  We analyse 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 spacetime 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 derive the temperature transport equation
in higher dimensions and find expressions for the causal and Eckart
temperatures showing their explicit dependance on the dimension. We analyse a
particular solution, obtained via group techniques, to show its physical
applicability.
