Abstract Carbon has many allotropes besides diamond and graphite. In this work, we studied the structure, elasticity, ideal strength and thermal conductivity of a pure three-dimensional sp2 carbon crystal by first principles calculation. The results reveal that this carbon allotrope is a rare crystal with negative linear compressibility: its a axis expands while c axis shortens under hydrostatic pressure. It is mechanically stable up to 60 GPa and its elastic constant C33 decreases at first and then increases and fluctuates with the rising pressure. Its ideal tensile, shear and compressive strengths are systematically studied and the results indicate that this pure sp2 carbon is superhard despite of its low density. It can endure strains of 0.622 and 0.470, respectively, when tensioned and compressed. These make it a very promising shock absorbing material once synthesized. Its cracked mechanisms at the critical strains are also investigated. Our study also finds that, contrary to the common view, the direction of its weakest shear strength has no relation with that of its weakest tensile strength. Proceeding from Slack's model, we also calculated its thermal conductivity in the temperature range from 200 to 1800 K, which gives a value of about 30 Wm− 1 K− 1 at 300 K. We hope our study will provide new insight on carbon allotropes and materials with negative linear compressibility.