Equation 677 Database

Size-35 Latin 677 magma with the same general 5-by-7 switched fibre-product shape as magma#ed4c392b, but with a different off-diagonal fibre twist. There is a congruence theta with five 7-element classes. In the following coordinate lists, the class index i is in F_5 and the listed entries are the fibre coordinates u=0,1,...,6 in F_7: C0: 30,0,23,28,13,8,18 C1: 33,1,20,26,10,6,15 C2: 32,3,22,27,12,7,17 C3: 31,2,21,25,11,5,16 C4: 34,4,24,29,14,9,19 In these coordinates the quotient M/theta is magma#e549b5f8, the F_5 affine law i ◇ j = 2i + 4j. The operation on the fibres is: (i,u) ◇ (j,v) = (2i+4j, 4u+3v) if i=j, (i,u) ◇ (j,v) = (2i+4j, q^{-1}(4q(u)+q(v))) if i≠j, where q=(1,2,4,0,5,3,6), meaning q(0)=1, q(1)=2, ..., q(6)=6. Thus each theta-class is a submagma isomorphic to magma#baf8b55c, the F_7 affine law u◇v=4u+3v. All off-diagonal products use a conjugate of the other size-7 affine law magma#7981e2df, namely u◇v=4u+v. This differs from magma#ed4c392b by replacing its off-diagonal permutation p=(1,2,4,0,6,3,5) with q=(1,2,4,0,5,3,6). This is not a direct product and has no size-5 complement submagma. The proper submagmas are exactly the five 7-element theta-classes above, together with the five idempotent singletons {30},{31},{32},{33},{34}. Any pair from distinct theta-classes generates the whole 35-element magma. The automorphism group has order 20. It induces the full affine automorphism group of the F_5 quotient and has seven 5-point orbits on M, namely the horizontal layers {0..4}, {5..9}, ..., {30..34}.