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=?



Central extension of A4 - Is center torsion-free, and what is rank?
Try the following GAP code:



```
> g:=FreeGroup("a","b","c","d");
<free group on the generators [ a, b, c, d ]>

> h:=g/ParseRelators(GeneratorsOfGroup(g),"ab=bc=ca,ac=cd=da,ad=db=ba,bd=dc=cb");
<fp group on the generators [ a, b, c, d ]>   

> z:=Centre(h);
Group(<37 generators>)

> IsAbelian(z);
true

> AbelianInvariants(z);
[ 0, 2 ]

```

