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This work provides a classification of string C-groups of order 2m andexponent at least 2^(m - 3). Prior to the classification, we complete the listof groups of exponent 2^(m - 3) and rank at least 3. The main result statesthat, aside from the cyclic and dihedral groups, only two groups of exponent atleast 2^(m - 3) are connected string C-groups. Both have rank 3 and arequotients of the string Coxeter group W = [4, 2^(m - 3)].

String C-groups from groups of order 2^m and exponent at least 2^(m - 3)