Symmetric generation and existence of J(3): 2, the automorphism group of the third Janko group

We make use of the primitive permutation action of degree 120 of the automorphism group of L-2(16), the special linear group of degree 2 over the field of order 16, to give an elementary definition of the automorphism group of the third Janko group J(3) and to prove its existence. This definition enables us to construct the 6156-point graph which is preserved by J(3):2, to obtain the order of J(3) and to prove its simplicity. A presentation for 4:2 follows from our definition, which also provides a concise notation for the elements of the group. We use this notation to give a representative of each of the conjugacy classes of J(3):2. (c) 2006 Elsevier Inc. All rights reserved.