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Efficient utilization of the radiansphere surrounding a small antenna could improve its bandwidth. The use of a fractal shape as antenna can be a potential solution to design efficient small antennas because its fractal dimension is greater than its topological, dimension. Besides, some fractal shapes are space-filling curves, a suitable property that helps miniaturization. However, we are limited to the use of prefractals. Although prefractals are considered to behave as fractals and retain its fractal dimension, the influence of the initiator and even the IFS that generates fractals with the same fractal dimension is important under the scope of radiation efficiency and quality factor. This relation is systematically investigated in this work.