Representing the automorphism group of an almost crystallographic group

Let E be an almost crystallographic (AC-) group, corresponding to the simply connected, connected, nilpotent Lie group L and with holonomy group F. If L(F) = {1}, there is a faithful representation Aut(E) hooked right arrow Aff(L). In case E is crystallographic, this condition L(F) = {1} is known to be equivalent to Z(E) = 1 or b(1) (E) = 0. We will show (Example 2.2) that, for AC-groups E, this is no longer valid and should be adapted. A generalised equivalent algebraic (and easier to verify) condition is presented (Theorem 2.3). Corresponding to an AC-group E and by factoring out subsequent centers we construct a series of AC-groups, which becomes constant after a finite number of terms. Under suitable conditions, this opens a way to represent Aut(E) faithfully in Gl(k, Z) x Aff(L(1)) (Theorem 4.1). We show how this can be used to calculate Out(E). This is of importance, especially, when E is almost Bieberbach and, hence, Out(E) is known to have an interesting geometric meaning.