Equation 677 Database

Size-35 Latin 677 magma with a split 5-by-7 switched fibre-product structure. 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: 30,3,10,16,7,20,12 C1: 31,1,9,17,5,22,14 C2: 32,2,8,18,6,23,15 C3: 34,28,27,29,26,24,25 C4: 33,0,11,19,4,21,13 The quotient M/theta is magma#e549b5f8, the F_5 affine law i ◇ j = 2i + 4j. 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=(0,1,4,6,2,3,5), i.e. q(0)=0, q(1)=1, ..., q(6)=5. Thus every theta-class is a copy of magma#7981e2df, the F_7 affine law u◇v=4u+v. Off the diagonal, the same F_7 law is used after the common nonlinear fibre relabeling q. Unlike several nearby size-35 switched fibre products, this one is split: the five idempotents {30,31,32,33,34} form a size-5 complement submagma to theta. In the above coordinates this complement is F_5 × {0}, namely 30,31,32,34,33 in quotient-coordinate order. The proper submagmas are exactly the five theta-classes, the complement {30,31,32,33,34}, and the five idempotent singletons. Two distinct elements in the same theta-class generate that whole 7-element class; two distinct idempotents generate the size-5 complement; every other pair from distinct theta-classes generates the whole magma. The automorphism group has order 20. It is induced by the full affine automorphism group of the F_5 quotient and fixes the fibre coordinate u. Its seven 5-point orbits are: {0,1,2,3,28}, {4,5,6,7,26}, {8,9,10,11,27}, {12,13,14,15,25}, {16,17,18,19,29}, {20,21,22,23,24}, {30,31,32,33,34}.