The characteristic 2 representation theory of the finite symplectic group G = Sp(4,q), for q odd, is investigated.;The defect group of each block is determined using a standard characterization of defect groups as S;The remaining non-principal blocks contain either 2 or 3 modular irreducibles. Using arithmetic properties of decomposition numbers and restrictions of characters to the S;The principal block contains 7 modular irreducibles. The same methods yield six of the seven columns of decomposition numbers. The unknown entries in the remaining column depend on two parameters for which bounds are given. At least one modular irreducible in the principal block is not the restriction of an ordinary irreducible to the elements of odd order.;Using the known ordinary (complex) character table of G and the corresponding central characters, the ordinary characters are distributed into 2-blocks. The blocks fall into nine natural families in addition to the principal block, and all blocks in a given family have the same defect group and decomposition matrix.;The complete decomposition matrix is determined in the case G = S
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