Exploiting simple types of symmetry common to many natural protein oligomers as a starting point, several recent studies have succeeded in engineering complex self-assembling protein architectures reminiscent but distinct from those evolved in the natural world. Designing symmetric protein cages with a wide range of properties has been of particular interest for potential applications in the fields of medicine, energy, imaging and more. In this study we genetically fused three naturally symmetric protein components together – a pentamer, trimer, and dimer – in a fashion designed to create a self-assembling icosahedral protein cage built from 60 copies of the protein subunit. The connection between the pentamer and dimer was based on a continuous shared alpha helix in order to control the relative orientation of those components. Following selection of suitable components by computational methods, a construct with favorable design properties was tested experimentally. Negative stain electron microscopy and solution-state methods indicated successful formation of a 60-subunit icosahedral cage, 2.5 MDa in mass and 30 nm in diameter. Diverse experimental studies also suggested substantial degrees of flexibility and asymmetric deformation of the assembled particle in solution. The results add further examples of successes and challenges in designing atomically precise protein materials.