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The companions of the exploding carbon-oxygen white dwarfs (CO WDs) for producing type Ia supernovae (SNe Ia) are still not conclusively confirmed. A red-giant (RG) star has been suggested to be the mass donor of the exploding WD, named as the symbiotic channel. However, previous studies on the this channel gave a relatively low rate of SNe Ia. Aims. We aim to systematically investigate the parameter space, Galactic rates and delay time distributions of SNe Ia from the symbiotic channel by employing a revised mass-transfer prescription. Methods. 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