Equation 677 Database

Size-35 Latin 677 magma in the non-split 5-by-7 switched fibre-product family. There is a unique nontrivial proper 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 coordinate u=0,1,...,6 in F_7: C0: 33,0,10,21,5,25,16 C1: 31,1,11,23,6,28,19 C2: 32,3,13,20,8,27,17 C3: 30,2,12,22,7,26,15 C4: 34,4,14,24,9,29,18 The quotient M/theta is the usual size-5 affine quotient i ◇ j = 2i + 4j over F_5. In these coordinates the operation is: (i,u) ◇ (j,v) = (2i+4j, 4u+v) if i=j, (i,u) ◇ (j,v) = (2i+4j, q^{-1}(4q(u)+q(v))) if i≠j, where q=(6,1,2,0,5,3,4), i.e. q(0)=6, q(1)=1, ..., q(6)=4. Thus every theta-class is a copy of the F_7 affine law u◇v=4u+v, while every off-diagonal fibre product uses the same law after the common fibre relabeling q. This places the magma in the same broad “7981e2df-on-the-diagonal” branch as some nearby size-35 examples, but with a different off-diagonal twist. This example is not split. The five idempotents {30,31,32,33,34} form a transversal to theta, but not a submagma; for instance 30◇31=10. In fact there is no size-5 complement submagma. The proper submagmas are exactly the five theta-classes above and the five idempotent singletons {30},{31},{32},{33},{34}. Two distinct elements in the same theta-class generate that whole 7-element class, while any pair from distinct theta-classes generates the whole 35-element magma. The automorphism group has order 20. It is induced by the full affine automorphism group of the F_5 quotient; the off-diagonal twist kills all nontrivial independent fibre scalings. Its seven 5-point orbits are the horizontal layers {0..4}, {5..9}, {10..14}, {15..19}, {20..24}, {25..29}, and {30..34}.