diff --git a/clifford.py b/clifford.py
index 2548963..4765438 100644
--- a/clifford.py
+++ b/clifford.py
@@ -13,17 +13,32 @@ Following the prescription of Anders (thesis pg. 26):
 
 from numpy import *
 
-i = matrix(eye(2, dtype=complex))
+def find_up_to_phase(u):
+    """ Find the index of a given u within a list of unitaries, up to a global phase  """
+    global unitaries
+    for i, t in enumerate(unitaries):
+        for phase in range(8):
+            if allclose(t, exp(1j*phase*pi/4.)*u):
+                return i, phase
+    raise IndexError
+
+id = matrix(eye(2, dtype=complex))
 px = matrix([[0, 1], [1, 0]], dtype=complex)
 py = matrix([[0, -1j], [1j, 0]], dtype=complex)
 pz = matrix([[1, 0], [0, -1]], dtype=complex)
-h = matrix([[1, 1], [1, -1]], dtype=complex) / sqrt(2)
-p = matrix([[1, 0], [0, 1j]], dtype=complex)
+ha = matrix([[1, 1], [1, -1]], dtype=complex) / sqrt(2)
+ph= matrix([[1, 0], [0, 1j]], dtype=complex)
 
-permutations = (i, h, p, h*p, h*p*h, h*p*h*p)
-signs = (i, px, py, pz)
+permutations = (id, ha, ph, ha*ph, ha*ph*ha, ha*ph*ha*ph)
+signs = (id, px, py, pz)
 unitaries = [p*s for p in permutations for s in signs]
 
+conjugation_table = []
+
+for i, u in enumerate(unitaries):
+    i, phase = find_up_to_phase(u.H)
+    conjugation_table.append(i)
+
 
 # TODO:
 # - check that we re-generate the table
diff --git a/tests/test_clifford.py b/tests/test_clifford.py
index ad4e43d..5a7aefa 100644
--- a/tests/test_clifford.py
+++ b/tests/test_clifford.py
@@ -11,14 +11,6 @@ msqx = sqrtm(-1j * lc.px)
 paulis = (lc.px, lc.py, lc.pz)
 
 
-def find_u(u, unitaries):
-    """ Find the index of a given u within a list of unitaries """
-    for i, t in enumerate(unitaries):
-        if allclose(t, u):
-            return i
-    return -1
-
-
 def identify_pauli(m):
     """ Given a signed Pauli matrix, name it. """
     for sign in (+1, -1):
@@ -27,6 +19,12 @@ def identify_pauli(m):
                 return sign, pauli_label
 
 
+def test_find_up_to_phase():
+    """ Test that slightly suspicious function """
+    assert lc.find_up_to_phase(lc.id) == (0, 0)
+    assert lc.find_up_to_phase(lc.px) == (1, 0)
+    assert lc.find_up_to_phase(exp(1j*pi/4.)*lc.ha) == (4, 7)
+
 def get_action(u):
     """ What does this unitary operator do to the Paulis? """
     return [identify_pauli(u * p * u.H) for p in paulis]
@@ -44,31 +42,22 @@ def test_we_have_24_matrices():
 
 def test_we_have_all_useful_gates():
     """ Check that all the interesting gates are included up to a global phase """
-    names = "i", "px", "py", "pz", "h", "p"
-    unitaries = lc.i, lc.px, lc.py, lc.pz, lc.h, lc.p
-    for name, unitary in zip(names, unitaries):
-        i = find_u(unitary, lc.unitaries)
-        assert i >= 0
-        print "{}\t=\tlc.unitaries[{}]".format(name, i)
-
-    names = "sqrt(ix)", "sqrt(-ix)", "sqrt(iy)", "sqrt(-iy)", "sqrt(iz)", "sqrt(-iz)",
-    unitaries = sqz, msqz, sqy, msqy, sqx, msqx
-    for name, unitary in zip(names, unitaries):
-        rotated = [exp(1j * phase * pi / 4.) * unitary for phase in range(8)]
-        results = [find_u(r, lc.unitaries) for r in rotated]
-        assert any(x > 0 for x in results)
-        phase, index = [(i, r) for i, r in enumerate(results) if r>=0][0]
-        print "exp(1j*{}*pi/4) . {}\t=\tlc.unitaries[{}]".format(phase, name, index)
+    common_us = lc.id, lc.px, lc.py, lc.pz, lc.ha, lc.ph, sqz, msqz, sqy, msqy, sqx, msqx
+    for u in common_us:
+        lc.find_up_to_phase(u)
 
 
 def test_group():
     """ Test we are really in a group """
+    matches = set()
     for a in lc.unitaries:
         for b in lc.unitaries:
-            unitary = a*b
-            rotated = [exp(1j * phase * pi / 4.) * unitary for phase in range(8)]
-            results = [find_u(r, lc.unitaries) for r in rotated]
-            assert len([x for x in results if x>=0])==1
+            i, phase = lc.find_up_to_phase(a*b)
+            matches.add(i)
+    assert len(matches)==24
 
 
+def test_conjugation_table():
+    """ Check that the table of Hermitian conjugates is okay """
+    assert len(set(lc.conjugation_table))==24