The fate of cooling gas in the centers of galaxy clusters and groups is still not well understood , as is also the case for the complex processes of triggering star formation in central dominant galaxies ( CDGs ) , re-heating of cooled gas by AGN , and the triggering/ ‘ ‘ feeding ’ ’ of supermassive black hole outbursts . We present CO observations of the early type galaxy NGC 5044 , which resides at the center of an X-ray bright group with a moderate cooling flow . For our analysis we combine CO ( 2-1 ) data from the 7m antennae of the Atacama Compact Array ( ACA ) , and the ACA total power array ( TP ) . We demonstrate , using the 7m array data , that we can recover the total flux inferred from IRAM 30m single dish observations , which corresponds to a total molecular mass of about \SI { 4 e 7 } { M _ { \odot } } . Most of the recovered flux is blueshifted with respect to the galaxy rest frame and is extended on kpc-scales , suggesting low filling factor dispersed clouds . We find 8 concentrations of molecular gas out to a radius of \SI { 10 } { arcsec } ( \SI { 1.5 } { kpc } ) , which we identify with giant molecular clouds . The total molecular gas mass is more centrally concentrated than the X-ray emitting gas , but extended in the north-east/south-west direction beyond the IRAM 30m beam . We also compare the spatial extent of the molecular gas to the H \alpha emission : The CO emission coincides with the very bright H \alpha region in the center . We do not detect CO emission in the fainter H \alpha regions . Furthermore , we find two CO absorption features spatially located at the center of the galaxy , within \SI { 5 } { pc } projected distance of the AGN , infalling at 255 and \SI { 265 } { km s ^ { -1 } } relative to the AGN . This indicates that the two giant molecular clouds seen in absorption are most likely within the sphere of influence of the supermassive black hole .