diff --git a/automaton.py b/automaton.py
index afb4e9f..ad8e62f 100755
--- a/automaton.py
+++ b/automaton.py
@@ -1,9 +1,9 @@
 #!/usr/bin/python
 
 # 0 is infinity
-coxeter_matrix = [[1, 2, 3],
-                  [2, 1, 0],
-                  [3, 0, 1]]
+coxeter_matrix = [[1, 5, 5],
+                  [5, 1, 5],
+                  [5, 5, 1]]
 
 import math
 from copy import copy
@@ -26,6 +26,18 @@ class Root:
 	def __copy__(self):
 		return Root(self.id, self.rank, self.depth, self.v.copy(), self.neighbors.copy())
 
+class Groupelement:
+	def __init__(self, id, rank, word):
+		self.id = id
+		self.rank = rank
+		self.word = word
+		self.length = len(word)
+		self.left = [None]*rank
+		self.right = [None]*rank
+		self.node = None
+		self.lex_node = None
+		self.inverse = None
+
 # compute <alpha_k, beta> where alpha_k is one of the simple roots and beta any root
 def form_gen_root(form, k, root):
 	rank = len(form)
@@ -152,71 +164,84 @@ def generate_automaton(small_roots, lex_reduced = False):
 				id += 1
 			edges.append((nodes[node], nodes[newnode], k))
 
-	return (nodes, levels, edges)
+	graph = [[None for i in range(rank)] for j in range(len(nodes))]
+	for (fr,to,gen) in edges:
+		graph[fr][gen] = to
+	return graph
 
 # main program
 
 form = [[-math.cos(math.pi/m) if m > 0 else -1 for m in row] for row in coxeter_matrix]
 rank = len(coxeter_matrix)
 small_roots = find_small_roots(form)
-nodes, levels, edges = generate_automaton(small_roots, lex_reduced = False)
-nodes_lex, levels_lex, edges_lex = generate_automaton(small_roots, lex_reduced = True)
+graph = generate_automaton(small_roots, lex_reduced = False)
+graph_lex = generate_automaton(small_roots, lex_reduced = True)
 
-#for r in small_roots:
-#	print((r.id,r.v,[n.id if n else -1 for n in r.neighbors]))
+group = [Groupelement(0, rank, tuple())]
+group[0].inverse = group[0]
+group[0].node = group[0].lex_node = 0
 
-revedges = sorted(edges, key = lambda x:x[1])
-
-adjlist = {}
-revadjlist = {}
-for efrom, eto, egen in edges:
-	if not efrom in adjlist:
-		adjlist[efrom] = [-1]*rank
-	adjlist[efrom][egen] = eto
-	if not eto in revadjlist:
-		revadjlist[eto] = [-1]*rank
-	revadjlist[eto][egen] = efrom
-
-words = [([], 0)]
-depth = 0
 i = 0
-while len(words[i][0]) < 10:
-	curword = words[i][0]
-	curnode = words[i][1]
-	for gen, nextnode in enumerate(adjlist[curnode]):
-		if nextnode < 0:
-			continue
-		nextword = curword.copy()
-		nextword.append(gen)
-		words.append((nextword, nextnode))
-	i += 1
+size = 1
+while True:
+	current = group[i]
+	i+=1
 
-#print(sorted([x[1] for x in words]))
-#print(["".join([chr(ord('a')+x) for x in w[0]]) for w in words])
+	if current.length >= 10:
+		break
 
-levelnodes = []
-for n,id in nodes.items():
-	level = levels[n]
-	if level >= len(levelnodes):
-		levelnodes.append([])
-	levelnodes[level].append(id)
+	for gen, new_lex_node in enumerate(graph_lex[current.lex_node]):
+		if new_lex_node:
+			new_element = Groupelement(size, rank, current.word + (gen,))
+			new_element.lex_node = new_lex_node
+			new_element.node = graph[current.node][gen]
+			group.append(new_element)
+			size += 1
 
-print("digraph test123 {")
-print('rankdir="TB"')
+			# w = w_1 t, w s = w_1
+			# right multiplication (and left in case it does the same as a right mult)
+			for k in range(rank):
 
-for (level,ns) in enumerate(levelnodes):
-	print('{rank = "same";', end = ' ')
-	for n in ns:
-		print("{id:d};".format(id=n), end = ' ')
-	print('}')
+				# if right multiplication by k decreases length
+				if not graph[new_element.node][k]:
+					word = list(new_element.word)
+					longer_suffix = group[0]
+					while len(word) > 0:
+						letter = word.pop()
+						shorter_suffix = longer_suffix
+						longer_suffix = shorter_suffix.left[letter]
 
+						# w = w_1 t w_2, w_2 s_k = t w_2
+						# in the case word = [] longer_suffix could be None
+						# found it
+						if len(word) == 0 or shorter_suffix.right[k] == longer_suffix:
+							# finish word
+							while len(word) > 0:
+								shorter_suffix = shorter_suffix.left[word.pop()]
+							new_element.right[k] = shorter_suffix
+							shorter_suffix.right[k] = new_element
 
-colors = ['red', 'darkgreen', 'blue', 'orange']
+			# find inverse and left multiply
+			inverse = group[0]
+			word = list(new_element.word)
+			while len(word) > 0:
+				inverse = inverse.right[word.pop()]
+				if not inverse:
+					break
+			if inverse:
+				new_element.inverse = inverse
+				inverse.inverse = new_element
+				for k in range(rank):
+					if inverse.right[k]:
+						other = inverse.right[k].inverse
+						new_element.left[k] = other
+						other.left[k] = new_element
+					if new_element.right[k]:
+						other = new_element.right[k].inverse
+						inverse.left[k] = other
+						other.left[k] = inverse
 
-for e in edges:
-	print("{fr:d} -> {to:d} [color={color}];".format(
-		fr = e[0],
-		to = e[1],
-		color = colors[e[2]]))
-
-print("}")
+length = 0
+for i in range(1,len(group)+1):
+	if i == len(group) or group[i].length > group[i-1].length:
+		print("{number:d} elements up to length {length:d}".format(number = i, length = group[i-1].length))