The group is generated by a,b,c,d with relations
ab=bc=ca
ac=cd=da
ad=db=ba
bd=dc=cb
I checked using GAP that it is $A_4$ mod its center. But what is the center? Torsion free of rank=?
The group is generated by a,b,c,d with relations ab=bc=ca ac=cd=da ad=db=ba bd=dc=cb I checked using GAP that it is $A_4$ mod its center. But what is the center? Torsion free of rank=? 


Try the following GAP code:


