, 2, 0)]', '[(1, 0, 1)]', '[(1, 1, 0)]', '[(2, 0, 0)]']
[OK]   G1: each Pfaffian is a unit times a monomial; monomials y1y2, y1y3, y2y3, y2^2, y1^2
[OK]   G1: M(0,0,1) has rank 2
[OK]   G1: the basis of R spans the radical of f_(0,0,1)
   equations of R on v(y): [y1*y2, omega*y1**2 + y2**2]
   charts: v(1,y2,y3) = [y2, -omega*y3, -y2*y3, y2**2, 1]    v(0,1,y3) = [0, 0, -y3, 1, 0]
[OK]   G1: N(v(y)) = (y1,y2)^T (row vector) has rank <= 1
[OK]   G1: v(1,t,s) = P_t + s Q_t
[OK]   G1: B(P_t, Q_t) = [t*(omega - t**2), -omega + t**2, 0]
[OK]   G1: v(0,1,s) = e4 + s Q_inf and B on l_inf is [-1, 0, 0]
[OK]   G1: B(e2, e3) = 0
[OK]   G1: B on the basis u1,u2,u3 of R is {(0, 1): [0, 0, 0], (0, 2): [0, -1, 0], (1, 2): [omega, 0, 0]}
[OK]   G1: B(x, v(y)) = (y2 L, -y1 L, 0) for x in R
[OK]   G1: B(v(z), v(y)) = Lambda(y,z) * kappa(y,z), Lambda = omega*y1*z1 - y2*z2
[OK]   G1: Lemma 3.3(b) parametrisation of the points of Z with X5 = 1
-- G2
[OK]   G2: M(y) is the matrix of Section 3
   Pfaffians Pf_1..Pf_5 = [y1*y2, y2*y3, y1*y3, -y2**2, y1**2]
[OK]   G2: v(y) = (Pf_1,-Pf_2,Pf_3,-Pf_4,Pf_5) is the vector of Section 3
[OK]   G2: M(y) v(y) = 0 identically
   exponent vectors of the Pfaffians: ['[(0, 1, 1)]', '[(0, 2, 0)]', '[(1, 0, 1)]', '[(1, 1, 0)]', '[(2, 0, 0)]']
[OK]   G2: each Pfaffian is a unit times a monomial; monomials y1y2, y1y3, y2y3, y2^2, y1^2
[OK]   G2: M(0,0,1) has rank 2
[OK]   G2: the basis of R spans the radical of f_(0,0,1)
   equations of R on v(y): [y2**2, y1**2]
   charts: v(1,y2,y3) = [y2, -y2*y3, y3, y2**2, 1]    v(0,1,y3) = [0, -y3, 0, 1, 0]
[OK]   G2: N(v(y)) = (y1,y2)^T (row vector) has rank <= 1
[OK]   G2: v(1,t,s) = P_t + s Q_t
[OK]   G2: B(P_t, Q_t) = [2*t**2, -2*t, 0]
[OK]   G2: v(0,1,s) = e4 + s Q_inf and B on l_inf is [0, 0, 0]
[OK]   G2: B(e2, e3) = 0
[OK]   G2: B on the basis u1,u2,u3 of R is {(0, 1): [-1, 0, 0], (0, 2): [0, -1, 0], (1, 2): [0, 0, 0]}
[OK]   G2: B(x, v(y)) = (y2 L, -y1 L, 0) for x in R
[OK]   G2: B(v(z), v(y)) = Lambda(y,z) * kappa(y,z), Lambda = y1*z2 + y2*z1
[OK]   G2: Lemma 3.3(b) parametrisation of the points of Z with X5 = 1
-- counting identities (polynomials in p)
[OK]   n3 = p^4 + p^3 - p^2 - p
[OK]   isotropic planes (n2 (p+1) + n3)/(p+1) = p^3 + 2p^2 + p + 1
[OK]   affine breadth-<=2 count 1 + (p-1)(2p^2+2p+1) = 2p^3 - p
[OK]   class number p^3 + n2 (p-1) p + n3 (p-1) = p^5 + 2p^4 - p^3 - p^2
[OK]   Theorem B: vertex counts p^2 + (p+1) + (p^2+p) = p^2 + (3p+1) + (p^2-p) = n2
[OK]   Theorem B: |E(G1)| = 3p(p+1)^2/2
[OK]   Theorem B: |E(G2)| = (3p^3+6p^2+p)/2 and |E(G1)| - |E(G2)| = p
-- Proposition C
[OK]   C(a): a, b^(1/s), c, d^(1/s), e^s satisfy the seven relations of G1(omega s^2)
[OK]   C(a): their images span G/G'
[OK]   C(b): with omega = s^2, a',...,e' satisfy the seven relations of G2
   [b',a'] = [-2*s, 2, 0]    [c',a'] = [2*s, 2, 0]
[OK]   C(b): [b',a'] = (-2s, 2, 0) and [c',a'] = (2s, 2, 0)
[OK]   C(b): det of the coefficient matrix of a',...,e' is 16 s^2
[OK]   Remark 4.2: <b + c/s, a + s d + e/s> = <P_t, Q_t> for t = s, omega = s^2
[OK]   Remark 4.2: <b - c/s, a - s d - e/s> = <P_t, Q_t> for t = -s, omega = s^2
[OK]   Remark 4.2: a - s d + e/s is NOT in <P_-s, Q_-s> (the sign misprint of the first write-up)
[OK]   Section 6: -B14(y) = M_{G1(1/omega)}(y1, y2, omega y3)
[OK]   Section 6: -B15(y) = M_{G2}(y)
Part B done (0.2 s)
====================================================================================================
PART C. Enumeration over F_p, p = 3, 5, 7
p=3  G1, omega=1 square         breadths {2: 25, 3: 96}  |S|=25  planes 1210  isotropic 49  iso3 0  lines in S 17  |A(G)|=3 ['<b, c>', '<a+d+e, b+c>', '<a+2d+2e, b+2c>']
       commuting graph: 69 edges, degrees {4: 9, 6: 10, 7: 6};  class number 369   (0.0 s)
[OK]   p=3 G1, omega=1 square: breadths: 25 points of breadth 2, the others of breadth 3
[OK]   p=3 G1, omega=1 square: breadth-2 locus = P(R) u Z(F_p), P(R) n Z(F_p) = D
[OK]   p=3 G1, omega=1 square: all 1210 planes enumerated; 49 isotropic (abelian A >= G' of order p^5)
[OK]   p=3 G1, omega=1 square: no isotropic 3-space (no abelian A >= G' of order p^6)
[OK]   p=3 G1, omega=1 square: lines inside the breadth-2 locus: p^2 + 2p + 2
[OK]   p=3 G1, omega=1 square: |A(G)| = 3 with the predicted members
[OK]   p=3 G1, omega=1 square: Theorem B degree pattern and 69 edges
[OK]   p=3 G1, omega=1 square: vertices of degree 2p = points on the members of A(G)
[OK]   p=3 G1, omega=1 square: class number p^5 + 2p^4 - p^3 - p^2 (Vaughan-Lee)
p=3  G1, omega=2 non-square     breadths {2: 25, 3: 96}  |S|=25  planes 1210  isotropic 49  iso3 0  lines in S 17  |A(G)|=1 ['<b, c>']
       commuting graph: 72 edges, degrees {4: 9, 6: 4, 7: 12};  class number 369   (0.0 s)
[OK]   p=3 G1, omega=2 non-square: breadths: 25 points of breadth 2, the others of breadth 3
[OK]   p=3 G1, omega=2 non-square: breadth-2 locus = P(R) u Z(F_p), P(R) n Z(F_p) = D
[OK]   p=3 G1, omega=2 non-square: all 1210 planes enumerated; 49 isotropic (abelian A >= G' of order p^5)
[OK]   p=3 G1, omega=2 non-square: no isotropic 3-space (no abelian A >= G' of order p^6)
[OK]   p=3 G1, omega=2 non-square: lines inside the breadth-2 locus: p^2 + 2p + 2
[OK]   p=3 G1, omega=2 non-square: |A(G)| = 1 with the predicted members
[OK]   p=3 G1, omega=2 non-square: Theorem B degree pattern and 72 edges
[OK]   p=3 G1, omega=2 non-square: vertices of degree 2p = points on the members of A(G)
[OK]   p=3 G1, omega=2 non-square: class number p^5 + 2p^4 - p^3 - p^2 (Vaughan-Lee)
p=3  G2                         breadths {2: 25, 3: 96}  |S|=25  planes 1210  isotropic 49  iso3 0  lines in S 17  |A(G)|=3 ['<c, e>', '<b, d>', '<b, c>']
       commuting graph: 69 edges, degrees {4: 9, 6: 10, 7: 6};  class number 369   (0.0 s)
[OK]   p=3 G2: breadths: 25 points of breadth 2, the others of breadth 3
[OK]   p=3 G2: breadth-2 locus = P(R) u Z(F_p), P(R) n Z(F_p) = D
[OK]   p=3 G2: all 1210 planes enumerated; 49 isotropic (abelian A >= G' of order p^5)
[OK]   p=3 G2: no isotropic 3-space (no abelian A >= G' of order p^6)
[OK]   p=3 G2: lines inside the breadth-2 locus: p^2 + 2p + 2
[OK]   p=3 G2: |A(G)| = 3 with the predicted members
[OK]   p=3 G2: Theorem B degree pattern and 69 edges
[OK]   p=3 G2: vertices of degree 2p = points on the members of A(G)
[OK]   p=3 G2: class number p^5 + 2p^4 - p^3 - p^2 (Vaughan-Lee)
p=5  G1, omega=1 square         breadths {2: 61, 3: 720}  |S|=61  planes 20306  isotropic 181  iso3 0  lines in S 37  |A(G)|=3 ['<b, c>', '<a+d+e, b+c>', '<a+4d+4e, b+4c>']
       commuting graph: 265 edges, degrees {6: 25, 10: 16, 11: 20};  class number 4225   (0.1 s)
[OK]   p=5 G1, omega=1 square: breadths: 61 points of breadth 2, the others of breadth 3
[OK]   p=5 G1, omega=1 square: breadth-2 locus = P(R) u Z(F_p), P(R) n Z(F_p) = D
[OK]   p=5 G1, omega=1 square: all 20306 planes enumerated; 181 isotropic (abelian A >= G' of order p^5)
[OK]   p=5 G1, omega=1 square: no isotropic 3-space (no abelian A >= G' of order p^6)
[OK]   p=5 G1, omega=1 square: lines inside the breadth-2 locus: p^2 + 2p + 2
[OK]   p=5 G1, omega=1 square: |A(G)| = 3 with the predicted members
[OK]   p=5 G1, omega=1 square: Theorem B degree pattern and 265 edges
[OK]   p=5 G1, omega=1 square: vertices of degree 2p = points on the members of A(G)
[OK]   p=5 G1, omega=1 square: class number p^5 + 2p^4 - p^3 - p^2 (Vaughan-Lee)
p=5  G1, omega=2 non-square     breadths {2: 61, 3: 720}  |S|=61  planes 20306  isotropic 181  iso3 0  lines in S 37  |A(G)|=1 ['<b, c>']
       commuting graph: 270 edges, degrees {6: 25, 10: 6, 11: 30};  class number 4225   (0.1 s)
[OK]   p=5 G1, omega=2 non-square: breadths: 61 points of breadth 2, the others of breadth 3
[OK]   p=5 G1, omega=2 non-square: breadth-2 locus = P(R) u Z(F_p), P(R) n Z(F_p) = D
[OK]   p=5 G1, omega=2 non-square: all 20306 planes enumerated; 181 isotropic (abelian A >= G' of order p^5)
[OK]   p=5 G1, omega=2 non-square: no isotropic 3-space (no abelian A >= G' of order p^6)
[OK]   p=5 G1, omega=2 non-square: lines inside the breadth-2 locus: p^2 + 2p + 2
[OK]   p=5 G1, omega=2 non-square: |A(G)| = 1 with the predicted members
[OK]   p=5 G1, omega=2 non-square: Theorem B degree pattern and 270 edges
[OK]   p=5 G1, omega=2 non-square: vertices of degree 2p = points on the members of A(G)
[OK]   p=5 G1, omega=2 non-square: class number p^5 + 2p^4 - p^3 - p^2 (Vaughan-Lee)
p=5  G1, omega=3 non-square     breadths {2: 61, 3: 720}  |S|=61  planes 20306  isotropic 181  iso3 0  lines in S 37  |A(G)|=1 ['<b, c>']
       commuting graph: 270 edges, degrees {6: 25, 10: 6, 11: 30};  class number 4225   (0.1 s)
[OK]   p=5 G1, omega=3 non-square: breadths: 61 points of breadth 2, the others of breadth 3
[OK]   p=5 G1, omega=3 non-square: breadth-2 locus = P(R) u Z(F_p), P(R) n Z(F_p) = D
[OK]   p=5 G1, omega=3 non-square: all 20306 planes enumerated; 181 isotropic (abelian A >= G' of order p^5)
[OK]   p=5 G1, omega=3 non-square: no isotropic 3-space (no abelian A >= G' of order p^6)
[OK]   p=5 G1, omega=3 non-square: lines inside the breadth-2 locus: p^2 + 2p + 2
[OK]   p=5 G1, omega=3 non-square: |A(G)| = 1 with the predicted members
[OK]   p=5 G1, omega=3 non-square: Theorem B degree pattern and 270 edges
[OK]   p=5 G1, omega=3 non-square: vertices of degree 2p = points on the members of A(G)
[OK]   p=5 G1, omega=3 non-square: class number p^5 + 2p^4 - p^3 - p^2 (Vaughan-Lee)
p=5  G1, omega=4 square         breadths {2: 61, 3: 720}  |S|=61  planes 20306  isotropic 181  iso3 0  lines in S 37  |A(G)|=3 ['<b, c>', '<a+2d+3e, b+3c>', '<a+3d+2e, b+2c>']
       commuting graph: 265 edges, degrees {6: 25, 10: 16, 11: 20};  class number 4225   (0.1 s)
[OK]   p=5 G1, omega=4 square: breadths: 61 points of breadth 2, the others of breadth 3
[OK]   p=5 G1, omega=4 square: breadth-2 locus = P(R) u Z(F_p), P(R) n Z(F_p) = D
[OK]   p=5 G1, omega=4 square: all 20306 planes enumerated; 181 isotropic (abelian A >= G' of order p^5)
[OK]   p=5 G1, omega=4 square: no isotropic 3-space (no abelian A >= G' of order p^6)
[OK]   p=5 G1, omega=4 square: lines inside the breadth-2 locus: p^2 + 2p + 2
[OK]   p=5 G1, omega=4 square: |A(G)| = 3 with the predicted members
[OK]   p=5 G1, omega=4 square: Theorem B degree pattern and 265 edges
[OK]   p=5 G1, omega=4 square: vertices of degree 2p = points on the members of A(G)
[OK]   p=5 G1, omega=4 square: class number p^5 + 2p^4 - p^3 - p^2 (Vaughan-Lee)
p=5  G2                         breadths {2: 61, 3: 720}  |S|=61  planes 20306  isotropic 181  iso3 0  lines in S 37  |A(G)|=3 ['<c, e>', '<b, d>', '<b, c>']
       commuting graph: 265 edges, degrees {6: 25, 10: 16, 11: 20};  class number 4225   (0.1 s)
[OK]   p=5 G2: breadths: 61 points of breadth 2, the others of breadth 3
[OK]   p=5 G2: breadth-2 locus = P(R) u Z(F_p), P(R) n Z(F_p) = D
[OK]   p=5 G2: all 20306 planes enumerated; 181 isotropic (abelian A >= G' of order p^5)
[OK]   p=5 G2: no isotropic 3-space (no abelian A >= G' of order p^6)
[OK]   p=5 G2: lines inside the breadth-2 locus: p^2 + 2p + 2
[OK]   p=5 G2: |A(G)| = 3 with the predicted members
[OK]   p=5 G2: Theorem B degree pattern and 265 edges
[OK]   p=5 G2: vertices of degree 2p = points on the members of A(G)
[OK]   p=5 G2: class number p^5 + 2p^4 - p^3 - p^2 (Vaughan-Lee)
p=7  G1, omega=1 square         breadths {2: 113, 3: 2688}  |S|=113  planes 140050  isotropic 449  iso3 0  lines in S 65  |A(G)|=3 ['<b, c>', '<a+d+e, b+c>', '<a+6d+6e, b+6c>']
       commuting graph: 665 edges, degrees {8: 49, 14: 22, 15: 42};  class number 21217   (0.9 s)
[OK]   p=7 G1, omega=1 square: breadths: 113 points of breadth 2, the others of breadth 3
[OK]   p=7 G1, omega=1 square: breadth-2 locus = P(R) u Z(F_p), P(R) n Z(F_p) = D
[OK]   p=7 G1, omega=1 square: all 140050 planes enumerated; 449 isotropic (abelian A >= G' of order p^5)
[OK]   p=7 G1, omega=1 square: no isotropic 3-space (no abelian A >= G' of order p^6)
[OK]   p=7 G1, omega=1 square: lines inside the breadth-2 locus: p^2 + 2p + 2
[OK]   p=7 G1, omega=1 square: |A(G)| = 3 with the predicted members
[OK]   p=7 G1, omega=1 square: Theorem B degree pattern and 665 edges
[OK]   p=7 G1, omega=1 square: vertices of degree 2p = points on the members of A(G)
[OK]   p=7 G1, omega=1 square: class number p^5 + 2p^4 - p^3 - p^2 (Vaughan-Lee)
p=7  G1, omega=2 square         breadths {2: 113, 3: 2688}  |S|=113  planes 140050  isotropic 449  iso3 0  lines in S 65  |A(G)|=3 ['<b, c>', '<a+3d+5e, b+5c>', '<a+4d+2e, b+2c>']
       commuting graph: 665 edges, degrees {8: 49, 14: 22, 15: 42};  class number 21217   (0.9 s)
[OK]   p=7 G1, omega=2 square: breadths: 113 points of breadth 2, the others of breadth 3
[OK]   p=7 G1, omega=2 square: breadth-2 locus = P(R) u Z(F_p), P(R) n Z(F_p) = D
[OK]   p=7 G1, omega=2 square: all 140050 planes enumerated; 449 isotropic (abelian A >= G' of order p^5)
[OK]   p=7 G1, omega=2 square: no isotropic 3-space (no abelian A >= G' of order p^6)
[OK]   p=7 G1, omega=2 square: lines inside the breadth-2 locus: p^2 + 2p + 2
[OK]   p=7 G1, omega=2 square: |A(G)| = 3 with the predicted members
[OK]   p=7 G1, omega=2 square: Theorem B degree pattern and 665 edges
[OK]   p=7 G1, omega=2 square: vertices of degree 2p = points on the members of A(G)
[OK]   p=7 G1, omega=2 square: class number p^5 + 2p^4 - p^3 - p^2 (Vaughan-Lee)
p=7  G1, omega=3 non-square     breadths {2: 113, 3: 2688}  |S|=113  planes 140050  isotropic 449  iso3 0  lines in S 65  |A(G)|=1 ['<b, c>']
       commuting graph: 672 edges, degrees {8: 49, 14: 8, 15: 56};  class number 21217   (0.9 s)
[OK]   p=7 G1, omega=3 non-square: breadths: 113 points of breadth 2, the others of breadth 3
[OK]   p=7 G1, omega=3 non-square: breadth-2 locus = P(R) u Z(F_p), P(R) n Z(F_p) = D
[OK]   p=7 G1, omega=3 non-square: all 140050 planes enumerated; 449 isotropic (abelian A >= G' of order p^5)
[OK]   p=7 G1, omega=3 non-square: no isotropic 3-space (no abelian A >= G' of order p^6)
[OK]   p=7 G1, omega=3 non-square: lines inside the breadth-2 locus: p^2 + 2p + 2
[OK]   p=7 G1, omega=3 non-square: |A(G)| = 1 with the predicted members
[OK]   p=7 G1, omega=3 non-square: Theorem B degree pattern and 672 edges
[OK]   p=7 G1, omega=3 non-square: vertices of degree 2p = points on the members of A(G)
[OK]   p=7 G1, omega=3 non-square: class number p^5 + 2p^4 - p^3 - p^2 (Vaughan-Lee)
p=7  G1, omega=4 square         breadths {2: 113, 3: 2688}  |S|=113  planes 140050  isotropic 449  iso3 0  lines in S 65  |A(G)|=3 ['<b, c>', '<a+2d+4e, b+4c>', '<a+5d+3e, b+3c>']
       commuting graph: 665 edges, degrees {8: 49, 14: 22, 15: 42};  class number 21217   (0.9 s)
[OK]   p=7 G1, omega=4 square: breadths: 113 points of breadth 2, the others of breadth 3
[OK]   p=7 G1, omega=4 square: breadth-2 locus = P(R) u Z(F_p), P(R) n Z(F_p) = D
[OK]   p=7 G1, omega=4 square: all 140050 planes enumerated; 449 isotropic (abelian A >= G' of order p^5)
[OK]   p=7 G1, omega=4 square: no isotropic 3-space (no abelian A >= G' of order p^6)
[OK]   p=7 G1, omega=4 square: lines inside the breadth-2 locus: p^2 + 2p + 2
[OK]   p=7 G1, omega=4 square: |A(G)| = 3 with the predicted members
[OK]   p=7 G1, omega=4 square: Theorem B degree pattern and 665 edges
[OK]   p=7 G1, omega=4 square: vertices of degree 2p = points on the members of A(G)
[OK]   p=7 G1, omega=4 square: class number p^5 + 2p^4 - p^3 - p^2 (Vaughan-Lee)
p=7  G1, omega=5 non-square     breadths {2: 113, 3: 2688}  |S|=113  planes 140050  isotropic 449  iso3 0  lines in S 65  |A(G)|=1 ['<b, c>']
       commuting graph: 672 edges, degrees {8: 49, 14: 8, 15: 56};  class number 21217   (0.9 s)
[OK]   p=7 G1, omega=5 non-square: breadths: 113 points of breadth 2, the others of breadth 3
[OK]   p=7 G1, omega=5 non-square: breadth-2 locus = P(R) u Z(F_p), P(R) n Z(F_p) = D
[OK]   p=7 G1, omega=5 non-square: all 140050 planes enumerated; 449 isotropic (abelian A >= G' of order p^5)
[OK]   p=7 G1, omega=5 non-square: no isotropic 3-space (no abelian A >= G' of order p^6)
[OK]   p=7 G1, omega=5 non-square: lines inside the breadth-2 locus: p^2 + 2p + 2
[OK]   p=7 G1, omega=5 non-square: |A(G)| = 1 with the predicted members
[OK]   p=7 G1, omega=5 non-square: Theorem B degree pattern and 672 edges
[OK]   p=7 G1, omega=5 non-square: vertices of degree 2p = points on the members of A(G)
[OK]   p=7 G1, omega=5 non-square: class number p^5 + 2p^4 - p^3 - p^2 (Vaughan-Lee)
p=7  G1, omega=6 non-square     breadths {2: 113, 3: 2688}  |S|=113  planes 140050  isotropic 449  iso3 0  lines in S 65  |A(G)|=1 ['<b, c>']
       commuting graph: 672 edges, degrees {8: 49, 14: 8, 15: 56};  class number 21217   (0.9 s)
[OK]   p=7 G1, omega=6 non-square: breadths: 113 points of breadth 2, the others of breadth 3
[OK]   p=7 G1, omega=6 non-square: breadth-2 locus = P(R) u Z(F_p), P(R) n Z(F_p) = D
[OK]   p=7 G1, omega=6 non-square: all 140050 planes enumerated; 449 isotropic (abelian A >= G' of order p^5)
[OK]   p=7 G1, omega=6 non-square: no isotropic 3-space (no abelian A >= G' of order p^6)
[OK]   p=7 G1, omega=6 non-square: lines inside the breadth-2 locus: p^2 + 2p + 2
[OK]   p=7 G1, omega=6 non-square: |A(G)| = 1 with the predicted members
[OK]   p=7 G1, omega=6 non-square: Theorem B degree pattern and 672 edges
[OK]   p=7 G1, omega=6 non-square: vertices of degree 2p = points on the members of A(G)
[OK]   p=7 G1, omega=6 non-square: class number p^5 + 2p^4 - p^3 - p^2 (Vaughan-Lee)
p=7  G2                         breadths {2: 113, 3: 2688}  |S|=113  planes 140050  isotropic 449  iso3 0  lines in S 65  |A(G)|=3 ['<c, e>', '<b, d>', '<b, c>']
       commuting graph: 665 edges, degrees {8: 49, 14: 22, 15: 42};  class number 21217   (0.9 s)
[OK]   p=7 G2: breadths: 113 points of breadth 2, the others of breadth 3
[OK]   p=7 G2: breadth-2 locus = P(R) u Z(F_p), P(R) n Z(F_p) = D
[OK]   p=7 G2: all 140050 planes enumerated; 449 isotropic (abelian A >= G' of order p^5)
[OK]   p=7 G2: no isotropic 3-space (no abelian A >= G' of order p^6)
[OK]   p=7 G2: lines inside the breadth-2 locus: p^2 + 2p + 2
[OK]   p=7 G2: |A(G)| = 3 with the predicted members
[OK]   p=7 G2: Theorem B degree pattern and 665 edges
[OK]   p=7 G2: vertices of degree 2p = points on the members of A(G)
[OK]   p=7 G2: class number p^5 + 2p^4 - p^3 - p^2 (Vaughan-Lee)
====================================================================================================
PART D. Hilbert function of F_p[y1,y2,y3]/I_4, degrees 0..8
   G1 p=3 omega=2   nonzero 4x4 minors: 25   Hilbert function: [1, 3, 6, 10, 3, 3, 3, 3, 3]
[OK]   G1 p=3 omega=2: Hilbert function 1,3,6,10,3,3,3,3,3 (a point of multiplicity 3)
   G1 p=5 omega=2   nonzero 4x4 minors: 25   Hilbert function: [1, 3, 6, 10, 3, 3, 3, 3, 3]
[OK]   G1 p=5 omega=2: Hilbert function 1,3,6,10,3,3,3,3,3 (a point of multiplicity 3)
   G1 p=5 omega=3   nonzero 4x4 minors: 25   Hilbert function: [1, 3, 6, 10, 3, 3, 3, 3, 3]
[OK]   G1 p=5 omega=3: Hilbert function 1,3,6,10,3,3,3,3,3 (a point of multiplicity 3)
   G1 p=7 omega=3   nonzero 4x4 minors: 25   Hilbert function: [1, 3, 6, 10, 3, 3, 3, 3, 3]
[OK]   G1 p=7 omega=3: Hilbert function 1,3,6,10,3,3,3,3,3 (a point of multiplicity 3)
   G1 p=7 omega=5   nonzero 4x4 minors: 25   Hilbert function: [1, 3, 6, 10, 3, 3, 3, 3, 3]
[OK]   G1 p=7 omega=5: Hilbert function 1,3,6,10,3,3,3,3,3 (a point of multiplicity 3)
   G1 p=7 omega=6   nonzero 4x4 minors: 25   Hilbert function: [1, 3, 6, 10, 3, 3, 3, 3, 3]
[OK]   G1 p=7 omega=6: Hilbert function 1,3,6,10,3,3,3,3,3 (a point of multiplicity 3)
   G2 p=3           nonzero 4x4 minors: 25   Hilbert function: [1, 3, 6, 10, 3, 3, 3, 3, 3]
[OK]   G2 p=3: Hilbert function 1,3,6,10,3,3,3,3,3 (a point of multiplicity 3)
   G2 p=5           nonzero 4x4 minors: 25   Hilbert function: [1, 3, 6, 10, 3, 3, 3, 3, 3]
[OK]   G2 p=5: Hilbert function 1,3,6,10,3,3,3,3,3 (a point of multiplicity 3)
   G2 p=7           nonzero 4x4 minors: 25   Hilbert function: [1, 3, 6, 10, 3, 3, 3, 3, 3]
[OK]   G2 p=7: Hilbert function 1,3,6,10,3,3,3,3,3 (a point of multiplicity 3)
====================================================================================================
summary (p, group, |A(G)|, edges of the commuting graph):
    (3, 'G1, omega=1 square', 3, 69)
    (3, 'G1, omega=2 non-square', 1, 72)
    (3, 'G2', 3, 69)
    (5, 'G1, omega=1 square', 3, 265)
    (5, 'G1, omega=2 non-square', 1, 270)
    (5, 'G1, omega=3 non-square', 1, 270)
    (5, 'G1, omega=4 square', 3, 265)
    (5, 'G2', 3, 265)
    (7, 'G1, omega=1 square', 3, 665)
    (7, 'G1, omega=2 square', 3, 665)
    (7, 'G1, omega=3 non-square', 1, 672)
    (7, 'G1, omega=4 square', 3, 665)
    (7, 'G1, omega=5 non-square', 1, 672)
    (7, 'G1, omega=6 non-square', 1, 672)
    (7, 'G2', 3, 665)
number of failed checks: 0
running time: 7 s
ALL CHECKS PASSED: True
