In previous works, the universal mapping class group was taken to be the group \({\mathrm {PPSL}}(2,{\mathbb Z})\) of all piecewise \({\mathrm {PSL}}(2,{\mathbb Z})\) homeomorphisms of the unit circle \(S^1\) with finitely many breakpoints among the rational points in \(S^1\) , and in fact, the Thompson group \(T\approx {\mathrm {PPSL}}(2,{\mathbb Z})\) . The new spin mapping class group P(SL(2, \({\mathbb Z}\) )) is given by all piecewise-constant maps \(S^1\to {\mathrm {SL}}(2,{\mathbb Z})\) which projectivize to an element of \({\mathrm {PPSL}}(2,{\mathbb Z})\) . We compute a finite presentation of \({\mathrm {PPSL}}(2,{\mathbb Z})\) from basic principles of general position as an orbifold fundamental group. The orbifold deck group of the spin cover is explicitly computed here, from which follows also a finite presentation of \(\mathrm {P}(\mathrm {SL}(2,{\mathbb Z}))\) . This is our main new achievement. Certain commutator relations in \(\mathrm {P}(\mathrm {SL}(2,{\mathbb Z}))\) seem to organize according to root lattices, which would be a novel development. We naturally wonder what is the automorphism group of \(\mathrm {P}(\mathrm {SL}(2,{\mathbb Z}))\) and speculate that it is a large sporadic group. There is a companion chapter to this one which explains the topological background from first principles and proves that the group studied here using combinatorial group theory is indeed \(\mathrm {P}(\mathrm {SL}(2,{\mathbb Z}))\) .