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We construct two small resolutions of singularities of the Coble fourfold(the double cover of the four-dimensional projective space branched over theIgusa quartic). We use them to show that all $S_6$-invariant three-dimensionalquartics are birational to conic bundles over the quintic del Pezzo surfacewith the discriminant curves from the Wiman-Edge pencil. As an application, wecheck that $S_6$-invariant three-dimensional quartics are unirational, obtainnew proofs of rationality of four special quartics among them and irrationalityof the others, and describe their Weil divisor class groups as$S_6$-representations.

Coble fourfold, $S_6$-invariant quartic threefolds, and Wiman-Edge sextics