Non-Abelian fracton order from gauging a mixture of subsystem and global symmetries

We demonstrate a general gauging procedure of a pure matter theory on a lattice with a mixture of subsystem and global symmetries. This mixed symmetry can be either a semidirect product of a subsystem symmetry and a global symmetry, or a non-trivial extension of them. We demonstrate this gauging procedure on a cubic lattice in three dimensions with four examples: $G={\mathbb{Z}}_3^{\text{sub}}⋊{\mathbb{Z}}_2^{\text{glo}}$, $G=($\mathbb{Z}$_2^{$\text{sub}$} \times $\mathbb{Z}$_2^{$\text{sub}$}) \rtimes $\mathbb{Z}$_2^{$\text{glo}$}$,$1\to \mathbb {Z}_2^\text {sub}\to G\to \mathbb {Z}_2^\text {glo}\to 1$,and$1\to \mathbb {Z}_2^\text {sub}\to G\to K_4^\text {glo}\to 1$. The former two cases and the last one produce the non-Abelian fracton orders. Our construction of the gauging procedure provides an identification of the electric charges of these fracton orders with irreducible representations of the symmetry. Furthermore, by constraining the local Hilbert space, the magnetic fluxes with different geometry (tube-like and plaquette-like) satisfy a subalgebra of the quantum double models (QDMs). This algebraic structure leads to an identification of the magnetic fluxes to the conjugacy classes of the symmetry.