diff --git a/tst/testinstall/grp/glzmodmz.tst b/tst/testinstall/grp/glzmodmz.tst
index dd980eae80..a47ac2323c 100644
--- a/tst/testinstall/grp/glzmodmz.tst
+++ b/tst/testinstall/grp/glzmodmz.tst
@@ -1,5 +1,33 @@
 gap> START_TEST("glzmodmz.tst");
 
+#
+gap> CheckGeneratorsInvertible := function(G)
+>   return ForAll(GeneratorsOfGroup(G),
+>               g -> not IsZero(Determinant(g)));
+> end;;
+gap> CheckGeneratorsSpecial := function(G)
+>   return ForAll(GeneratorsOfGroup(G),
+>               g -> IsOne(Determinant(g)));
+> end;;
+gap> CheckBilinearForm := function(G)
+>   local M;
+>   M := InvariantBilinearForm(G).matrix;
+>   return ForAll(GeneratorsOfGroup(G),
+>               g -> g*M*TransposedMat(g) = M);
+> end;;
+gap> frob := function(g,aut)
+>   return List(g,row->List(row,x->x^aut));
+> end;;
+gap> CheckSesquilinearForm := function(G)
+>   local M, F, aut;
+>   M := InvariantSesquilinearForm(G).matrix;
+>   F := FieldOfMatrixGroup(G);
+>   aut := FrobeniusAutomorphism(F);
+>   aut := aut^(DegreeOverPrimeField(F)/2);
+>   return ForAll(GeneratorsOfGroup(G),
+>               g -> g*M*TransposedMat(frob(g,aut)) = M);
+> end;;
+
 #
 gap> SizeOfGLdZmodmZ(2,2);
 6
@@ -12,10 +40,16 @@ Error, GL(2,Integers mod 1) is not a well-defined group, resp. not supported.
 #
 gap> G:=GL(4,Integers mod 4);
 GL(4,Z/4Z)
+gap> CheckGeneratorsInvertible(G);
+true
 gap> H:=SL(4,Integers mod 4);
 SL(4,Z/4Z)
+gap> CheckGeneratorsSpecial(H);
+true
 gap> K:=Sp(4,Integers mod 4);
 Sp(4,Z/4Z)
+gap> CheckGeneratorsSpecial(K) and CheckBilinearForm(K);
+true
 
 #
 gap> NrMovedPoints(Image(NiceMonomorphism(G : cheap)));;
@@ -34,5 +68,43 @@ gap> Size(G); Size(H); Size(K);
 660602880
 737280
 
+#
+gap> K:=Sp(4,Integers mod 9);
+Sp(4,Z/9Z)
+gap> CheckGeneratorsSpecial(K) and CheckBilinearForm(K);
+true
+gap> K:=Sp(6,Integers mod 16);
+Sp(6,Z/16Z)
+gap> CheckGeneratorsSpecial(K) and CheckBilinearForm(K);
+true
+
+#
+gap> K:=GO(3,Integers mod 9);
+GO(3,Z/9Z)
+gap> CheckGeneratorsInvertible(K) and CheckBilinearForm(K);
+true
+gap> K:=GO(+1,4,Integers mod 9);
+GO(1,4,Z/9Z)
+gap> CheckGeneratorsInvertible(K) and CheckBilinearForm(K);
+true
+gap> K:=GO(-1,4,Integers mod 9);
+GO(-1,4,Z/9Z)
+gap> CheckGeneratorsInvertible(K) and CheckBilinearForm(K);
+true
+
+#
+gap> K:=SO(3,Integers mod 9);
+SO(3,Z/9Z)
+gap> CheckGeneratorsSpecial(K) and CheckBilinearForm(K);
+true
+gap> K:=SO(+1,4,Integers mod 9);
+SO(1,4,Z/9Z)
+gap> CheckGeneratorsSpecial(K) and CheckBilinearForm(K);
+true
+gap> K:=SO(-1,4,Integers mod 9);
+SO(-1,4,Z/9Z)
+gap> CheckGeneratorsSpecial(K) and CheckBilinearForm(K);
+true
+
 #
 gap> STOP_TEST("glzmodmz.tst", 10000);