We consider affine extensions of geometries of Petersen and of tilde type, i.e., of flag-transitive geometries 1 belonging to a diagram (c.X ) b
where either bwwb
, i.e., the geometry of edges and vertices of the Petersen graph, or bwwb X =b= = =b t , i.e., the 3-fold cover of the generalized Sp 4 (2)-quadrangle. We call such geometries (c.P)-resp. (c.T )-geometries because the residue of an element of type 1 is a P-resp. T-geometry. The class of flag-transitive P-and T-geometries has been classified by different authors and an overview of the results is given in [7]. In particular, there exist only two flag-transitive P-geometries of rank 3 with automorphism groups Aut(M 22 ) and 3Aut(M 22 ) and three flag-transitive T-geometries of rank 3 with automorphism groups M 24 , He, and 3 7 Sp 6 (2). In this paper we will assume: (V) If F is a flag of 1 of type [1, ..., n&3] then the residue res(F ) of F in 1 is the P-geometry for M 22 or the T-geometry for M 24 .
Since there are no P-geometries of rank 5 and no T-geometries of rank 5 satisfying (V), this forces n 6. On the other hand, it is known that for n=3 the universal covers of the rank 3 (c.P)-and (c.T )-geometry are both infinite; so a general classification for n=3 will not be possible. The case of flag-transitive (c.P)-geometries 1 of rank 4 and satisfying (V) has been considered in [4]. There it is shown that Aut(1 ) is isomorphic to a factorgroup of one of the groups 2 11 : Aut(M 22 ), M 24 or 2 } U 6 (2): 2 (where by H : K and H } K we denote, as usual, a split resp. non-split extension of a group K by a group H). In the first case, the normal 2-subgroup is the universal representation group (see below) of the P-geometry for M 22 . It is isomorphic to a submodule of index two (the even half) of the