We present two calculations for a class of robust homoclinic cycles with symmetry Z(n) (sic) Z(2)(n), for which the sufficient conditions for asymptotic stability given by Krupa and Melbourne are not optimal. Firstly, we compute optimal conditions for asymptotic stability using transition matrix techniques which make explicit use of the geometry of the group action. Secondly we consider a specific polynomial vector field that contains a robust heteroclinic cycle with this symmetry. Through an explicit computation of the global parts of the Poincare map near the cycle we show that, generically, the resonance bifurcation from the cycle is supercritical: a unique branch of asymptotically stable periodic orbits emerges from the resonance bifurcation and exists for coefficient values where the cycle has lost stability. This second calculation is of a novel kind: it is the first calculation that explicitly computes the criticality of a resonance bifurcation, and it answers a conjecture of Field and Swift in a particular limiting case. Moreover, we are able to obtain an asymptotically correct analytic expression for the period of the bifurcating orbit, with no adjustable parameters, which has not previously been achieved. We show that the asymptotic analysis compares very favourably with numerical results.