Equation 677 Database

Size-35 Latin 677 magma, again a 5-by-7 switched fibre product, but with a third off-diagonal fibre permutation. 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 have fibre coordinates u=0,1,...,6 in F_7: C0: 31,0,21,26,10,9,16 C1: 32,3,23,28,13,7,18 C2: 33,1,22,25,12,6,17 C3: 34,4,24,29,14,8,19 C4: 30,2,20,27,11,5,15 In these coordinates the quotient M/theta is the F_5 affine magma i ◇ j = 2i + 4j. The fibre operation is: (i,u) ◇ (j,v) = (2i+4j, 4u+3v) if i=j, (i,u) ◇ (j,v) = (2i+4j, r^{-1}(4r(u)+r(v))) if i≠j, where r=(1,2,5,0,6,3,4), i.e. r(0)=1, r(1)=2, ..., r(6)=4. Thus every theta-class is a copy of the F_7 affine law u◇v=4u+3v, while every off-diagonal fibre product is a conjugate of the F_7 affine law u◇v=4u+v. This is the same general construction type as magma#ed4c392b and magma#113fd524, but with off-diagonal twist r=(1,2,5,0,6,3,4) rather than their previously observed twists. This is not the direct product: there is no size-5 complement submagma. The proper submagmas are exactly the five theta-classes above, together with the five idempotent singletons {30},{31},{32},{33},{34}. Any pair of elements from distinct theta-classes generates the whole 35-element magma. The automorphism group has order 20. In the displayed coordinates it acts as the full affine automorphism group of the F_5 quotient and does not introduce independent fibre scalings. Its seven 5-point orbits are the horizontal layers {0..4}, {5..9}, ..., {30..34}.