<p>The dynamics of a stack of M5 branes probing a transverse multi-centered Taub-NUT space are described by a class of 6d <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N} \)</EquationSource> </InlineEquation> = (1, 0) superconformal field theories known as the M-string orbifold SCFTs. We determine the equivariant partition functions for this class of theories on a geometric background of type <i>T</i> ² × <i>ℂ</i>²/Γ, where <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="normal">Γ</mi> <mo>∈</mo> <mfenced close="}" open="{" separators=",,,,"> <msub> <mi mathvariant="script">C</mi> <mi>N</mi> </msub> <msub> <mi mathvariant="script">Q</mi> <mi>N</mi> </msub> <mi mathvariant="script">T</mi> <mi mathvariant="script">O</mi> <mi mathvariant="script">I</mi> </mfenced> </math></EquationSource> <EquationSource Format="TEX">\( \Gamma \in \left\{{\mathcal{C}}_N,{\mathcal{Q}}_N,\mathcal{T},\mathcal{O},\mathcal{I}\right\} \)</EquationSource> </InlineEquation> is an arbitrary finite subgroup of <i>SU</i> (2). The partition functions are built out of contributions from BPS strings as well as BPS particles that arise upon putting the 6d theory on a circle. We find that BPS particle contributions can be expressed in terms of Γ<i>-covariant Hilbert series</i> which count holomorphic sections of vector bundles on the orbifold singularity with monodromy specified by an irreducible representation of Γ. The BPS string contributions, on the other hand, are given by the elliptic genera of 2d <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">\( \mathcal{N} \)</EquationSource> </InlineEquation> = (0, 4) Γ-<i>dressed quiver gauge theories</i>, obtained by stacking Kronheimer-Nakajima quivers of type Γ between interfaces that support current algebras for the McKay dual affine Lie algebra <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math display="inline"> <mover accent="true"> <mi mathvariant="fraktur">g</mi> <mo stretchy="true">̂</mo> </mover> </math></EquationSource> <EquationSource Format="TEX">\( \hat{\mathfrak{g}} \)</EquationSource> </InlineEquation>. We obtain explicit expressions for the elliptic genera of arbitrary BPS string configurations corresponding to fractional instanton strings on <i>ℂ</i>²/Γ, and for the case of <i>star-shaped</i> quivers of type <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="normal">Γ</mi> <mo>∈</mo> <mfenced close="}" open="{" separators=",,,"> <msub> <mi mathvariant="script">Q</mi> <mn>4</mn> </msub> <mi mathvariant="script">T</mi> <mi mathvariant="script">O</mi> <mi mathvariant="script">I</mi> </mfenced> </math></EquationSource> <EquationSource Format="TEX">\( \Gamma \in \left\{{\mathcal{Q}}_4,\mathcal{T},\mathcal{O},\mathcal{I}\right\} \)</EquationSource> </InlineEquation> we give a prescription to compute the elliptic genera by gluing 2d analogues of Gaiotto and Witten’s <i>T</i> [<i>SU</i> (<i>N</i>)] theories.</p>