The new Mn14Al56+xGe3-x(x = 0-0.6) compounds of a new structure type have been synthesized and characterized by physical property measurements and electronic structure calculations. In contrast to their well-known silicon analogues, their unique structure (P3) exhibits unprecedented partially destroyed Mackay icosahedra that retain the icosahedral symmetry only in half of the individual polyhedra. The electronic band-structure analysis indicates that the chemical bonding in the structure is still optimized despite the destruction of the Mackay icosahedra and that a further valence electron concentration (VEC) optimization is achieved by the partial occupation of aluminum on a germanium site. The electronic band-structure calculation results were in agreement with the poor metallicity observed for the samples. While the Mn14Al56Ge3 is metallic, the resistivity of Mn14Al56.6Ge2.4 shows a minimum around 20 K and a maximum around 100 K. Both of the samples are Pauli-paramagnetic with an additional small Curie component.