Starting by the multiplication table of a finite group G, for each odd prime p a special p-group P of exponent p is constructed. Some connections between the structures of G and P, concerning subgroups and automorphisms, are given. When p does not divide the order of G, for every normal subgroup of G a direct decomposition of P is done. Besides, the order of the derived subgroup of G is proved to be connected with the existence of some abelian subgroups of maximal order in P. Finally, the elements of P with large centralizers are characterized in terms of algebraic properties of the multiplication table of G.
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