Order 587 = 55 + 4*133, glued along a group-divisible design developed from base blocks under the translations of Z/133; blocks of five points.
Points: 0..54 are fixed points; 55 + 133r + a is the point (r, a), r in 0..3, a in Z/133. The fixed points form one group and carry A(55;32,24,0): x*y = 32x + 24y mod 55. For each r and each c in 0..18, the 7 points (r, c + 19i), i in Z/7, form a group and carry A(7;4,1,0) on i: (r, c + 19i)*(r, c + 19j) = (r, c + 19k), k = 4i + 1j mod 7. 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/133, of the base blocks. Pure base blocks: for each representative B below, each s in {0, 2} and each m in {1, 121, 11}, 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, 121, 11}, v = m w, where w_(r + s mod 4) = u_r - u_(-s mod 4).
Pure representatives: {(0,0), (0,60), (0,129), (0,49), (1,17)} {(0,0), (0,24), (0,130), (0,115), (3,72)} {(0,0), (0,40), (0,119), (0,6), (2,5)} {(0,0), (0,10), (1,28), (2,42), (3,71)} {(0,0), (0,8), (1,77), (2,31), (3,30)} {(0,0), (0,16), (1,120), (2,64), (3,58)} {(1,0), (1,14), (1,4), (1,87), (2,33)} {(0,0), (1,70), (1,112), (1,129), (1,78)} {(1,0), (1,36), (1,111), (1,30), (3,88)} {(0,0), (1,68), (1,56), (2,104), (3,124)} {(0,0), (1,81), (1,34), (2,115), (3,1)} {(0,0), (1,16), (1,9), (2,99), (3,7)}
Vectors: (0,10,117,21); (0,5,20,24); (0,27,72,53); (0,46,88,110); (0,71,56,23); (0,58,3,128); (0,15,54,93); (0,1,2,68); (0,3,40,38)
Equation 255: satisfied. Idempotent elements: 131.
qawbecrdtey · 2026-10-08 15:15:46