The new title compound was synthesized by high-temperature means and its X-ray structure refined in the cubic space group Pm3, Z = 3, a = 8.278(1) Angstrom. The structure exhibits a 3-D framework made from a Ga-14 and Mg network within which large and small cavities are occupied by centered GaCu12 icosahedral and Cu-6 octahedral clusters, respectively. The clusters are well bonded within the network. Electronic structure calculations show that a pseudogap exists just above the Fermi energy, and nearly all pairwise covalent interactions remain bonding over a range of energy above that point. Analysis suggests that the compound is hypoelectronic with a four-electron deficiency per unit cell, and such a derivative with Sc substituting for Mg is an appropriate quasicrystal approximant (Im3). Such characteristics seem to be key factors in the formation of icosahedral quasicrystals.