diff --git a/gtsam/base/tests/testGroup.cpp b/gtsam/base/tests/testGroup.cpp
index 2cee30070..dadf2896c 100644
--- a/gtsam/base/tests/testGroup.cpp
+++ b/gtsam/base/tests/testGroup.cpp
@@ -16,36 +16,56 @@
  **/
 
 #include <gtsam/base/Group.h>
+#include <gtsam/base/Testable.h>
 #include <Eigen/Core>
 #include <iostream>
+#include <boost/foreach.hpp>
 
 namespace gtsam {
 
+/// Symmetric group
 template<int N>
-class Permutation: public Eigen::PermutationMatrix<N> {
-public:
-  Permutation() {
-    Eigen::PermutationMatrix<N>::setIdentity();
-  }
-  Permutation(const Eigen::PermutationMatrix<N>& P) :
+class Symmetric: private Eigen::PermutationMatrix<N> {
+  Symmetric(const Eigen::PermutationMatrix<N>& P) :
       Eigen::PermutationMatrix<N>(P) {
   }
-  Permutation inverse() const {
+public:
+  Symmetric() {
+    Eigen::PermutationMatrix<N>::setIdentity();
+  }
+  static Symmetric Transposition(int i, int j) {
+    Symmetric g;
+    return g.applyTranspositionOnTheRight(i, j);
+  }
+  Symmetric operator*(const Symmetric& other) const {
+    return Eigen::PermutationMatrix<N>::operator*(other);
+  }
+  bool operator==(const Symmetric& other) const {
+    for (size_t i = 0; i < N; i++)
+      if (this->indices()[i] != other.indices()[i])
+        return false;
+    return true;
+  }
+  Symmetric inverse() const {
     return Eigen::PermutationMatrix<N>(Eigen::PermutationMatrix<N>::inverse());
   }
+  friend std::ostream &operator<<(std::ostream &os, const Symmetric& m) {
+    for (size_t i = 0; i < N; i++)
+      os << m.indices()[i] << " ";
+    return os;
+  }
+  void print(const std::string& s = "") const {
+    std::cout << s << *this << std::endl;
+  }
+  bool equals(const Symmetric<N>& other, double tol = 0) const {
+    return this->indices() == other.indices();
+  }
 };
 
 /// Define permutation group traits to be a model of the Multiplicative Group concept
 template<int N>
-struct traits<Permutation<N> > : internal::MultiplicativeGroupTraits<
-    Permutation<N> > {
-  static void Print(const Permutation<N>& m, const std::string& str = "") {
-    std::cout << m << std::endl;
-  }
-  static bool Equals(const Permutation<N>& m1, const Permutation<N>& m2,
-      double tol = 1e-8) {
-    return m1.indices() == m2.indices();
-  }
+struct traits<Symmetric<N> > : internal::MultiplicativeGroupTraits<Symmetric<N> >,
+    Testable<Symmetric<N> > {
 };
 
 } // namespace gtsam
@@ -56,17 +76,61 @@ struct traits<Permutation<N> > : internal::MultiplicativeGroupTraits<
 using namespace std;
 using namespace gtsam;
 
-typedef Permutation<3> G; // Let's use the permutation group of order 3
-
 //******************************************************************************
-TEST(Group, Concept) {
-  BOOST_CONCEPT_ASSERT((IsGroup<G>));
+typedef Symmetric<2> S2;
+TEST(Group, S2) {
+  S2 e, s1 = S2::Transposition(0, 1);
+  BOOST_CONCEPT_ASSERT((IsGroup<S2>));
+  EXPECT(check_group_invariants(e, s1));
 }
 
 //******************************************************************************
-TEST(Group , Invariants) {
-  G g, h;
-  EXPECT(check_group_invariants(g, h));
+typedef Symmetric<3> S3;
+TEST(Group, S3) {
+  S3 e, s1 = S3::Transposition(0, 1), s2 = S3::Transposition(1, 2);
+  BOOST_CONCEPT_ASSERT((IsGroup<S3>));
+  EXPECT(check_group_invariants(e, s1));
+  EXPECT(assert_equal(s1, s1 * e));
+  EXPECT(assert_equal(s1, e * s1));
+  EXPECT(assert_equal(e, s1 * s1));
+  S3 g = s1 * s2; // 1 2 0
+  EXPECT(assert_equal(s1, g * s2));
+  EXPECT(assert_equal(e, compose_pow(g, 0)));
+  EXPECT(assert_equal(g, compose_pow(g, 1)));
+  EXPECT(assert_equal(e, compose_pow(g, 3))); // g is generator of Z3 subgroup
+}
+
+//******************************************************************************
+// The direct product of S2=Z2 and S3 is the symmetry group of a hexagon,
+// i.e., the dihedral group of order 12 (denoted Dih6 because 6-sided polygon)
+typedef DirectProduct<S2, S3> Dih6;
+
+std::ostream &operator<<(std::ostream &os, const Dih6& m) {
+  os << "( " << m.first << ", " << m.second << ")";
+  return os;
+}
+
+// Provide traits with Testable
+namespace gtsam {
+template<>
+struct traits<Dih6> : internal::MultiplicativeGroupTraits<Dih6> {
+  static void Print(const Dih6& m, const string& s = "") {
+    cout << s << m << endl;
+  }
+  static bool Equals(const Dih6& m1, const Dih6& m2, double tol = 1e-8) {
+    return m1 == m2;
+  }
+};
+} // namespace gtsam
+
+TEST(Group, Dih6) {
+  Dih6 e, g(S2::Transposition(0, 1),
+      S3::Transposition(0, 1) * S3::Transposition(1, 2));
+  BOOST_CONCEPT_ASSERT((IsGroup<Dih6>));
+  EXPECT(check_group_invariants(e, g));
+  EXPECT(assert_equal(e, compose_pow(g, 0)));
+  EXPECT(assert_equal(g, compose_pow(g, 1)));
+  EXPECT(assert_equal(e, compose_pow(g, 6))); // g is generator of Z6 subgroup
 }
 
 //******************************************************************************