  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
4x10^7 Msun. 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
10 arcsec (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 Halpha emission:
The CO emission coincides with the very bright Halpha region in the center. We
do not detect CO emission in the fainter Halpha regions. Furthermore, we find
two CO absorption features spatially located at the center of the galaxy,
within 5 pc projected distance of the AGN, infalling at 255 and 265 km/s
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.
