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The obstruction to spin$^c$ is often denoted by $W_3(M) \in H^3(M,\mathbb{Z})$. It is obtained as follows: Let $$ \beta \colon H^2(M,\mathbb{Z}/2\mathbb{Z}) \to H^3(M,\mathbb{Z}) $$ be the Bockstein homomorphism obtained from the short exact sequence $$ 0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0 $$ Then we have $W_3(M) = \beta(w_2(M))$, i.e. the class obtained by applying the Bockstein homomorphism to the second Stiefel-Whitney class. This agrees with the Dixmier-Douady class of the (stabilisation of the) complex Clifford bundle $\mathbb{C}l(M)$. This is probably all contained in the book by Lawson and Michelsohn somewhere, but I don't have the exact reference at the moment.