Equation 677 Database

This is a Latin 677 magma with a 5-by-7 congruence decomposition. its diagonal 7-element fibres are copies of magma#7981e2df, the F_7 affine law u◇v=4u+v. 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 coordinate u=0,1,...,6 in F_7: C0: 33,0,13,23,5,28,17 C1: 30,1,11,20,7,25,15 C2: 31,2,10,21,6,26,16 C3: 32,3,12,22,8,27,18 C4: 34,4,14,24,9,29,19 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=(6,2,5,0,4,3,1), i.e. q(0)=6, q(1)=2, ..., q(6)=1. Thus Labels C and D form the “7981e2df-on-the-diagonal” variant of the construction, distinguished by their different off-diagonal twists: C uses (6,0,2,5,1,3,4), while this magma uses (6,2,5,0,4,3,1). Proper submagmas are exactly the five theta-classes above and the five idempotent singletons {30},{31},{32},{33},{34}. Any pair from distinct theta-classes generates the whole 35-element magma, so there is no size-5 complement submagma. The automorphism group has order 20. It induces the full affine automorphism group of the F_5 quotient, with seven 5-point orbits on M: {0..4}, {5..9}, ..., {30..34}.