We have studied properties of novel thermotropic mesogenic materials that exhibit both an achiral double gyroid (Ia3[combining macron]d symmetry) and chiral cubic phase (previously assigned the Im3[combining macron]m symmetry). We argue that in the chiral cubic phase molecules form micelles and channels arranged into continuously interconnected hexagons. From the X-ray diffraction experiment supported by modelling, exact positions of hexagons and their connections were deduced and showed to be embedded on a WP (degenerated Neovius) minimal primitive surface. The elastic energy of such a structure is close to the one of the double gyroid phase, which is in agreement with a very low enthalpy change observed at the phase transition. We also argue that the chirality of the phase is related to the lack of mirror symmetry of non-flat hexagons accompanied by an alternating inclination of molecules in the neighbouring segments of hexagon; the chirality of individual hexagon is amplified on the whole hexagon network by steric effects.