Equation 677 Database

Size-35 Latin 677 magma with a nontrivial 5-by-7 fibre structure. There is a congruence theta with five 7-element classes C0={0,9,11,16,21,26,31}, C1={1,6,12,17,20,25,32}, C2={2,5,10,15,22,27,30}, C3={3,7,13,18,23,28,33}, C4={4,8,14,19,24,29,34}. The quotient M/theta is the size-5 affine magma magma#e549b5f8, i.e. i◇j = 2i+4j over F_5. Each theta-class is a submagma isomorphic to magma#baf8b55c, the F_7 affine law u◇v = 4u+3v. More explicitly, use base coordinate i in F_5 by membership in Ci. Use the seven horizontal layers {0..4},{5..9},...,{30..34}; relabel their layer numbers 0,1,2,3,4,5,6 by u=1,5,4,6,2,3,0 in F_7. Then the operation is: (i,u)◇(j,v) = (2i+4j, 4u+3v) if i=j, (i,u)◇(j,v) = (2i+4j, p^{-1}(4p(u)+p(v))) if i≠j, where p=(1,2,4,0,6,3,5), i.e. p(0)=1, p(1)=2, ..., p(6)=5. Thus the diagonal fibres use the size-7 law 4u+3v, while all off-diagonal fibre products use the other size-7 law, isomorphic to magma#7981e2df, the F_7 law x◇y=4x+y. This is not the direct product magma#e549b5f8 × magma#baf8b55c: there are no size-5 complement submagmas. 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, namely the horizontal layers {0..4}, {5..9}, ..., {30..34}.