We analyze shear-free spherically symmetric relativistic models of gravitating fluids with heat flow and electric charge defined on higher-dimensional manifolds. The solution to the Einstein-Maxwell system is governed by the pressure isotropy condition, which depends on the space-time dimension. We study this highly nonlinear partial differential equation using Lie's group theoretic approach. The Lie symmetry generators that leave the equation invariant are determined. We provide exact solutions to the gravitational potentials using the first symmetry admitted by the equation. Our new exact solutions contain the earlier results for the four-dimensional case. Using the other Lie generators, we are able to provide solutions to the gravitational potentials or reduce the order of the master equation to a first order nonlinear differential equation. We also find expressions for the causal and Eckart temperatures and show their dependence on the dimension.