Classic two-dimensional graphene possesses outstanding properties due to Dirac cone structures so that many Dirac cone materials had been predicted. Using the first principle symmetric search algorithm, a new graphene-like carbon allotrope with P2m space group, named azugraphene, was predicted and its 38 atoms in the unit cell can be fragmented into three 5–7 rings of azulene, one hexagon, and two remainder atoms. Azugraphene is a low-energy graphene-like hexagonal carbon allotrope with six Dirac cones in the first Brillouin zone. Due to its stability and the existence of its elementary fragments, azugraphene is potentially synthesizable. In addition, the stable AB stacking bilayer azugraphene is also a Dirac cone material with a band gap of 2.5 meV. Therefore, both the monolayer and bilayer azugraphenes have great potential in physics, chemistry, and nanoelectronics.