Log24

The transformation group G that scrambles the 4×4 array in the Diamond Theorem consists of 322,560 permutations and is isomorphic to the affine group AGL(4,2). Within the framework of the MOG, Curtis demonstrated that this exact symmetry group (which he denoted as 2⁴.A₈) functions as the octad stabilizer subgroup of the sporadic simple group M₂₄. Thus, the group defining the symmetries of the 4×4 diamond configurations is mathematically identical to the subgroup that stabilizes an 8-element "octad" within the 24-point Steiner system S(5,8,24). 

The mathematical architecture bridging these concepts is the Klein correspondence (and Conwell's 1910 correspondence), which establishes a mapping between the 35 lines of PG(3,2) and the 35 ways to partition an 8-set into two 4-sets. The MOG physically illustrates this correspondence by pairing partitions of an 8-set (represented by a 4×2 array) with the 35 partitions of the 4×4 affine space. The Diamond Theorem translates these abstract combinatorial partitions into concrete symmetric properties governed by binary arithmetic (where components sum to zero via XOR).