  One-dimensional systems with topological order are intimately related to the
appearance of zero-energy modes localized on their boundaries. The most common
example is the Kitaev chain, which displays Majorana zero-energy modes and it
is characterized by a two-fold ground state degeneracy related to the global
$\mathbb{Z}_2$ symmetry associated with fermionic parity. By extending the
symmetry to the $\mathbb{Z}_N$ group, it is possible to engineer systems
hosting topological parafermionic modes. In this work, we address
one-dimensional systems with a generic discrete symmetry group $G$. We define a
ladder model of gauge fluxes that generalizes the Ising and Potts models and
displays a symmetry broken phase. Through a non-Abelian Jordan-Wigner
transformation, we map this flux ladder into a model of dyonic operators,
defined by the group elements and irreducible representations of $G$. We show
that the so-obtained dyonic model has topological order, with zero-energy modes
localized at its boundary. These dyonic zero-energy modes are in general weak
topological modes, but strong dyonic zero modes appear when suitable
position-dependent couplings are considered.
