Equation 677 Database

The most symmetric Eq 677 Steiner-line magma at size 61. Size 61, fully idempotent, RC. 183 size-5 sub-magmas (each = the F_5 affine line magma#e549b5f8). Every pair of distinct elements lies in exactly one block; each element is in 15 blocks. |Aut(M)| = 915 = 15 * 61. Aut is the Frobenius-style group Z_61 : Z_15: the regular Z_61 acts by translation, and the point-stabilizer is the order-15 subgroup of F_61* = {1, 9, 12, 13, 15, 16, 20, 22, 25, 34, 42, 47, 56, 57, 58} (= (F_61*)^4, the unique cyclic subgroup of order 15), acting by multiplication. In the suggested reorder, F_61 labeling makes translation x -> x+1 an automorphism. The operation is then x*y = x + f(y - x) mod 61 with f(y) = alpha(y) * y, where alpha takes each of the 4 Phi_10 roots {3, 27, 41, 52} on exactly one coset of (F_61*)^4 in F_61*. Specifically (in the chosen F_61 labeling): coset 0 (= (F_61*)^4 itself, containing 1): alpha = 41 coset 1 (containing 2): alpha = 52 coset 2 (containing 3): alpha = 3 coset 3 (containing 6): alpha = 27 Each Phi_10 root is assigned to exactly one of the 4 (F_61*)^4-cosets, and the slope class sizes are 15+15+15+15 = 60 (= |F_61*|). This is the analog of magma#b7e8bf90 at size 41, with the same recipe: take the unique multiplicative subgroup of (F_q*)^4 in F_q* (q ≡ 1 mod 10), assign the 4 Phi_10 roots to its 4 cosets, and define f(y) = alpha(coset of y) * y. The compatibility conditions for Eq 677 force a specific assignment of Phi_10 roots to cosets (essentially unique up to Galois conjugation). Among the 36 size-61 Eq 677 magmas, this is the unique one with |Aut| = 915. The Z_15 stabilizer is the highest non-trivial multiplicative stabilizer; the only other multi-stabilizer magma at size 61 is magma#05fbff2c with Z_3 stabilizer (|Aut| = 183). The other 30 Steiner-line magmas have only Z_61 translation symmetry (|Aut| = 61). And the 4 linear F_61 magmas have |Aut| = 3660. [text written by Claude]