Equation 677 Database

This is a Latin 677 magma with a 5-by-7 congruence decomposition. 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: 30,0,22,27,12,7,17 C1: 33,1,20,26,10,5,15 C2: 31,4,21,25,11,6,16 C3: 34,3,24,29,14,9,19 C4: 32,2,23,28,13,8,18 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+3v) if i=j, (i,u) ◇ (j,v) = (2i+4j, s^{-1}(4s(u)+s(v))) if i≠j, where s=(6,4,2,0,5,1,3), i.e. s(0)=6, s(1)=4, ..., s(6)=3. Thus the diagonal fibres are copies of magma#baf8b55c, the F_7 affine law u◇v=4u+3v, while every off-diagonal fibre product uses a conjugate of magma#7981e2df, the F_7 law u◇v=4u+v. 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}.