Connect each design to its 14 rows. Connect disjoint designs to each other (each design is disjoint with 8 others). Connect rows to each other if they have exactly one element in common (there are 4x4 = 16 such neighbors). The resulting graph is the Higman–Sims graph. Rows are connected to 16 other rows and to 6 designs == degree 22. Designs are connected to 14 rows and 8 disjoint designs == degree 22. Thus all 100 vertices have degree 22 each. The automorphism group of the Higman–Sims graph is a group of order isomorphic to the semidirect product of the Higman–Sims group of order with the cyclic group of order 2. It has automorphisms that take any edge to any other edge, making the Higman–Sims graph an edge-transitive graph. The characteristic polynomial of the Higman–Sims graph is ("x" − 22)("x" − 2)77("x" + 8)22. Therefore the Higman–Sims graph is an integral graph: its spectrum consists entirely of integers. It is also the only graph with this characteristic polynomial, making it a graph determined by its spectrum. The Higman–Sims graph naturally occurs inside the Leech lattice: if "X", "Y" and "Z" are three points in the Leech lattice such that the distances "XY", "XZ" and "YZ" are formula_1 respectively, then there are exactly 100 Leech lattice points "T" such that all the distances "XT", "YT" and "ZT" are equal to 2, and if we connect two such points "T" and "T"′ when the distance between them is formula_2, the resulting graph is isomorphic to the Higman–Sims graph. Furthermore, the set of all automorphisms of the Leech lattice (that is, Euclidean congruences fixing it) which fix each of "X", "Y" and "Z" is the Higman–Sims group (if we allow exchanging "X" and "Y", the order 2 extension of all graph automorphisms is obtained). This shows that the Higman–Sims group occurs inside the Conway groups Co2 (with its order 2 extension) and Co3, and consequently also Co1.