diff --git a/abp/clifford.py b/abp/clifford.py
index ca489a8..c93cf56 100644
--- a/abp/clifford.py
+++ b/abp/clifford.py
@@ -8,7 +8,7 @@ This program generates lookup tables
 import os, json
 from functools import reduce
 import itertools as it
-from . import qi
+import qi
 import numpy as np
 import tempfile
 from tqdm import tqdm
diff --git a/abp/qi.py b/abp/qi.py
index 204ae0d..ddb05c3 100644
--- a/abp/qi.py
+++ b/abp/qi.py
@@ -41,3 +41,5 @@ nobond = np.kron(plus, plus)
 common_us = id, px, py, pz, ha, ph, sqz, msqz, sqy, msqy, sqx, msqx
 names = "identity", "px", "py", "pz", "hadamard", "phase", "sqz", "msqz", "sqy", "msqy", "sqx", "msqx"
 by_name = dict(zip(names, common_us))
+
+paulis = px, py, pz
diff --git a/tests/test_clifford.py b/tests/test_clifford.py
index 4adf935..db1371d 100644
--- a/tests/test_clifford.py
+++ b/tests/test_clifford.py
@@ -2,7 +2,58 @@ from numpy import *
 from scipy.linalg import sqrtm
 from tqdm import tqdm
 import itertools as it
-
 from abp import clifford
+from abp import qi
+
+
+def identify_pauli(m):
+    """ Given a signed Pauli matrix, name it. """
+    for sign in (+1, -1):
+        for pauli_label, pauli in zip("xyz", qi.paulis):
+            if allclose(sign * pauli, m):
+                return sign, pauli_label
+
+
+def _test_find():
+    """ Test that slightly suspicious function """
+    assert lc.find(id, lc.unitaries) == 0
+    assert lc.find(px, lc.unitaries) == 1
+    assert lc.find(exp(1j*pi/4.)*ha, lc.unitaries) == 4
+
+def get_action(u):
+    """ What does this unitary operator do to the Paulis? """
+    return [identify_pauli(u.dot(p.dot(qi.hermitian_conjugate(u)))) for p in qi.paulis]
+
+
+def format_action(action):
+    return "".join("{}{}".format("+" if s >= 0 else "-", p) for s, p in action)
+
+
+def test_we_have_24_matrices():
+    """ Check that we have 24 unique actions on the Bloch sphere """
+    actions = set(tuple(get_action(u)) for u in clifford.unitaries)
+    assert len(set(actions)) == 24
+
+
+def test_we_have_all_useful_gates():
+    """ Check that all the interesting gates are included up to a global phase """
+    for name, u in qi.by_name.items():
+        clifford.find_clifford(u, clifford.unitaries)
+
+
+def _test_group():
+    """ Test we are really in a group """
+    matches = set()
+    for a, b in tqdm(it.combinations(clifford.unitaries, 2), "Testing this is a group"):
+        i, phase = clifford.find_clifford(a.dot(b), clifford.unitaries)
+        matches.add(i)
+    assert len(matches)==24
+
 
+def test_conjugation_table():
+    """ Check that the table of Hermitian conjugates is okay """
+    assert len(set(clifford.conjugation_table))==24
 
+def test_times_table():
+    """ Check the times table """
+    assert clifford.times_table[0][4]==4