We analyzed the atomic and electronic structures of relaxed Si_{1- x-y} Ge _x C _y crystals tor small carbon concentrations, such as y = 0.0625, by first-principle total energy calculations. The lattice constants linearly reduce as functions of y and the concentration-ratio of Ge to C (x/y) for exact strain-compensation on Si is 9.3, which is larger than the 8.0 as obtained from Vegard's law. The C atoms prefer sites surrounded by Si atoms (Type A) to sites surrounded by Ge atoms (Type B). The incorporation of C reduces the energy gap. While “ Type A” structures have finite gaps (semiconductors), “ Type B” structures have no band gap at all. In the case of semiconductors (Type A), nonlinear effective masses do not change for electrons and become smaller for holes due to the incorporation of C atoms. Wealso found that the Si _1-x-y Ge _x C _y , crystals with y = 0.0625 have direct band gaps and the life time of their spontaneous emission during the transition between the band edge states is much smaller than for Si _1-x Ge _x systems. These results show that Si _1-x-y Ge _x C _y systems offer high potential for optical and electrical applications.