The normalizer of the level (2,2)-Heisenberg Group

The level (2, 2)-Heisenberg GroupG(2, 2) as first introduced by Mumford in [Mu] is a subgroup inSL(4,ℂ) of order 32. LetN be the normalizer ofG(2, 2) inSL(4,ℂ). This note describes explicitely the two natural isomorphisms fromN/G(2, 2) to the symmetric group of 6 elements. These identifications clarify the computations in the classical treatises as for example in the theory of Kummer surfaces in [Hu] and the theory of the quadric line complex as in [Je] and will be used in [Nie] to describe the moduli space for abelian surfaces with a level (2, 6)-structure.