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We demonstrate silicon-based $p \text{-} n \text{-} p$p-n-p floating-base GeSn heterojunction phototransistors with enhanced optical responsivity for efficient short-wave infrared (SWIR) photodetection. The narrow-bandgap GeSn active layer sandwiched between the $p \text{-} {\rm Ge}$p-Ge collector and $n \text{-} {\rm Ge}$n-Ge base effectively extends the photodetection range in the SWIR range, and the internal gain amplifies the optical response by a factor of more than three at a low driving voltage of 0.4 V compared to that of a reference GeSn $p \text{-} i \text{-} n$p-i-n photodetector (PD). We anticipate that our findings will be leveraged to realize complementary metal-oxide-semiconductor-compatible, sensitive, low driving voltage SWIR PDs in a wide range of applications.