The simply connected Chevalley group G(E-7, R) of type E-7 is considered in the 56-dimensional representation. The main objective is to prove that the following four groups coincide: the normalizer of the elementary Chevalley group E(E-7, R), the normalizer of the Chevalley group G(E-7, R) itself, the transporter of E(E-7, R) into G(E-7, R), and the extended Chevalley group G (E-7, R). This holds over an arbitrary commutative ring R, with all normalizers and transporters being calculated in GL(56, R). Moreover, G (E-7, R) is characterized as the stabilizer of a system of quadrics. This last result is classically known over algebraically closed fields, here it is proved that the corresponding group scheme is smooth over Z, which implies that it holds over arbitrary commutative rings. These results are a key step in a subsequent paper, devoted to overgroups of exceptional groups in minimal representations.