Interpreting semidirect product in GAP in terms of underlying groups

Question

I obtained the group $G$ which is the semidirect product $(\mathbb{Z}_5 \times \mathbb{Z}_5) \rtimes \mathbb{Z}_3$ in GAP as below.

gap> m:=[[4,0],[0,2]]*One(GF(5));
[ [ Z(5)^2, 0*Z(5) ], [ 0*Z(5), Z(5) ] ]
gap> s1:=SemidirectProduct(Group(m),GF(5)^2);
<matrix group of size 100 with 3 generators>

And when I obtained the elements of $G$ I get a list which continues as follows.

gap> Elements(s1);
[ [ [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5) ], [ 0*Z(5), 0*Z(5), Z(5)^0 ] ],
  [ [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5) ], [ 0*Z(5), Z(5)^0, Z(5)^0 ] ],
  [ [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5) ], [ 0*Z(5), Z(5), Z(5)^0 ] ],
  [ [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5) ], [ 0*Z(5), Z(5)^2, Z(5)^0 ] ],...

I want to write the elements as $(\bar0,(\bar0,\bar0)),(\bar0,(\bar0,\bar1)),(\bar0,(\bar0,\bar2))$,....

Can someone please tell how to interpret the elements in the above list which are present as [ Z(5)^0, 0*Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^0, 0*Z(5) ] ... to obtain the form I want.

What does Z(5)^0, 0*Z(5) mean?

Thanks a lot in advance.

Answer 1

The notation Z(5) stands for a generator of the unit group of the finite field with five elements. So Z(5)^0 is $1$, and 0*Z(5) is $0$ is the finite field. The command IntFFE will convert these numbers for you.

gap> IntFFE(Z(5));            
2 
gap> IntFFE(Z(5)^2);            
4