#### ch 1

sage: True and True
True
sage: True and False
False
sage: False and True
False
sage: False and False
False

sage: def implies(p,q): return not(p and not(q))
sage: implies(True,True)
True
sage: implies(True,False)
False
sage: implies(False,True)
True
sage: implies(False,False)
True

sage: S1 = Set([1,ZZ,"Zeus"])
sage: S2 = Set([S1,1,"Hera"])
sage: S1; S2
{1, 'Zeus', Integer Ring}
{{1, 'Zeus', Integer Ring}, 1, 'Hera'}
sage: S2.cardinality()
3
sage: S1.difference(S2)
{'Zeus', Integer Ring}
sage: S2.difference(S1)
{{1, 'Zeus', Integer Ring}, 'Hera'}
sage: "Zeus" in S1
True
sage: S2.symmetric_difference(S1)
{{1, 'Zeus', Integer Ring}, 'Hera', 'Zeus', Integer Ring}
sage: S1.symmetric_difference(S2)
{{1, 'Zeus', Integer Ring}, 'Hera', 'Zeus', Integer Ring}
sage: S1.intersection(S2)
{1}
sage: S1.union(S2)
{{1, 'Zeus', Integer Ring}, 1, 'Hera', 'Zeus', Integer Ring}


### ch 2

sage: euler_phi(3)                                            
2
sage: pi = lambda x: pari(x).primepi()
sage: pi(4)
2
sage: f1 = lambda x: sqrt(x)
sage: f1(3)
sqrt(3)
sage: f1(4)
2
sage: f2 = lambda x: RR(sqrt(x))
sage: f2(3)
1.73205080756888
sage: f2(4)
2.00000000000000

sage: f = x^4 - 16                                      
sage: f.roots()
[(2*I, 1), (-2, 1), (-2*I, 1), (2, 1)]

sage: list(cartesian_product_iterator([[1,2], ['a','b']]))
[(1, 'a'), (1, 'b'), (2, 'a'), (2, 'b')]

sage: A = matrix(3,3,[1,2,3,4,5,6,7,8,9])                  
sage: (-1)*A
[-1 -2 -3]
[-4 -5 -6]
[-7 -8 -9]
sage: v = vector([1,1,1])
sage: A*v
(6, 15, 24)
sage: x = var("x"); y = var("y"); z = var("z")
sage: w = vector([x,y,z])
sage: A*w
(  3 z + 2 y + x, 6 z + 5 y + 4 x, 9 z + 8 y + 7 x)

sage: A = matrix(3,3,[1,2,3,4,5,6,1,0,0])             
sage: det(A)
-3
sage: A^(-1)
[   0    0    1]
[  -2    1   -2]
[ 5/3 -2/3    1]
sage: B = matrix(4,3,[1,2,3,4,5,6,7,8,9,1,1,1])
sage: B*A
[12 12 15]
[30 33 42]
[48 54 69]
[ 6  7  9]

sage: A = matrix(3,3,[1,2,3,4,5,6,7,8,9])                       
sage: det(A)
0
sage: a = var("a"); b = var("b"); c = var("c"); d = var("d")
sage: B = matrix(2,2,[a,b,c,d])
sage: det(B)
a*d - b*c

sage: C0 = [x for x in range(-10,10) if x%3 == 0]; print C0
[-9, -6, -3, 0, 3, 6, 9]
sage: C1 = [x for x in range(-10,10) if x%3 == 1]; print C1
[-8, -5, -2, 1, 4, 7]
sage: C2 = [x for x in range(-10,10) if x%3 == 2]; print C2
[-10, -7, -4, -1, 2, 5, 8]

sage: expand((x+y)^(5))
 y^5 + 5*x*y^4 + 10*x^2*y^3 + 10*x^3*y^2 + 5*x^4*y + x^5             
sage: binomial(5,2)
10
sage: terms = ["x","x","y","y","y"]
sage: number_of_permutations(terms)
10
sage: permutations(terms)
['xxyyy', 'xyxyy', 'xyyxy', 'xyyyx', 'yxxyy', 
        'yxyxy', 'yxyyx', 'yyxxy', 'yyxyx', 'yyyxx']
sage: terms = [1,2,3,4,5]
sage: number_of_permutations(terms)
120
sage: factorial(5)
120

### ch 3

def swap(g):
    N = parent(g).degree()
    return sum([len([i2 for i2 in range(i1+1,N+1) 
                if g(i2)<g(i1)]) for i1 in range(1,N)])
  

sage: G = SymmetricGroup(4)
sage: g = G.random_element(); g     ## random output
(1,2,3)
sage: swap(g)
2
sage: G = SymmetricGroup(3)
sage: g = G([(1,2)])
sage: swap(g)
1

sage: G = SymmetricGroup(6)
sage: G.order()
720
sage: swaps = [swap(g) for g in G]
sage: M = max(swaps); M
15
sage: L = [swaps.count(i) for i in range(16)]
sage: list_plot(L).show()

sage: G = SymmetricGroup(4)                                  
sage: g1 = G([(2,4),(1,3)])
sage: g2 = G([(1,2,3,4)])
sage: matrix([[1,2,3,4],[g1(i) for i in [1,2,3,4]]])
[1 2 3 4]
[3 4 1 2]
sage: matrix([[1,2,3,4],[g2(i) for i in [1,2,3,4]]])
[1 2 3 4]
[2 3 4 1]
sage: g1*g2
(1,4,3,2)
sage: matrix([[1,2,3,4],[(g1*g2)(i) for i in [1,2,3,4]]])
[1 2 3 4]
[4 1 2 3]
sage: g2*g1
(1,4,3,2)

sage: G = SymmetricGroup(4)                                    
sage: g1 = G([(2,4),(1,3)])
sage: g2 = G([(1,2,3,4)])
sage: g2*g1
(1,4,3,2)
sage: g1.sign()
1
sage: g2.sign()
-1
sage: (g1*g2).sign()
-1
sage: g1.order()
2
sage: g2.order()
4
sage: (g1*g2).order()
4


#### ch 4

sage: f= [(17,19,24,22),(18,21,23,20),(6,25,43,16),\          
                (7,28,42,13),(8,30,41,11)]
sage: b=[(33,35,40,38),(34,37,39,36),( 3, 9,46,32),\
                   ( 2,12,47,29),( 1,14,48,27)]
sage: l=[( 9,11,16,14),(10,13,15,12),( 1,17,41,40),\
                    ( 4,20,44,37),( 6,22,46,35)]
sage: r=[(25,27,32,30),(26,29,31,28),( 3,38,43,19),\
                   ( 5,36,45,21),( 8,33,48,24)]
sage: u=[( 1, 3, 8, 6),( 2, 5, 7, 4),( 9,33,25,17),\
                   (10,34,26,18),(11,35,27,19)]
sage: d=[(41,43,48,46),(42,45,47,44),(14,22,30,38),\
                   (15,23,31,39),(16,24,32,40)]
sage: cube = PermutationGroup([f,b,l,r,u,d])
sage: F,B,L,R,U,D = cube.gens()
sage: cube.order()
43252003274489856000
sage: F.order()
4

sage: rubik = CubeGroup()
sage: rubik.move("")
[(), [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 
17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 
31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 
45, 46, 47, 48]]
sage: rubik.faces("R*L")
{'back': [[48, 34, 6], [45, 0, 4], [43, 39, 1]],
 'down': [[40, 42, 19], [37, 0, 21], [35, 47, 24]],
 'front': [[41, 18, 3], [44, 0, 5], [46, 23, 8]],
 'left': [[11, 13, 16], [10, 0, 15], [9, 12, 14]],
 'right': [[27, 29, 32], [26, 0, 31], [25, 28, 30]],
 'up': [[17, 2, 38], [20, 0, 36], [22, 7, 33]]}
sage: P = rubik.plot_cube("R^2*U^2*R^2*U^2*R^2*U^2")
sage: show(P)

### ch 5

sage: G = PermutationGroup(['(1,2,3)', '(2,3)'])        
sage: G.cayley_table(names="abcdef")
       [a b c d e f]
       [b a d c f e]
       [c e a f b d]
       [d f b e a c]
       [e c f a d b]
       [f d e b c a]

sage: G = SymmetricGroup(6)
sage: g1 = G.random_element()
sage: g2 = G.random_element()
sage: H = G.subgroup([g1,g2])
sage: G.order()/H.order() in ZZ
True

sage: R2 = PolynomialRing(QQ,2,"xy")
sage: F2 = FractionField(R2)
sage: x,y = R2.gens()
sage: f = lambda n: F2((x+1/x+y+1/y+x*y+y/x+1/(x*y)+x/y)^n)
sage: a = f(10).numerator()
sage: b = f(10).denominator()
sage: b
x^10*y^10
sage: a.coefficient(x^(11)*y^(11))
19246920

sage: rubik = CubeGroup()
sage: r = rubik.R()
sage: u = rubik.U()
sage: G = PermutationGroup([r^2,u^2])
sage: G.order()
12
sage: D6 = DihedralGroup(6)
sage: G.is_isomorphic(D6)
True

sage: rubik = CubeGroup()
sage: r = rubik.R()
sage: u = rubik.U()
sage: G = PermutationGroup([r^2,u^2])
sage: G.is_solvable()
True

sage: g = r*u*r^(-1)*u^(-1)
sage: g^2
(1,35,9)(2,5,21)(3,27,33)(8,25,19)(24,30,43)(26,28,34)
sage: g^3
(1,33)(3,35)(8,43)(9,27)(19,30)(24,25)

sage: rubik = CubeGroup()
sage: r = rubik.R()
sage: u = rubik.U()
sage: G = PermutationGroup([r^2,u^2])
sage: CG = G.conjugacy_classes_representatives(); CG
[(),
 (3,43)(5,45)(8,48)(19,38)(21,36)(24,33)(25,32)(26,31)(27,30)
   (28,29),
 (2,7)(18,34)(21,36)(28,29),
 (2,7)(3,43)(5,45)(8,48)(18,34)(19,38)(24,33)(25,32)(26,31)
   (27,30),
 (1,8,48)(3,43,6)(4,5,45)(9,25,32)(10,26,31)(11,27,30)
   (17,33,24)(19,38,35),
 (1,8,48)(2,7)(3,43,6)(4,5,45)(9,25,32)(10,26,31)(11,27,30)
   (17,33,24)(18,34)(19,38,35)(21,36)(28,29)]
sage: t = PolynomialRing(QQ,"t").gen()
sage: p_G = sum([t^(g.order()) for g in CG]); p_G
t^6 + t^3 + 3*t^2 + t


sage: G = SymmetricGroup(6)
sage: CG = G.conjugacy_classes_representatives()
sage: class_eqn = sum([G.order()/(G0.Centralizer(x)).Order()\
                       for x in CG])
sage: class_eqn == G.order()
True

######## ch 6

sage: R = PolynomialRing(QQ,"x")
sage: x = R.gen()
sage: def f(t,n):
          if n<2: return R(n)
          else:
              return f(t,n-2)+t*f(t,n-1)
sage: [f(x,n) for n in [0,1,2,3,4,5]]
[0,1, x, x^2 + 1, x^3 + 2*x, x^4 + 3*x^2 + 1]

sage: MS = MatrixSpace(QQ, 3, 5)
sage: A1 = MS([[1,2,3,4,5],[1,1,-1,0,1],[6,7,8,9,10]])
sage: A1.echelon_form()
[   1    0    0 -3/2   -3]
[   0    1    0    2    4]
[   0    0    1  1/2    0]
sage: A2 = MS([[1,2,3,4,5],[1,1,1,1,1],[6,7,8,9,10]])
sage: A2.echelon_form()
[ 1  0 -1 -2 -3]
[ 0  1  2  3  4]
[ 0  0  0  0  0]

sage: MS = MatrixSpace(GF(2), 12, 12)
sage: lights = [[1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0],\
....: [1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0],\
....: [1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0],\
....: [1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0],\
....: [1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0],\
....: [1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0],\
....: [0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1],\
....: [0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1],\
....: [0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1],\
....: [0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1],\
....: [0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1],\
....: [0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1]]
sage: A = MS(lights)
sage: A.echelon_form()

[1 0 0 0 0 1 0 1 0 0 0 1]
[0 1 0 0 0 1 0 1 1 0 0 0]
[0 0 1 0 0 1 0 1 0 1 0 0]
[0 0 0 1 0 1 0 1 0 0 1 0]
[0 0 0 0 1 1 0 0 1 1 1 1]
[0 0 0 0 0 0 1 1 1 1 1 1]
[0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0]
sage: A.rank()
6

###### ch 7

sage: G = graphs.CycleGraph(3)                           
sage: show(G.plot())

sage: G = Graph({"a":["b","c"], "b":["a","c"],\                
                 "c":["a","b"]})
sage: show(G.plot())

sage: gens = [(1,2),(2,3)]
sage: gap.eval("gens := "+str(gens))
'[ (1,2), (2,3) ]'
sage: gap.eval("C := CayleyGraph(Group(gens),gens)")
'rec( isGraph := true, order := 6, 
      group := Group([ (1,3)(2,4)(5,6), (1,2)(3,5)(4,6) ]), 
      schreierVector := [ -1, 2, 1, 1, 2, 1 ], 
      adjacencies := [ [ 2, 3 ] ], 
      representatives := [ 1 ], 
      names := [ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ], 
      isSimple := true )'
sage: E = eval(gap.eval("UndirectedEdges(C)"))
sage: V = eval(gap.eval("Elements(Vertices(C))"))
sage: L = [sum([[y[i] for y in [x for x in E if v in x]\
                 if y[i]!=v]  for i in range(len(gens))],[])\
                 for v in V]
sage: d = dict(zip(V,L))
sage: G = Graph(d)
sage: show(G.plot())

sage: gens = [(1,2),(2,3)]
sage: G = PermutationGroup(gens)
sage: C = G.cayley_graph()
sage: show(C.plot())

### ch 8

sage: G = graphs.DodecahedralGraph()
sage: show(G.plot())

sage: G = graphs.IcosahedralGraph()
sage: show(G.plot())

######### ch 9

sage: G = SymmetricGroup(4)
sage: gensG = G.gens()
sage: h = G.random_element()
sage: gensG_h = [h*g*h^(-1) for g in gensG]
sage: phi = PermutationGroupMorphism_im_gens(G,G,gensG,gensG_h)     
sage: phi.image(G)
'Group([ (1,3,4,2), (1,3) ])'
sage: phi.range()
 Symmetric group of order 4! as a permutation group

sage: MS = MatrixSpace(GF(2), 3, 3)
sage: A = MS([0, 1 ,0, 1, 0, 0, 0, 0, 1])
sage: B = MS([1, 0 ,0, 0, 0, 1, 0, 1, 0])
sage: G = MatrixGroup([A,B])
sage: G1 = SymmetricGroup(3)
sage: imG = G._gap_().IsomorphismPermGroup().Image(G._gap_())
sage: gens = imG.GeneratorsOfGroup()
sage: G2 = PermutationGroup(gens)
sage: G2.is_isomorphic(G1)
True

sage: rubik = CubeGroup()
sage: r = rubik.R()
sage: l = rubik.L()
sage: u = rubik.U()
sage: d = rubik.D()
sage: f = rubik.F()
sage: b = rubik.B()
sage: mr = r^(-1)*l
sage: a = f^2*mr*f^(-1)*mr^(-1)*u^(-1)*mr*f*mr^(-1)*u*f^2         
sage: b = f*u^2*f^(-1)*u^(-1)*l^(-1)*b^(-1)*u^2*b*u*l
sage: G = PermutationGroup([a,b])
sage: G.order()
8
sage: G.is_abelian()
False
sage: b*a*b == a
True
sage: b^4
()
sage: a^2 == b^2
True

sage: G = SymmetricGroup(5)
sage: CG = G.conjugacy_classes_representatives()        
sage: len(CG)
7
sage: number_of_partitions(5, algorithm='bober')
7

sage: G = CyclicPermutationGroup(10)
sage: g0 = G.gens()[0]
sage: phi = PermutationGroupMorphism_im_gens( G, G, [g0], [g0^2])
sage: phi.kernel()
Group([ (1,6)(2,7)(3,8)(4,9)(5,10) ])
sage: ker = PermutationGroup(["(1,6)(2,7)(3,8)(4,9)(5,10)"])
sage: ker.order()
2

sage: G = SymmetricGroup(5)                         
sage: H = AlternatingGroup(5)
sage: H.is_subgroup(G)
True
sage: H.is_normal(G)
True
sage: H.is_simple()
True
sage: G.quotient_group(H)
Permutation Group with generators [(1,2)]

sage: C2 = CyclicPermutationGroup(2)
sage: C3 = CyclicPermutationGroup(3)
sage: G = C2.direct_product(C3)
sage: G
(Permutation Group with generators [(1,2), (3,4,5)],
 Homomorphism : Cyclic group of order 2 as a permutation group 
  --> Permutation Group with generators [(1,2), (3,4,5)],
 Homomorphism : Cyclic group of order 3 as a permutation group 
  --> Permutation Group with generators [(1,2), (3,4,5)],
 Homomorphism : Permutation Group with generators [(1,2), (3,4,5)] 
  --> Cyclic group of order 2 as a permutation group,
 Homomorphism : Permutation Group with generators [(1,2), (3,4,5)] 
  --> Cyclic group of order 3 as a permutation group)
sage: G[0].order()
6

sage: C3 = CyclicPermutationGroup(3)
sage: S8 = SymmetricGroup(8)
sage: W = C3._gap_().WreathProduct(S8._gap_())              
sage: W.Order()
264539520
sage: W.Order() == 3^8*factorial(8)
True

### ch 10

sage: p = lambda n: \
          prod([(t^(j+1)-1)/(t-1) for j in range(1,n+1)])
sage: p(2)
t^3 + 2*t^2 + 2*t + 1

sage: G = SymmetricGroup(3)
sage: p = sum([t^swap(g) for g in G])
sage: p
t^3 + 2*t^2 + 2*t + 1

sage: rubik = CubeGroup()
sage: b = rubik.B()
sage: d = rubik.D()
sage: f = rubik.F()
sage: l = rubik.L()
sage: r = rubik.R()
sage: u = rubik.U()
sage: r*u*r^(-1)
(1,19,24,6)(2,21,7,4)(8,30,17,9)(10,34,28,18)(11,35,25,43)
sage: G = PermutationGroup([b,d,f,l,r,u])
sage: g = G(str(r*u*r^(-1)))
sage: g.word_problem([b,d,f,l,r,u])
  x5*x6*x5^-1
  [['(25,27,32,30)(26,29,31,28)(3,38,43,19)(5,36,45,21)
       *(8,33,48,24)', 1], 
   ['(1,3,8,6)(2,5,7,4)(9,33,25,17)(10,34,26,18)
       *(11,35,27,19)', 1], 
   ['(25,27,32,30)(26,29,31,28)(3,38,43,19)(5,36,45,21)
       *(8,33,48,24)', -1]]
'(25,27,32,30)(26,29,31,28)(3,38,43,19)(5,36,45,21)
    *(8,33,48,24)(1,3,8,6)(2,5,7,4)(9,33,25,17)(10,34,26,18)
    *(11,35,27,19) (25,27,32,30)(26,29,31,28)(3,38,43,19)
    *(5,36,45,21)(8,33,48,24)^-1'

sage: rubik = CubeGroup()
sage: state = rubik.faces("R*U")
sage: rubik.solve(state)
'R*U'
sage: state = rubik.faces("R*U*R^(-1)")
sage: state
{'back': [[33, 28, 25], [36, 0, 37], [38, 39, 40]],
 'down': [[41, 42, 11], [44, 0, 45], [46, 47, 48]],
 'front': [[9, 10, 24], [20, 0, 7], [22, 23, 6]],
 'left': [[8, 34, 35], [12, 0, 13], [14, 15, 16]],
 'right': [[43, 26, 27], [18, 0, 29], [17, 31, 32]],
 'up': [[19, 21, 3], [2, 0, 5], [1, 4, 30]]}
sage: rubik.solve(state)
'R*U*R^-1'

## ch 11

sage: u2 = '(1,3)(2,4)'; f2 = '(1,8)(4,5)'; d2 = '(5,7)(6,8)'   
sage: b2 = '(3,6)(2,7)'; r2 = '(2,5)(1,6)'; l2 = '(4,7)(3,8)'
sage: H = PermutationGroup([u2,f2,d2,b2,r2,l2])
sage: H.order()
96
sage: H.conjugacy_classes_representatives()

[(),
 (5,7)(6,8),
 (3,6,8)(4,5,7),
 (2,4)(5,7),
 (2,4,7,5)(3,6),
 (1,3)(6,8),
 (1,3,6,8)(4,7),
 (1,3)(2,4)(5,7)(6,8),
 (1,3,6,8)(2,4,5,7),
 (1,3)(2,5)(4,7)(6,8)]

sage: P = PolynomialRing(GF(5),"x")
sage: x = P.gen()
sage: K.<a> = GF(5^2, name='a', modulus=x^2 - 2 )
sage: K
Finite Field in a of size 5^2
sage: L = GF(25,"b")
sage: L
Finite Field in b of size 5^2

sage: rubik = CubeGroup()                                  
sage: f = rubik.F()
sage: u = rubik.U()
sage: T = PermutationGroup([f,u])
sage: T.order()
73483200

sage: P = PGL(2,5)                                        
sage: sT = PermutationGroup(['(1,8,5,3)','(4,3,5,6)'])
sage: sT.order()
120
sage: sT.is_isomorphic(P)
True

sage: rT = PermutationGroup(['(1,6,9,5)','(1,4,3,2)'])    
sage: rT.is_isomorphic(S7)
True

########## ch 14:



sage: r = lambda t: 11-t
sage: s = lambda t: min(2*t,23-2*t)
sage: R = [str((i+1,r(i)+1)) for i in range(12)]
sage: S = [str((i+1,s(i)+1)) for i in range(12)]
sage: print R
['(1, 12)', '(2, 11)', '(3, 10)', '(4, 9)', '(5, 8)', 
 '(6, 7)', '(7, 6)', '(8, 5)', '(9, 4)', '(10, 3)', 
 '(11, 2)', '(12, 1)']
sage: print S
['(1, 1)', '(2, 3)', '(3, 5)', '(4, 7)', '(5, 9)', 
 '(6, 11)', '(7, 12)', '(8, 10)', '(9, 8)', '(10, 6)', 
 '(11, 4)', '(12, 2)']
sage: g1 = G('(1, 12)(2, 11)(3, 10)(4, 9)(5, 8)(6, 7)')
sage: g2 = G('(2, 3, 5, 9, 8, 10, 6, 11, 4, 7, 12)')
sage: M = PermutationGroup([g1,g2])
sage: M.order()
95040
sage: MathieuGroup(12).order()
95040
sage: MathieuGroup(12).is_isomorphic(M)
True

sage: G = PGL(2,13)
sage: G._gap_().LargestMovedPoint()
14
sage: G._gap_().SmallerDegreePermutationRepresentation()
IdentityMapping( Group([ (3,14,13,12,11,10,9,8,7,6,5,4),
  (1,2,9)(3,8,10)(4,5,12)(6,13,14) ]) )

sage: Set([x^2 for x in GF(23)])
{0, 1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18}
sage: MathieuGroup(12).order()
95040
sage: MathieuGroup(24).order()
244823040

sage: C = HammingCode(3,GF(2))
sage: C
Linear code of length 7, dimension 4 over Finite Field of size 2
sage: C.gen_mat()
[1 1 1 0 0 0 0]
[1 0 0 1 1 0 0]
[0 1 0 1 0 1 0]
[1 1 0 1 0 0 1]
sage: C.check_mat()
[0 1 1 1 1 0 0]
[1 0 1 1 0 1 0]
[1 1 0 1 0 0 1]

sage: C = TernaryGolayCode(); C
Linear code of length 11, dimension 6 over Finite Field of size 3
sage: C = BinaryGolayCode(); C
Linear code of length 23, dimension 12 over Finite Field of size 2

sage: C = ExtendedBinaryGolayCode(); C
Linear code of length 24, dimension 12 over 
    Finite Field of size 2
sage: print C.weight_distribution()
[1, 0, 0, 0, 0, 0, 0, 0, 759, 0, 0, 0, 2576, 0, 0, 0, 759, 0, 
 0, 0, 0, 0, 0, 0, 1]
sage: c = C.random(); c
(0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 
 0, 1, 0, 1, 1, 1)
sage: hamming_weight(C.random())
12
sage: c in C
True
sage: d = c.nonzero_positions(); d
[2, 4, 8, 9, 11, 14, 15, 17, 19, 21, 22, 23]

sage: C = ExtendedBinaryGolayCode()
The weight  8  codewords of C form a t-(v,k,lambda) design, 
 where
      t =  5  , v =  24  , k =  8  , lambda =  1
      There are  759  blocks of this design.
The weight  12  codewords of C form a t-(v,k,lambda) design, 
 where
      t =  5  , v =  24  , k =  12  , lambda =  48
      There are  2576  blocks of this design.
The weight  16  codewords of C form a t-(v,k,lambda) design, 
 where
      t =  5  , v =  24  , k =  16  , lambda =  78
      There are  759  blocks of this design.
The weight  24  codewords of C form a t-(v,k,lambda) design, 
 where
      t =  5  , v =  24  , k =  24  , lambda =  1
      There are  1  blocks of this design.
The weight  8  codewords of C* form a t-(v,k,lambda) design, 
 where
      t =  5  , v =  24  , k =  8  , lambda =  1
      There are  759  blocks of this design.
The weight  12  codewords of C* form a t-(v,k,lambda) design, 
 where
      t =  5  , v =  24  , k =  12  , lambda =  48
      There are  2576  blocks of this design.
The weight  16  codewords of C* form a t-(v,k,lambda) design, 
 where
      t =  5  , v =  24  , k =  16  , lambda =  78
      There are  759  blocks of this design.

['weights from C: ',
 [8, 12, 16, 24],
 'designs from C: ',
 [[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)], 
  [5, (24, 24, 1)]],
 'weights from C*: ',
 [8, 12, 16],
 'designs from C*: ',
 [[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)]]]

sage: MS = MatrixSpace(QQ,12,12)
sage: A = MS(sage_eval(gap.eval("HadamardMat( 12 )")))
sage: A
[ 1  1  1  1  1  1  1  1  1  1  1  1]
[ 1 -1  1 -1  1  1  1 -1 -1 -1  1 -1]
[ 1 -1 -1  1 -1  1  1  1 -1 -1 -1  1]
[ 1  1 -1 -1  1 -1  1  1  1 -1 -1 -1]
[ 1 -1  1 -1 -1  1 -1  1  1  1 -1 -1]
[ 1 -1 -1  1 -1 -1  1 -1  1  1  1 -1]
[ 1 -1 -1 -1  1 -1 -1  1 -1  1  1  1]
[ 1  1 -1 -1 -1  1 -1 -1  1 -1  1  1]
[ 1  1  1 -1 -1 -1  1 -1 -1  1 -1  1]
[ 1  1  1  1 -1 -1 -1  1 -1 -1  1 -1]
[ 1 -1  1  1  1 -1 -1 -1  1 -1 -1  1]
[ 1  1 -1  1  1  1 -1 -1 -1  1 -1 -1]