this block, we obtain the other blocks of the (5,6,12) system by repeatedly applying the linear fractional transformations: where are in 11 and . With the usual conventions of defining and , these functions map the set onto itself. In geometric language, they are projectivities of the projective line. They form a group under composition which is the projective special linear group (2,11) of order 660. There are exactly five elements of this group that leave the starting block fixed setwise, namely those such that and so that . So there will be 660/5 = 132 images of that block. As a consequence of the multiply transitive property of this group acting on this set, any subset of five elements of will appear in exactly one of these 132 images of size six. Kitten construction An alternative construction of W12 is obtained by use of the 'kitten' of R.T. Curtis, which was intended as a "hand calculator" to write down blocks one at a time. The kitten method is based on completing patterns in a 3x3 grid of numbers, which represent an affine geometry on the vector space F3xF3, an S(2,3,9) system. Construction from K6 graph factorization The relations between the graph factors of the complete graph K6 generate an S(5,6,12). A K6 graph has 6 vertices, 15 edges, 15 perfect matchings, and 6 different 1-factorizations (ways to partition the edges into disjoint perfect matchings). The set of vertices (labeled 123456) and the set of factorizations (labeled ABCDEF) provide one block each. Every pair of factorizations has exactly one perfect matching in common. Suppose factorizations A and B have the common matching with edges 12, 34 and 56. Add three new blocks AB3456, 12AB56, and 1234AB, replacing each edge in the common matching with the factorization labels in turn. Similarly add three more blocks 12CDEF, 34CDEF, and 56CDEF, replacing the factorization labels by the corresponding edge labels of the common matching. Do this for all 15 pairs of factorizations to add 90 new blocks. Finally, take the full set of combinations of 6 objects out of 12, and discard any combination that has 5 or more objects in common with any of the 92 blocks generated so far. Exactly 40 blocks remain, resulting in blocks of the S(5,6,12). This method works because there is an outer automorphism on the symmetric group S6, which maps the vertices to factorizations and the