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We study the moduli space $\fM^s(6;3,6,4)$ of simple rank 6 vector bundles$\E$ on $\PP^3$ with Chern polynomial $1+3t+6t^2+4t^3$ and properties of thesebundles, especially we prove some partial results concerning their stability.We first recall how these bundles are related to the construction of sexticnodal surfaces in $\PP^3$ having an even set of 56 nodes (cf. \cite{CaTo}). Weprove that there is an open set, corresponding to the simple bundles withminimal cohomology, which is irreducible of dimension 19 and bimeromorphic toan open set $\fA^0$ of the G.I.T. quotient space of the projective space$\fB:=\{B\in \PP(U^\vee\otimes W\otimes V^\vee)\}$ of triple tensors of type$(3,3,4)$ by the natural action of $SL(W)\times SL(U)$. We give severalconstructions for these bundles, which relate them to cubic surfaces in 3-space$\PP^3$ and to cubic surfaces in the dual space $(\PP^3)^{\vee}$. One of theseconstructions, suggested by Igor Dolgachev, generalizes to other types oftensors. Moreover, we relate the socalled {\em cross-product involution} for$(3,3,4)$-tensors, introduced in \cite{CaTo}, with the Schur quadric associatedto a cubic surface in $\PP^3$ and study further properties of this involution.

A remarkable moduli space of rank 6 vector bundles related to cubic surfaces