Order 557 = 49 + 4*127, glued along a group-divisible design developed from base blocks under the translations of Z/127; blocks of five points.
Points: 0..48 are fixed points; 49 + 127r + a is the point (r, a), r in 0..3, a in Z/127. The fixed points form one group, the hole, and carry A(49;18,8,0): x*y = 18x + 8y mod 49; every other point is a group of its own and squares to itself. Two points of different groups multiply in the block through them, which carries A(5;2,4,0) on its points as listed: if the block lists b_0, ..., b_4, then b_i*b_j = b_k with k = 2i + 4j mod 5.
Blocks: the translates (r, a) -> (r, a + t), t in Z/127, of the base blocks. Pure base blocks: for each representative B below, each s in {0, 2} and each m in {1, 19, 107}, the block listing (r + s mod 4, m a) for (r, a) in B, in order. Base blocks through fixed points: the fixed point h lists h, then (r, v_r) for r = 0..3, where fixed point 0 has v = 0, and the fixed points from 1 on take, in turn, for each vector u below, each s in {0, 2} and each m in {1, 19, 107}, v = m w, where w_(r + s mod 4) = u_r - u_(-s mod 4).
Pure representatives: {(0,0), (0,39), (0,14), (0,89), (1,67)} {(0,0), (0,59), (0,124), (0,114), (3,49)} {(0,0), (0,36), (0,116), (0,107), (2,124)} {(0,0), (0,46), (1,1), (2,82), (3,23)} {(0,0), (0,32), (1,63), (2,6), (3,44)} {(0,0), (0,58), (1,106), (2,123), (3,121)} {(1,0), (1,32), (1,111), (1,121), (2,120)} {(0,0), (1,45), (1,94), (1,41), (1,42)} {(1,0), (1,30), (1,55), (1,69), (3,66)} {(0,0), (1,20), (1,9), (2,5), (3,26)} {(0,0), (1,72), (1,26), (2,55), (3,8)} {(0,0), (1,33), (1,101), (2,90), (3,21)}
Vectors: (0,78,83,126); (0,13,48,43); (0,71,10,80); (0,5,108,33); (0,4,15,2); (0,66,61,91); (0,64,125,59); (0,65,103,22)
Equation 255: satisfied. Idempotent elements: 509.
qawbecrdtey · 2026-10-08 15:26:12