### Abstract

Groups with commuting inner mappings are of nilpotency class at most 2, but there exist loops with commuting inner mappings and of nilpotency class higher than 2, called loops of Csörgő type . In order to obtain small loops of Csörgő type, we expand our programme from [Drápal and Vojtěchovský, European J. Combin. 29: 1662–1681, 2008] and analyze the following set-up in groups: Let G be a group, Z ⩽ Z ( G ), and suppose that δ : G / Z × G / Z → Z satisfies δ ( x, x ) = 1, δ ( x, y ) = δ ( y, x ) –1 , z yx δ ([ z, y ], x ) = z xy δ ([ z, x ], y ) for all x, y, z ∈ G , and δ ( xy, z ) = δ ( x, z ) δ ( y, z ) whenever { x, y, z } ∩ G ′ is not empty. Then there is μ : G / Z × G / Z → Z with δ ( x, y ) = μ ( x, y ) μ ( y, x ) –1 such that the multiplication x ∗ y = xyμ ( x, y ) defines a loop with commuting inner mappings, and this loop is of Csörgoő type (of nilpotency class 3) if and only if g ( x, y, z ) = δ ([ x, y ], z ) δ ([ y, z ], x ) δ ([ z, x ], y ) is nontrivial. Moreover, G has nilpotency class at most 3, and if g is nontrivial then |G| ⩾ 128, |G| is even, and g induces a trilinear alternating form. We describe all nontrivial set-ups ( G, Z, δ ) with |G| = 128. This allows us to construct for the first time a loop of Csörgő type with an inner mapping group that is not elementary abelian.