Jim Caprioli at The Finland Station

Yes!

Solved my problem, at last. It took me more than a week. Nothing special of course, but for me it's a thrill, feels like I made a discovery.

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#GAP code to list 16 4x4 matrices representing Z4x|Z4
m1:=[[0,1] ,[1,0]];
m2:=[[E(4),0] ,[0,-E(4)]];
m3:=[[-1,0] ,[0,-1]];
m4:=[[1,0] ,[0,1]];

n1:=[[0,-1] ,[1,0]];
n2:=[[E(4),0] ,[0,-E(4)]];
n3:=[[-1,0] ,[0,-1]];
n4:=[[-1,0] ,[0,-1]];

M:=Group(m1,m2,m3,m4);
Display(StructureDescription(M));

N:=Group(n1,n2,n3,n4);
Display(StructureDescription(N));

f1:=DirectSumMat(m1,n1);
f2:=DirectSumMat(m2,n2);
f3:=DirectSumMat(m3,n3);
f4:=DirectSumMat(m4,n4);
G:=Group(f1,f2,f3,f4);
Display(StructureDescription(G));

x:=1;
while(x<17) do
Display(Elements(G)[x]);
x:=x+1;
od;

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(c) 2005, Nilo de Roock