Hot gas fraction as a function of the halo mass. The points are colour-coded for the residuals about the median of different properties as a function of the halo mass. The properties are: the BH mass Δlog M_BH, the BCG stellar mass Δlog M_⋆, CEN, the mean stellar formation redshift Δz_⋆, CEN, the magnitude gap Δlog M_1, 2, the richness Δlog N_SAT, the total metallicity Δlog(Z₅₀₀/Z_⊙), and the total temperature Δlog(k_BT₅₀₀). In the bottom panels, we plot the Spearman coefficient between the hot gas fraction and the property in halo mass bins. The shaded band regions represent the bins where the p-value< 0.05. The analogue plot with X-ray luminosity in the soft band is in Fig. A.1.