FIX: wrong Generators of Matrix Wreath Product with Intransitive To…

…p Group.

It was implicitely assumed that the top group is transitive,
resulting in insufficient generators for the base group of the matrix wreath product
if the top group was given intransitive.

Add test to `testbugfix`.

lib/gprdmat.gi

@@ -253,7 +253,7 @@ end);
 DeclareGlobalFunction("MatWreathProduct");
 
 InstallGlobalFunction(MatWreathProduct,function(A,B)
-local f,n,m,Agens,Bgens,emb,i,j,a,g,dim,rans,range;
+local f,n,m,Agens,Bgens,emb,i,j,a,g,dim,rans,range,orbs;
   f:=DefaultFieldOfMatrixGroup(A);
   n:=DimensionOfMatrixGroup(A);
   m:=LargestMovedPoint(B);
@@ -271,7 +271,8 @@ local f,n,m,Agens,Bgens,emb,i,j,a,g,dim,rans,range;
     od;
     emb[j]:=Agens;
   od;
-  Agens:=emb[1];
+  orbs := OrbitsDomain(B);
+  Agens := Concatenation(List(orbs, orb -> emb[orb[1]]));
 
   Bgens:=List(GeneratorsOfGroup(B),
 	  x->KroneckerProduct(PermutationMat(x,m,f),One(A)));

tst/testbugfix/2021-09-22-MatWreathProduct.tst

@@ -0,0 +1,6 @@
+# Generators of Matrix Wreath Product with Intransitive Top Group # 4664
+gap> K := GL(3,2);;
+gap> H := Group((1,2,3)(4,5));;
+gap> G := WreathProduct(K, H);;
+gap> Size(Group(GeneratorsOfGroup(G))) = 802966929408;
+true