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String and particle braiding statistics are examined in a class oftopological orders described by discrete gauge theories with a gauge group $G$and a 4-cocycle twist $\omega_4$ of $G$'s cohomology group$\mathcal{H}^4(G,\mathbb{R}/\mathbb{Z})$ in 3 dimensional space and 1dimensional time (3+1D). We establish the topological spin and thespin-statistics relation for the closed strings, and their multi-stringbraiding statistics. The 3+1D twisted gauge theory can be characterized by arepresentation of a modular transformation group SL$(3,\mathbb{Z})$. We expressthe SL$(3,\mathbb{Z})$ generators $\mathsf{S}^{xyz}$ and $\mathsf{T}^{xy}$ interms of the gauge group $G$ and the 4-cocycle $\omega_4$. As we compactify oneof the spatial directions $z$ into a compact circle with a gauge flux $b$inserted, we can use the generators $\mathsf{S}^{xy}$ and $\mathsf{T}^{xy}$ ofan SL$(2,\mathbb{Z})$ subgroup to study the dimensional reduction of the 3Dtopological order $\mathcal{C}^{3\text{D}}$ to a direct sum of degeneratestates of 2D topological orders $\mathcal{C}_b^{2\text{D}}$ in different flux$b$ sectors: $\mathcal{C}^{3\text{D}} = \oplus_b \mathcal{C}_b^{2\text{D}}$.The 2D topological orders $\mathcal{C}_b^{2\text{D}}$ are described by 2D gaugetheories of the group $G$ twisted by the 3-cocycles $\omega_{3(b)}$,dimensionally reduced from the 4-cocycle $\omega_4$. We show that theSL$(2,\mathbb{Z})$ generators, $\mathsf{S}^{xy}$ and $\mathsf{T}^{xy}$, fullyencode a particular type of three-string braiding statistics with a patternthat is the connected sum of two Hopf links. With certain 4-cocycle twists, wediscover that, by threading a third string through two-string unlink intothree-string Hopf-link configuration, Abelian two-string braiding statistics ispromoted to non-Abelian three-string braiding statistics.

Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory