Preparation of single-walled carbon nanotubes (SWNTs) with diameters ranging from 1.2 to 1.5 nm has been highly demanded for field-effect transistors with superior properties. In the work, we report the chemical vapor deposition synthesis of SWNTs with an average diameter of 1.2 nm from a rationally designed nickel catalyst supported by magnesia. The high metal dispersion, the suitable reaction temperature, and the use of a methane carbon source co-determine the SWNT diameter distribution. Furthermore, through the synergistic application of poly(9,9-dioctylfluorene-2,7-diylalt-pyridine-2,6-diyl) wrapping and ultracentrifugation, SWNTs with enriched (10,8) species of 1.24 nm are successfully extracted, which is attributed to the selective interaction between the polymer molecules and the targeted SWNTs. This work not only sheds light on the growth of SWNTs with relatively large diameter, but also paves the way towards the preparation of high purity (n,m) species that could meet the requirements of advanced high-tech applications.