Embeddings of L(2,13) in groups of Lie type of E6

In this paper we show there is exactly one conjugacy class of subgroups ofE 6(C) isomorphic toL(2, 13) with each of the characters 13+14 and 1+12+14 on a 27-dimensional module forE 6. The one with character 13+14 is a subgroup of the irreducible closed subgroup of typeG 2. There is a unique conjugacy class for each of the three algebraic conjugate characters 1+12+14. Our arguments have applications to fields of characteristic prime to |L(2, 13)|.