Let I be an ideal of a regular local ring Q with residue field k. The minimal free resolution of R=Q/I carries a structure of a differential graded algebra. If the length of the resolution, which is called the codepth of R, is at most 3, then the induced algebra structure on TorQ* (R,k) is unique and provides for a classification of such local rings.
According to the multiplicative structure on TorQ* (R,k), a non-zero local ring R of codepth at most 3 belongs to exactly one of the (parametrized) classes designated B, C(c), G(r), H(p,q), S, or T. An overview of the theory can be found in L.L. Avramov, A cohomological study of local rings of embedding codepth 3, http://arxiv.org/abs/1105.3991.
Version 1.0 of this package was accepted for publication in volume 6 of the journal The Journal of Software for Algebra and Geometry on 2014-07-11, in the article Local rings of embedding codepth 3: A classification algorithm. That version can be obtained from the journal or from the Macaulay2 source code repository, http://github.com/Macaulay2/M2/blob/master/M2/Macaulay2/packages/CodepthThree.m2, commit number 4b2e83cd591e7dca954bc0dd9badbb23f61595c0.