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corrected bug in code and added new data
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def perm(p):
    X = list(range(1,p))
    s = Permutation([(1/X[i])%p for i in range(len(X))])
    return s

def tmat(p,q):
    G = PermutationGroup([perm(nth_prime(r)) for r in [p,q]])
    #print(G.order())
    M = matrix([[ (g*h.inverse()).order() for g in G] for h in G])
    return G,M

def printAttr(B):
    for d in dir(B):
        if d.startswith("__"):
            continue
        if d.startswith("is_"):
            try:
                if getattr(B,d)():
                    print(d,getattr(B,d)())
            except:
                pass
K=7
for M in range(1,K+1):
    for N in range(1,K+1):
        print(nth_prime(N),nth_prime(M))
        G,A=(tmat(nth_prime(N),nth_prime(M)))
        print((A))
        ev = sorted(A.eigenvalues())
        print(G.order(),ev)
        print([(g.order()) for g in G])
        plot(A).show()

p,q =  2 2
Matrix G_(p,q):
[1]
Order of dihedral group =  1 , Eigenvalues =  [1]
order of elements in group =  [1]
   
p,q =  3 2
Matrix G_(p,q):
[1 2]
[2 1][1]
Order of dihedral group =  21 , Eigenvalues =  [-1, 3][1]
order of elements in group =  [1, 2][1]
   
p,q =  5 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  7 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  11 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  13 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  17 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  219 32
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  323 32
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  2 3
Matrix G_(p,q):
[1]
Order of dihedral group =  1 , Eigenvalues =  [1]
order of elements in group =  [1]
   
p,q =  3 3
Matrix G_(p,q):
[1]
Order of dihedral group =  1 , Eigenvalues =  [1]
order of elements in group =  [1]
   
p,q =  5 3
Matrix G_(p,q):
[1 22]
[2 1]
Order of dihedral group =  2 4, Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  7 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  11 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 4, Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  13 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 43]
order of elements in group =  [1, 2]
   
p,q =  17 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  19 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  23 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  2 4]5
Matrix G_(p,q):
[1 2]
[2 41]
Order 1of dihedral group =  2 4, Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  3 5
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  5 5
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
[4   
p,q =  7 5
Matrix G_(p,q):
[1 2 2 12 2 4 24 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 42 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 24 2 42 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  711 35
Matrix G_(p,q):
[1 2 2 4 2 2 4 4 2]
[2 1 4 2 2 4 2 2 4]
[2 42 1 2 4 2 2 2]4]
[4[2 2 2 1 2 4 24 2]
[2 24 4 2 1 2 2 4]2]
[2[4 2 2 4 2 1 42 2]
[4 2 2 24 2 42 1 2]
[2 4 24 2 42 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  1113 35
Matrix G_(p,q):
[1 2 2 4 2 2 4 4 2]
[2 1 4 2 2 4 2 2 4]
[2 42 1 2 4 2 2 2]4]
[4[2 2 2 1 2 4 24 2]
[2 24 4 2 1 2 2 4]2]
[2[4 2 2 4 2 1 42 2]
[4 2 2 24 2 42 1 2]
[2 4 24 2 42 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  1317 35
Matrix G_(p,q):
[1 2 2 4 2 2 4 4 2]
[2 1 4 2 2 4 2 2 4]
[2 42 1 2 4 2 2 2]4]
[4[2 2 2 1 2 4 24 2]
[2 24 4 2 1 2 2 4]2]
[2[4 2 2 4 2 1 42 2]
[4 2 2 24 2 42 1 2]
[2 4 24 2 42 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  1719 35
Matrix G_(p,q):
[1 2 2 4 2 2 4 4 2]
[2 1 4 2 2 4 2 2 4]
[2 42 1 2 4 2 2 2]4]
[4[2 2 2 1 2 4 24 2]
[2 24 4 2 1 2 2 4]2]
[2[4 2 2 4 2 1 42 2]
[4 2 2 24 2 42 1 2]
[2 4 24 2 42 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  223 5
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  2 7
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  3 57
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  5 7
Matrix G_(p,q):
[1 2 42 2 2 4 4 2]
[2 1 4 2 2 4 2 2 4]
[2 42 1 2 4 2 2 2]4]
[4[2 2 2 1 2 4 24 2]
[2 24 4 2 1 2 2 4]2]
[2[4 2 2 4 2 1 42 2]
[4 2 2 24 2 42 1 2]
[2 4 24 2 42 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  57 57
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  711 57
Matrix G_(p,q):
[ 1  5  5  5  5  2  2  2  2  2  2  2  2  2  2  2 10 10 10 10]
[ 5  1  5  5  5  2  2  2  2  2  2  2  2  2  2 10 10 10 10  2]
[ 5  5  1  5  5  2  2  2  2  2  2  2  2  2  2 10 10 10  2 10]
[ 5  5  5  1  5  2  2  2  2  2  2  2  2  2  2 10 10  2 10 10]
[ 5  5  5  5  1  2  2  2  2  2  2  2  2  2  2 10  2 10 10 10]
[ 2  2  2  2  2  1  5  5  5  5  2 10 10 10 10  2  2  2  2  2]
[ 2  2  2  2  2  5  1  5  5  5 10 10 10 10  2  2  2  2  2  2]
[ 2  2  2  2  2  5  5  1  5  5 10 10 10  2 10  2  2  2  2  2]
[ 2  2  2  2  2  5  5  5  1  5 10 10  2 10 10  2  2  2  2  2]
[ 2  2  2  2  2  5  5  5  5  1 10  2 10 10 10  2  2  2  2  2]
[ 2  2  2  2  2  2 10 10 10 10  1  5  5  5  5  2  2  2  2  2]
[ 2  2  2  2  2 10 10 10 10  2  5  1  5  5  5  2  2  2  2  2]
[ 2  2  2  2  2 10 10 10  2 10  5  5  1  5  5  2  2  2  2  2]
[ 2  2  2  2  2 10 10  2 10 10  5  5  5  1  5  2  2  2  2  2]
[ 2  2  2  2  2 10  2 10 10 10  5  5  5  5  1  2  2  2  2  2]
[ 2 10 10 10 10  2  2  2  2  2  2  2  2  2  2  1  5  5  5  5]
[10 10 10 10  2  2  2  2  2  2  2  2  2  2  2  5  1  5  5  5]
[10 10 10  2 10  2  2  2  2  2  2  2  2  2  2  5  5  1  5  5]
[10 10  2 10 10  2  2  2  2  2  2  2  2  2  2  5  5  5  1  5]
[10  2 10 10 10  2  2  2  2  2  2  2  2  2  2  5  5  5  5  1]
[1 2 2 6 2 3 2 2 6 2 2 3]
[2 1 6 2 3 2 2 2 2 6 3 2]
[2 6 1 2 2 2 2 3 2 3 6 2]
[6 2 2 1 2 2 3 2 3 2 2 6]
[2 3 2 2 1 2 2 6 2 6 3 2]
[3 2 2 2 2 1 6 2 6 2 2 3]
[2 2 2 3 2 6 1 2 3 2 2 6]
[2 2 3 2 6 2 2 1 2 3 6 2]
[6 2 2 3 2 6 3 2 1 2 2 2]
[2 6 3 2 6 2 2 3 2 1 2 2]
[2 3 6 2 3 2 2 6 2 2 1 2]
[3 2 2 6 2 3 6 2 2 2 2 1]
Order of dihedral group =  12 , Eigenvalues =  [-7, -7, -6, -6, -6, -6, 2, 2, 2, 2, 9, 33]
order of elements in group =  [1, 3, 3, 2, 2, 2, 2, 2, 2, 6, 6, 2]
   
p,q =  13 7
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  17 7
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  19 7
Matrix G_(p,q):
[1 2 2 2 2 3 2 6 6 2 3 2]
[2 1 2 2 3 2 6 2 2 6 2 3]
[2 2 1 2 2 6 2 3 3 2 6 2]
[2 2 2 1 6 2 3 2 2 3 2 6]
[2 3 2 6 1 2 2 2 2 6 2 3]
[3 2 6 2 2 1 2 2 6 2 3 2]
[2 6 2 3 2 2 1 2 2 3 2 6]
[6 2 3 2 2 2 2 1 3 2 6 2]
[6 2 3 2 2 6 2 3 1 2 2 2]
[2 6 2 3 6 2 3 2 2 1 2 2]
[3 2 6 2 2 3 2 6 2 2 1 2]
[2 3 2 6 3 2 6 2 2 2 2 1]
Order of dihedral group =  12 , Eigenvalues =  [-7, -7, -6, -6, -6, -6, 2, 2, 2, 2, 9, 33]
order of elements in group =  [1, 3, 3, 2, 2, 2, 2, 2, 2, 6, 6, 2]
   
p,q =  23 7
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  2 11
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  3 11
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  5 11
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  7 11
Matrix G_(p,q):
[1 2 2 6 2 3 2 2 6 2 2 3]
[2 1 6 2 3 2 2 2 2 6 3 2]
[2 6 1 2 2 2 2 3 2 3 6 2]
[6 2 2 1 2 2 3 2 3 2 2 6]
[2 3 2 2 1 2 2 6 2 6 3 2]
[3 2 2 2 2 1 6 2 6 2 2 3]
[2 2 2 3 2 6 1 2 3 2 2 6]
[2 2 3 2 6 2 2 1 2 3 6 2]
[6 2 2 3 2 6 3 2 1 2 2 2]
[2 6 3 2 6 2 2 3 2 1 2 2]
[2 3 6 2 3 2 2 6 2 2 1 2]
[3 2 2 6 2 3 6 2 2 2 2 1]
Order of dihedral group =  12 , Eigenvalues =  [-7, -7, -6, -6, -6, -6, 2, 2, 2, 2, 9, 33]
order of elements in group =  [1, 3, 3, 2, 2, 2, 2, 2, 2, 6, 6, 2]
   
p,q =  11 11
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  13 11
Matrix G_(p,q):
[ 1  2  2  2  2 10  2  5  2  5 10  2  2 10  5  2  5  2  2 10]
[ 2  1  2  2 10  2  5  2  5  2  2 10 10  2  2  5  2  5 10  2]
[ 2  2  1  2  2  5  2 10  2 10  5  2  2  5 10  2 10  2  2  5]
[ 2  2  2  1  5  2 10  2 10  2  2  5  5  2  2 10  2 10  5  2]
[ 2 10  2  5  1  2 10  2  2  2  2  5  5  2  2 10  2 10  5  2]
[10  2  5  2  2  1  2 10  2  2  5  2  2  5 10  2 10  2  2  5]
[ 2  5  2 10 10  2  1  2  5  2  2 10  2  2  2  5  2  5 10  2]
[ 5  2 10  2  2 10  2  1  2  5 10  2  2  2  5  2  5  2  2 10]
[ 2  5  2 10  2  2  5  2  1  2  2 10 10  2  2  5  2  5 10  2]
[ 5  2 10  2  2  2  2  5  2  1 10  2  2 10  5  2  5  2  2 10]
[10  2  5  2  2  5  2 10  2 10  1  2  2  5  2  2 10  2  2  5]
[ 2 10  2  5  5  2 10  2 10  2  2  1  5  2  2  2  2 10  5  2]
[ 2 10  2  5  5  2  2  2 10  2  2  5  1  2  2 10  2 10  5  2]
[10  2  5  2  2  5  2  2  2 10  5  2  2  1 10  2 10  2  2  5]
[ 5  2 10  2  2 10  2  5  2  5  2  2  2 10  1  2  5  2  2 10]
[ 2  5  2 10 10  2  5  2  5  2  2  2 10  2  2  1  2  5 10  2]
[ 5  2 10  2  2 10  2  5  2  5 10  2  2 10  5  2  1  2  2  2]
[ 2  5  2 10 10  2  5  2  5  2  2 10 10  2  2  5  2  1  2  2]
[ 2 10  2  5  5  2 10  2 10  2  2  5  5  2  2 10  2  2  1  2]
[10  2  5  2  2  5  2 10  2 10  5  2  2  5 10  2  2  2  2  1]
Order of dihedral group =  20 , Eigenvalues =  [-21, -21, -12, -12, -12, -12, -12, -12, -12, -12, 4, 4, 4, 4, 4, 4, 4, 4, 43, 83]
order of elements in group =  [1, 5, 5, 5, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 2, 10, 10]
   
p,q =  17 11
Matrix G_(p,q):
[ 1  2  2 10 10  2 10  2  2  5  2  5  5  2  2  2  2  5  2 10]
[ 2  1 10  2  2 10  2 10  5  2  5  2  2  5  2  2  5  2 10  2]
[ 2 10  1  2  2  5  2  5 10  2  2  2  2 10  5  2 10  2  5  2]
[10  2  2  1  5  2  5  2  2 10  2  2 10  2  2  5  2 10  2  5]
[10  2  2  5  1  2  5  2  2 10  2 10  2  2  2  5  2 10  2  5]
[ 2 10  5  2  2  1  2  5 10  2 10  2  2  2  5  2 10  2  5  2]
[10  2  2  5  5  2  1  2  2  2  2 10 10  2  2  5  2 10  2  5]
[ 2 10  5  2  2  5  2  1  2  2 10  2  2 10  5  2 10  2  5  2]
[ 2  5 10  2  2 10  2  2  1  2  5  2  2  5 10  2  5  2 10  2]
[ 5  2  2 10 10  2  2  2  2  1  2  5  5  2  2 10  2  5  2 10]
[ 2  5  2  2  2 10  2 10  5  2  1  2  2  5 10  2  5  2 10  2]
[ 5  2  2  2 10  2 10  2  2  5  2  1  5  2  2 10  2  5  2 10]
[ 5  2  2 10  2  2 10  2  2  5  2  5  1  2  2 10  2  5  2 10]
[ 2  5 10  2  2  2  2 10  5  2  5  2  2  1 10  2  5  2 10  2]
[ 2  2  5  2  2  5  2  5 10  2 10  2  2 10  1  2 10  2  5  2]
[ 2  2  2  5  5  2  5  2  2 10  2 10 10  2  2  1  2 10  2  5]
[ 2  5 10  2  2 10  2 10  5  2  5  2  2  5 10  2  1  2  2  2]
[ 5  2  2 10 10  2 10  2  2  5  2  5  5  2  2 10  2  1  2  2]
[ 2 10  5  2  2  5  2  5 10  2 10  2  2 10  5  2  2  2  1  2]
[10  2  2  5  5  2  5  2  2 10  2 10 10  2  2  5  2  2  2  1]
Order of dihedral group =  20 , Eigenvalues =  [-21, -21, -12, -12, -12, -12, -12, -12, -12, -12, 4, 4, 4, 4, 4, 4, 4, 4, 43, 83]
order of elements in group =  [1, 5, 5, 5, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 10, 10]
   
p,q =  19 11
Matrix G_(p,q):
[ 1  2  2  3  3  2  2  2  2  6  6  2  2 12  4  2  2 12  2 12  4  2  2 12]
[ 2  1  3  2  2  3  2  2  6  2  2  6 12  2  2  4 12  2 12  2  2  4 12  2]
[ 2  3  1  2  2  3  2  6  2  2  2  6  4  2  2 12 12  2  4  2  2 12 12  2]
[ 3  2  2  1  3  2  6  2  2  2  6  2  2  4 12  2  2 12  2  4 12  2  2 12]
[ 3  2  2  3  1  2  6  2  2  6  2  2  2 12 12  2  2  4  2 12 12  2  2  4]
[ 2  3  3  2  2  1  2  6  6  2  2  2 12  2  2 12  4  2 12  2  2 12  4  2]
[ 2  2  2  6  6  2  1  2  2  3  3  2  2 12  4  2  2 12  2 12  4  2  2 12]
[ 2  2  6  2  2  6  2  1  3  2  2  3 12  2  2  4 12  2 12  2  2  4 12  2]
[ 2  6  2  2  2  6  2  3  1  2  2  3  4  2  2 12 12  2  4  2  2 12 12  2]
[ 6  2  2  2  6  2  3  2  2  1  3  2  2  4 12  2  2 12  2  4 12  2  2 12]
[ 6  2  2  6  2  2  3  2  2  3  1  2  2 12 12  2  2  4  2 12 12  2  2  4]
[ 2  6  6  2  2  2  2  3  3  2  2  1 12  2  2 12  4  2 12  2  2 12  4  2]
[ 2 12  4  2  2 12  2 12  4  2  2 12  1  2  2  3  3  2  2  2  2  6  6  2]
[12  2  2  4 12  2 12  2  2  4 12  2  2  1  3  2  2  3  2  2  6  2  2  6]
[ 4  2  2 12 12  2  4  2  2 12 12  2  2  3  1  2  2  3  2  6  2  2  2  6]
[ 2  4 12  2  2 12  2  4 12  2  2 12  3  2  2  1  3  2  6  2  2  2  6  2]
[ 2 12 12  2  2  4  2 12 12  2  2  4  3  2  2  3  1  2  6  2  2  6  2  2]
[12  2  2 12  4  2 12  2  2 12  4  2  2  3  3  2  2  1  2  6  6  2  2  2]
[ 2 12  4  2  2 12  2 12  4  2  2 12  2  2  2  6  6  2  1  2  2  3  3  2]
[12  2  2  4 12  2 12  2  2  4 12  2  2  2  6  2  2  6  2  1  3  2  2  3]
[ 4  2  2 12 12  2  4  2  2 12 12  2  2  6  2  2  2  6  2  3  1  2  2  3]
[ 2  4 12  2  2 12  2  4 12  2  2 12  6  2  2  2  6  2  3  2  2  1  3  2]
[ 2 12 12  2  2  4  2 12 12  2  2  4  6  2  2  6  2  2  3  2  2  3  1  2]
[12  2  2 12  4  2 12  2  2 12  4  2  2  6  6  2  2  2  2  3  3  2  2  1]
Order of dihedral group =  24 , Eigenvalues =  [-35, -35, -22, -22, -22, -22, -7, -7, -7, -7, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 10, 10, 53, 101]
order of elements in group =  [1, 3, 3, 2, 2, 2, 2, 2, 2, 12, 12, 4, 2, 2, 2, 12, 12, 4, 2, 6, 6, 2, 2, 2]
   
p,q =  23 11
Matrix G_(p,q):
[1 2 2 8 8 2 2 4 2 2 2 4 8 2 8 2]
[2 1 8 2 2 8 4 2 2 2 4 2 2 8 2 8]
[2 8 1 2 2 2 8 2 2 8 8 2 2 4 2 4]
[8 2 2 1 2 2 2 8 8 2 2 8 4 2 4 2]
[8 2 2 2 1 2 2 8 8 2 2 8 4 2 4 2]
[2 8 2 2 2 1 8 2 2 8 8 2 2 4 2 4]
[2 4 8 2 2 8 1 2 2 4 2 2 2 8 2 8]
[4 2 2 8 8 2 2 1 4 2 2 2 8 2 8 2]
[2 2 2 8 8 2 2 4 1 2 2 4 8 2 8 2]
[2 2 8 2 2 8 4 2 2 1 4 2 2 8 2 8]
[2 4 8 2 2 8 2 2 2 4 1 2 2 8 2 8]
[4 2 2 8 8 2 2 2 4 2 2 1 8 2 8 2]
[8 2 2 4 4 2 2 8 8 2 2 8 1 2 2 2]
[2 8 4 2 2 4 8 2 2 8 8 2 2 1 2 2]
[8 2 2 4 4 2 2 8 8 2 2 8 2 2 1 2]
[2 8 4 2 2 4 8 2 2 8 8 2 2 2 2 1]
Order of dihedral group =  16 , Eigenvalues =  [-21, -21, -5, -5, -5, -5, -1, -1, -1, -1, -1, -1, -1, -1, 27, 59]
order of elements in group =  [1, 2, 2, 2, 2, 2, 8, 8, 2, 2, 4, 4, 8, 8, 2, 2]
   
p,q =  2 13
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  3 13
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  5 13
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  7 13
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  11 13
Matrix G_(p,q):
[ 1  2  2  2  2 10  2  5  2  5 10  2  2 10  5  2  5  2  2 10]
[ 2  1  2  2 10  2  5  2  5  2  2 10 10  2  2  5  2  5 10  2]
[ 2  2  1  2  2  5  2 10  2 10  5  2  2  5 10  2 10  2  2  5]
[ 2  2  2  1  5  2 10  2 10  2  2  5  5  2  2 10  2 10  5  2]
[ 2 10  2  5  1  2 10  2  2  2  2  5  5  2  2 10  2 10  5  2]
[10  2  5  2  2  1  2 10  2  2  5  2  2  5 10  2 10  2  2  5]
[ 2  5  2 10 10  2  1  2  5  2  2 10  2  2  2  5  2  5 10  2]
[ 5  2 10  2  2 10  2  1  2  5 10  2  2  2  5  2  5  2  2 10]
[ 2  5  2 10  2  2  5  2  1  2  2 10 10  2  2  5  2  5 10  2]
[ 5  2 10  2  2  2  2  5  2  1 10  2  2 10  5  2  5  2  2 10]
[10  2  5  2  2  5  2 10  2 10  1  2  2  5  2  2 10  2  2  5]
[ 2 10  2  5  5  2 10  2 10  2  2  1  5  2  2  2  2 10  5  2]
[ 2 10  2  5  5  2  2  2 10  2  2  5  1  2  2 10  2 10  5  2]
[10  2  5  2  2  5  2  2  2 10  5  2  2  1 10  2 10  2  2  5]
[ 5  2 10  2  2 10  2  5  2  5  2  2  2 10  1  2  5  2  2 10]
[ 2  5  2 10 10  2  5  2  5  2  2  2 10  2  2  1  2  5 10  2]
[ 5  2 10  2  2 10  2  5  2  5 10  2  2 10  5  2  1  2  2  2]
[ 2  5  2 10 10  2  5  2  5  2  2 10 10  2  2  5  2  1  2  2]
[ 2 10  2  5  5  2 10  2 10  2  2  5  5  2  2 10  2  2  1  2]
[10  2  5  2  2  5  2 10  2 10  5  2  2  5 10  2  2  2  2  1]
Order of dihedral group =  20 , Eigenvalues =  [-21, -21, -12, -12, -12, -12, -12, -12, -12, -12, 4, 4, 4, 4, 4, 4, 4, 4, 43, 83]
order of elements in group =  [1, 5, 5, 5, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 2, 10, 10]
 

def perm(p):
    X = list(range(1,p))
    s = Permutation([(1/X[i])%p for i in range(len(X))])
    return s

def tmat(p,q):
    G = PermutationGroup([perm(nth_prime(r)) for r in [p,q]])
    #print(G.order())
    M = matrix([[ (g*h.inverse()).order() for g in G] for h in G])
    return G,M

def printAttr(B):
    for d in dir(B):
        if d.startswith("__"):
            continue
        if d.startswith("is_"):
            try:
                if getattr(B,d)():
                    print(d,getattr(B,d)())
            except:
                pass
K=7
for M in range(1,K+1):
    for N in range(1,K+1):
        print(nth_prime(N),nth_prime(M))
        G,A=(tmat(nth_prime(N),nth_prime(M)))
        print((A))
        ev = sorted(A.eigenvalues())
        print(G.order(),ev)
        print([(g.order()) for g in G])
        plot(A).show()

p,q =  2 2
Matrix G_(p,q):
[1]
Order of dihedral group =  1 , Eigenvalues =  [1]
order of elements in group =  [1]
   
p,q =  3 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  5 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  7 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  11 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  13 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  17 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  2 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  3 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  5 3
Matrix G_(p,q):
[1 2 2 4 2 2 4 2]
[2 1 4 2 2 2 2 4]
[2 4 1 2 4 2 2 2]
[4 2 2 1 2 4 2 2]
[2 2 4 2 1 2 2 4]
[2 2 2 4 2 1 4 2]
[4 2 2 2 2 4 1 2]
[2 4 2 2 4 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  7 3
Matrix G_(p,q):
[1 2 2 4 2 2 4 2]
[2 1 4 2 2 2 2 4]
[2 4 1 2 4 2 2 2]
[4 2 2 1 2 4 2 2]
[2 2 4 2 1 2 2 4]
[2 2 2 4 2 1 4 2]
[4 2 2 2 2 4 1 2]
[2 4 2 2 4 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  11 3
Matrix G_(p,q):
[1 2 2 4 2 2 4 2]
[2 1 4 2 2 2 2 4]
[2 4 1 2 4 2 2 2]
[4 2 2 1 2 4 2 2]
[2 2 4 2 1 2 2 4]
[2 2 2 4 2 1 4 2]
[4 2 2 2 2 4 1 2]
[2 4 2 2 4 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  13 3
Matrix G_(p,q):
[1 2 2 4 2 2 4 2]
[2 1 4 2 2 2 2 4]
[2 4 1 2 4 2 2 2]
[4 2 2 1 2 4 2 2]
[2 2 4 2 1 2 2 4]
[2 2 2 4 2 1 4 2]
[4 2 2 2 2 4 1 2]
[2 4 2 2 4 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  17 3
Matrix G_(p,q):
[1 2 2 4 2 2 4 2]
[2 1 4 2 2 2 2 4]
[2 4 1 2 4 2 2 2]
[4 2 2 1 2 4 2 2]
[2 2 4 2 1 2 2 4]
[2 2 2 4 2 1 4 2]
[4 2 2 2 2 4 1 2]
[2 4 2 2 4 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  2 5
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  3 5
Matrix G_(p,q):
[1 2 2 4 2 2 4 2]
[2 1 4 2 2 2 2 4]
[2 4 1 2 4 2 2 2]
[4 2 2 1 2 4 2 2]
[2 2 4 2 1 2 2 4]
[2 2 2 4 2 1 4 2]
[4 2 2 2 2 4 1 2]
[2 4 2 2 4 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  5 5
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  7 5
Matrix G_(p,q):
[ 1  5  5  5  5  2  2  2  2  2  2  2  2  2  2  2 10 10 10 10]
[ 5  1  5  5  5  2  2  2  2  2  2  2  2  2  2 10 10 10 10  2]
[ 5  5  1  5  5  2  2  2  2  2  2  2  2  2  2 10 10 10  2 10]
[ 5  5  5  1  5  2  2  2  2  2  2  2  2  2  2 10 10  2 10 10]
[ 5  5  5  5  1  2  2  2  2  2  2  2  2  2  2 10  2 10 10 10]
[ 2  2  2  2  2  1  5  5  5  5  2 10 10 10 10  2  2  2  2  2]
[ 2  2  2  2  2  5  1  5  5  5 10 10 10 10  2  2  2  2  2  2]
[ 2  2  2  2  2  5  5  1  5  5 10 10 10  2 10  2  2  2  2  2]
[ 2  2  2  2  2  5  5  5  1  5 10 10  2 10 10  2  2  2  2  2]
[ 2  2  2  2  2  5  5  5  5  1 10  2 10 10 10  2  2  2  2  2]
[ 2  2  2  2  2  2 10 10 10 10  1  5  5  5  5  2  2  2  2  2]
[ 2  2  2  2  2 10 10 10 10  2  5  1  5  5  5  2  2  2  2  2]
[ 2  2  2  2  2 10 10 10  2 10  5  5  1  5  5  2  2  2  2  2]
[ 2  2  2  2  2 10 10  2 10 10  5  5  5  1  5  2  2  2  2  2]
[ 2  2  2  2  2 10  2 10 10 10  5  5  5  5  1  2  2  2  2  2]
[ 2 10 10 10 10  2  2  2  2  2  2  2  2  2  2  1  5  5  5  5]
[10 10 10 10  2  2  2  2  2  2  2  2  2  2  2  5  1  5  5  5]
[10 10 10  2 10  2  2  2  2  2  2  2  2  2  2  5  5  1  5  5]
[10 10  2 10 10  2  2  2  2  2  2  2  2  2  2  5  5  5  1  5]
[10  2 10 10 10  2  2  2  2  2  2  2  2  2  2  5  5  5  5  1]
Order of dihedral group =  20 , Eigenvalues =  [-21, -21, -12, -12, -12, -12, -12, -12, -12, -12, 4, 4, 4, 4, 4, 4, 4, 4, 43, 83]
order of elements in group =  [1, 5, 5, 5, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 10, 10]
 

def perm(p):
    X = list(range(1,p))
    s = Permutation([(1/X[i])%p for i in range(len(X))])
    return s

def tmat(p,q):
    G = PermutationGroup([perm(r) for r in [p,q]])
    #print(G.order())
    M = matrix([[ (g*h.inverse()).order() for g in G] for h in G])
    return G,M

def printAttr(B):
    for d in dir(B):
        if d.startswith("__"):
            continue
        if d.startswith("is_"):
            try:
                if getattr(B,d)():
                    print(d,getattr(B,d)())
            except:
                pass
K=7
for M in range(1,K+1):
    for N in range(1,K+1):
        print(nth_prime(N),nth_prime(M))
        G,A=(tmat(nth_prime(N),nth_prime(M)))
        print((A))
        ev = sorted(A.eigenvalues())
        print(G.order(),ev)
        print([(g.order()) for g in G])
        plot(A).show()

p,q =  2 2
Matrix G_(p,q):
[1]
Order of dihedral group =  1 , Eigenvalues =  [1]
order of elements in group =  [1]
   
p,q =  3 2
Matrix G_(p,q):
[1]
Order of dihedral group =  1 , Eigenvalues =  [1]
order of elements in group =  [1]
   
p,q =  5 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  7 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  11 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  13 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  17 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  19 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  23 2
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  2 3
Matrix G_(p,q):
[1]
Order of dihedral group =  1 , Eigenvalues =  [1]
order of elements in group =  [1]
   
p,q =  3 3
Matrix G_(p,q):
[1]
Order of dihedral group =  1 , Eigenvalues =  [1]
order of elements in group =  [1]
   
p,q =  5 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  7 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  11 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  13 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  17 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  19 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  23 3
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  2 5
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  3 5
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  5 5
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  7 5
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  11 5
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  13 5
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  17 5
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  19 5
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  23 5
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  2 7
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  3 7
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  5 7
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  7 7
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  11 7
Matrix G_(p,q):
[1 2 2 6 2 3 2 2 6 2 2 3]
[2 1 6 2 3 2 2 2 2 6 3 2]
[2 6 1 2 2 2 2 3 2 3 6 2]
[6 2 2 1 2 2 3 2 3 2 2 6]
[2 3 2 2 1 2 2 6 2 6 3 2]
[3 2 2 2 2 1 6 2 6 2 2 3]
[2 2 2 3 2 6 1 2 3 2 2 6]
[2 2 3 2 6 2 2 1 2 3 6 2]
[6 2 2 3 2 6 3 2 1 2 2 2]
[2 6 3 2 6 2 2 3 2 1 2 2]
[2 3 6 2 3 2 2 6 2 2 1 2]
[3 2 2 6 2 3 6 2 2 2 2 1]
Order of dihedral group =  12 , Eigenvalues =  [-7, -7, -6, -6, -6, -6, 2, 2, 2, 2, 9, 33]
order of elements in group =  [1, 3, 3, 2, 2, 2, 2, 2, 2, 6, 6, 2]
   
p,q =  13 7
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  17 7
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  19 7
Matrix G_(p,q):
[1 2 2 2 2 3 2 6 6 2 3 2]
[2 1 2 2 3 2 6 2 2 6 2 3]
[2 2 1 2 2 6 2 3 3 2 6 2]
[2 2 2 1 6 2 3 2 2 3 2 6]
[2 3 2 6 1 2 2 2 2 6 2 3]
[3 2 6 2 2 1 2 2 6 2 3 2]
[2 6 2 3 2 2 1 2 2 3 2 6]
[6 2 3 2 2 2 2 1 3 2 6 2]
[6 2 3 2 2 6 2 3 1 2 2 2]
[2 6 2 3 6 2 3 2 2 1 2 2]
[3 2 6 2 2 3 2 6 2 2 1 2]
[2 3 2 6 3 2 6 2 2 2 2 1]
Order of dihedral group =  12 , Eigenvalues =  [-7, -7, -6, -6, -6, -6, 2, 2, 2, 2, 9, 33]
order of elements in group =  [1, 3, 3, 2, 2, 2, 2, 2, 2, 6, 6, 2]
   
p,q =  23 7
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  2 11
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  3 11
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  5 11
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  7 11
Matrix G_(p,q):
[1 2 2 6 2 3 2 2 6 2 2 3]
[2 1 6 2 3 2 2 2 2 6 3 2]
[2 6 1 2 2 2 2 3 2 3 6 2]
[6 2 2 1 2 2 3 2 3 2 2 6]
[2 3 2 2 1 2 2 6 2 6 3 2]
[3 2 2 2 2 1 6 2 6 2 2 3]
[2 2 2 3 2 6 1 2 3 2 2 6]
[2 2 3 2 6 2 2 1 2 3 6 2]
[6 2 2 3 2 6 3 2 1 2 2 2]
[2 6 3 2 6 2 2 3 2 1 2 2]
[2 3 6 2 3 2 2 6 2 2 1 2]
[3 2 2 6 2 3 6 2 2 2 2 1]
Order of dihedral group =  12 , Eigenvalues =  [-7, -7, -6, -6, -6, -6, 2, 2, 2, 2, 9, 33]
order of elements in group =  [1, 3, 3, 2, 2, 2, 2, 2, 2, 6, 6, 2]
   
p,q =  11 11
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  13 11
Matrix G_(p,q):
[ 1  2  2  2  2 10  2  5  2  5 10  2  2 10  5  2  5  2  2 10]
[ 2  1  2  2 10  2  5  2  5  2  2 10 10  2  2  5  2  5 10  2]
[ 2  2  1  2  2  5  2 10  2 10  5  2  2  5 10  2 10  2  2  5]
[ 2  2  2  1  5  2 10  2 10  2  2  5  5  2  2 10  2 10  5  2]
[ 2 10  2  5  1  2 10  2  2  2  2  5  5  2  2 10  2 10  5  2]
[10  2  5  2  2  1  2 10  2  2  5  2  2  5 10  2 10  2  2  5]
[ 2  5  2 10 10  2  1  2  5  2  2 10  2  2  2  5  2  5 10  2]
[ 5  2 10  2  2 10  2  1  2  5 10  2  2  2  5  2  5  2  2 10]
[ 2  5  2 10  2  2  5  2  1  2  2 10 10  2  2  5  2  5 10  2]
[ 5  2 10  2  2  2  2  5  2  1 10  2  2 10  5  2  5  2  2 10]
[10  2  5  2  2  5  2 10  2 10  1  2  2  5  2  2 10  2  2  5]
[ 2 10  2  5  5  2 10  2 10  2  2  1  5  2  2  2  2 10  5  2]
[ 2 10  2  5  5  2  2  2 10  2  2  5  1  2  2 10  2 10  5  2]
[10  2  5  2  2  5  2  2  2 10  5  2  2  1 10  2 10  2  2  5]
[ 5  2 10  2  2 10  2  5  2  5  2  2  2 10  1  2  5  2  2 10]
[ 2  5  2 10 10  2  5  2  5  2  2  2 10  2  2  1  2  5 10  2]
[ 5  2 10  2  2 10  2  5  2  5 10  2  2 10  5  2  1  2  2  2]
[ 2  5  2 10 10  2  5  2  5  2  2 10 10  2  2  5  2  1  2  2]
[ 2 10  2  5  5  2 10  2 10  2  2  5  5  2  2 10  2  2  1  2]
[10  2  5  2  2  5  2 10  2 10  5  2  2  5 10  2  2  2  2  1]
Order of dihedral group =  20 , Eigenvalues =  [-21, -21, -12, -12, -12, -12, -12, -12, -12, -12, 4, 4, 4, 4, 4, 4, 4, 4, 43, 83]
order of elements in group =  [1, 5, 5, 5, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 2, 10, 10]
   
p,q =  17 11
Matrix G_(p,q):
[ 1  2  2 10 10  2 10  2  2  5  2  5  5  2  2  2  2  5  2 10]
[ 2  1 10  2  2 10  2 10  5  2  5  2  2  5  2  2  5  2 10  2]
[ 2 10  1  2  2  5  2  5 10  2  2  2  2 10  5  2 10  2  5  2]
[10  2  2  1  5  2  5  2  2 10  2  2 10  2  2  5  2 10  2  5]
[10  2  2  5  1  2  5  2  2 10  2 10  2  2  2  5  2 10  2  5]
[ 2 10  5  2  2  1  2  5 10  2 10  2  2  2  5  2 10  2  5  2]
[10  2  2  5  5  2  1  2  2  2  2 10 10  2  2  5  2 10  2  5]
[ 2 10  5  2  2  5  2  1  2  2 10  2  2 10  5  2 10  2  5  2]
[ 2  5 10  2  2 10  2  2  1  2  5  2  2  5 10  2  5  2 10  2]
[ 5  2  2 10 10  2  2  2  2  1  2  5  5  2  2 10  2  5  2 10]
[ 2  5  2  2  2 10  2 10  5  2  1  2  2  5 10  2  5  2 10  2]
[ 5  2  2  2 10  2 10  2  2  5  2  1  5  2  2 10  2  5  2 10]
[ 5  2  2 10  2  2 10  2  2  5  2  5  1  2  2 10  2  5  2 10]
[ 2  5 10  2  2  2  2 10  5  2  5  2  2  1 10  2  5  2 10  2]
[ 2  2  5  2  2  5  2  5 10  2 10  2  2 10  1  2 10  2  5  2]
[ 2  2  2  5  5  2  5  2  2 10  2 10 10  2  2  1  2 10  2  5]
[ 2  5 10  2  2 10  2 10  5  2  5  2  2  5 10  2  1  2  2  2]
[ 5  2  2 10 10  2 10  2  2  5  2  5  5  2  2 10  2  1  2  2]
[ 2 10  5  2  2  5  2  5 10  2 10  2  2 10  5  2  2  2  1  2]
[10  2  2  5  5  2  5  2  2 10  2 10 10  2  2  5  2  2  2  1]
Order of dihedral group =  20 , Eigenvalues =  [-21, -21, -12, -12, -12, -12, -12, -12, -12, -12, 4, 4, 4, 4, 4, 4, 4, 4, 43, 83]
order of elements in group =  [1, 5, 5, 5, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 10, 10]
   
p,q =  19 11
Matrix G_(p,q):
[ 1  2  2  3  3  2  2  2  2  6  6  2  2 12  4  2  2 12  2 12  4  2  2 12]
[ 2  1  3  2  2  3  2  2  6  2  2  6 12  2  2  4 12  2 12  2  2  4 12  2]
[ 2  3  1  2  2  3  2  6  2  2  2  6  4  2  2 12 12  2  4  2  2 12 12  2]
[ 3  2  2  1  3  2  6  2  2  2  6  2  2  4 12  2  2 12  2  4 12  2  2 12]
[ 3  2  2  3  1  2  6  2  2  6  2  2  2 12 12  2  2  4  2 12 12  2  2  4]
[ 2  3  3  2  2  1  2  6  6  2  2  2 12  2  2 12  4  2 12  2  2 12  4  2]
[ 2  2  2  6  6  2  1  2  2  3  3  2  2 12  4  2  2 12  2 12  4  2  2 12]
[ 2  2  6  2  2  6  2  1  3  2  2  3 12  2  2  4 12  2 12  2  2  4 12  2]
[ 2  6  2  2  2  6  2  3  1  2  2  3  4  2  2 12 12  2  4  2  2 12 12  2]
[ 6  2  2  2  6  2  3  2  2  1  3  2  2  4 12  2  2 12  2  4 12  2  2 12]
[ 6  2  2  6  2  2  3  2  2  3  1  2  2 12 12  2  2  4  2 12 12  2  2  4]
[ 2  6  6  2  2  2  2  3  3  2  2  1 12  2  2 12  4  2 12  2  2 12  4  2]
[ 2 12  4  2  2 12  2 12  4  2  2 12  1  2  2  3  3  2  2  2  2  6  6  2]
[12  2  2  4 12  2 12  2  2  4 12  2  2  1  3  2  2  3  2  2  6  2  2  6]
[ 4  2  2 12 12  2  4  2  2 12 12  2  2  3  1  2  2  3  2  6  2  2  2  6]
[ 2  4 12  2  2 12  2  4 12  2  2 12  3  2  2  1  3  2  6  2  2  2  6  2]
[ 2 12 12  2  2  4  2 12 12  2  2  4  3  2  2  3  1  2  6  2  2  6  2  2]
[12  2  2 12  4  2 12  2  2 12  4  2  2  3  3  2  2  1  2  6  6  2  2  2]
[ 2 12  4  2  2 12  2 12  4  2  2 12  2  2  2  6  6  2  1  2  2  3  3  2]
[12  2  2  4 12  2 12  2  2  4 12  2  2  2  6  2  2  6  2  1  3  2  2  3]
[ 4  2  2 12 12  2  4  2  2 12 12  2  2  6  2  2  2  6  2  3  1  2  2  3]
[ 2  4 12  2  2 12  2  4 12  2  2 12  6  2  2  2  6  2  3  2  2  1  3  2]
[ 2 12 12  2  2  4  2 12 12  2  2  4  6  2  2  6  2  2  3  2  2  3  1  2]
[12  2  2 12  4  2 12  2  2 12  4  2  2  6  6  2  2  2  2  3  3  2  2  1]
Order of dihedral group =  24 , Eigenvalues =  [-35, -35, -22, -22, -22, -22, -7, -7, -7, -7, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 10, 10, 53, 101]
order of elements in group =  [1, 3, 3, 2, 2, 2, 2, 2, 2, 12, 12, 4, 2, 2, 2, 12, 12, 4, 2, 6, 6, 2, 2, 2]
   
p,q =  23 11
Matrix G_(p,q):
[1 2 2 8 8 2 2 4 2 2 2 4 8 2 8 2]
[2 1 8 2 2 8 4 2 2 2 4 2 2 8 2 8]
[2 8 1 2 2 2 8 2 2 8 8 2 2 4 2 4]
[8 2 2 1 2 2 2 8 8 2 2 8 4 2 4 2]
[8 2 2 2 1 2 2 8 8 2 2 8 4 2 4 2]
[2 8 2 2 2 1 8 2 2 8 8 2 2 4 2 4]
[2 4 8 2 2 8 1 2 2 4 2 2 2 8 2 8]
[4 2 2 8 8 2 2 1 4 2 2 2 8 2 8 2]
[2 2 2 8 8 2 2 4 1 2 2 4 8 2 8 2]
[2 2 8 2 2 8 4 2 2 1 4 2 2 8 2 8]
[2 4 8 2 2 8 2 2 2 4 1 2 2 8 2 8]
[4 2 2 8 8 2 2 2 4 2 2 1 8 2 8 2]
[8 2 2 4 4 2 2 8 8 2 2 8 1 2 2 2]
[2 8 4 2 2 4 8 2 2 8 8 2 2 1 2 2]
[8 2 2 4 4 2 2 8 8 2 2 8 2 2 1 2]
[2 8 4 2 2 4 8 2 2 8 8 2 2 2 2 1]
Order of dihedral group =  16 , Eigenvalues =  [-21, -21, -5, -5, -5, -5, -1, -1, -1, -1, -1, -1, -1, -1, 27, 59]
order of elements in group =  [1, 2, 2, 2, 2, 2, 8, 8, 2, 2, 4, 4, 8, 8, 2, 2]
   
p,q =  2 13
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  3 13
Matrix G_(p,q):
[1 2]
[2 1]
Order of dihedral group =  2 , Eigenvalues =  [-1, 3]
order of elements in group =  [1, 2]
   
p,q =  5 13
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  7 13
Matrix G_(p,q):
[1 2 2 2 2 4 4 2]
[2 1 2 2 4 2 2 4]
[2 2 1 2 4 2 2 4]
[2 2 2 1 2 4 4 2]
[2 4 4 2 1 2 2 2]
[4 2 2 4 2 1 2 2]
[4 2 2 4 2 2 1 2]
[2 4 4 2 2 2 2 1]
Order of dihedral group =  8 , Eigenvalues =  [-5, -5, -1, -1, -1, -1, 3, 19]
order of elements in group =  [1, 2, 2, 4, 2, 2, 4, 2]
   
p,q =  11 13
Matrix G_(p,q):
[ 1  2  2  2  2 10  2  5  2  5 10  2  2 10  5  2  5  2  2 10]
[ 2  1  2  2 10  2  5  2  5  2  2 10 10  2  2  5  2  5 10  2]
[ 2  2  1  2  2  5  2 10  2 10  5  2  2  5 10  2 10  2  2  5]
[ 2  2  2  1  5  2 10  2 10  2  2  5  5  2  2 10  2 10  5  2]
[ 2 10  2  5  1  2 10  2  2  2  2  5  5  2  2 10  2 10  5  2]
[10  2  5  2  2  1  2 10  2  2  5  2  2  5 10  2 10  2  2  5]
[ 2  5  2 10 10  2  1  2  5  2  2 10  2  2  2  5  2  5 10  2]
[ 5  2 10  2  2 10  2  1  2  5 10  2  2  2  5  2  5  2  2 10]
[ 2  5  2 10  2  2  5  2  1  2  2 10 10  2  2  5  2  5 10  2]
[ 5  2 10  2  2  2  2  5  2  1 10  2  2 10  5  2  5  2  2 10]
[10  2  5  2  2  5  2 10  2 10  1  2  2  5  2  2 10  2  2  5]
[ 2 10  2  5  5  2 10  2 10  2  2  1  5  2  2  2  2 10  5  2]
[ 2 10  2  5  5  2  2  2 10  2  2  5  1  2  2 10  2 10  5  2]
[10  2  5  2  2  5  2  2  2 10  5  2  2  1 10  2 10  2  2  5]
[ 5  2 10  2  2 10  2  5  2  5  2  2  2 10  1  2  5  2  2 10]
[ 2  5  2 10 10  2  5  2  5  2  2  2 10  2  2  1  2  5 10  2]
[ 5  2 10  2  2 10  2  5  2  5 10  2  2 10  5  2  1  2  2  2]
[ 2  5  2 10 10  2  5  2  5  2  2 10 10  2  2  5  2  1  2  2]
[ 2 10  2  5  5  2 10  2 10  2  2  5  5  2  2 10  2  2  1  2]
[10  2  5  2  2  5  2 10  2 10  5  2  2  5 10  2  2  2  2  1]
Order of dihedral group =  20 , Eigenvalues =  [-21, -21, -12, -12, -12, -12, -12, -12, -12, -12, 4, 4, 4, 4, 4, 4, 4, 4, 43, 83]
order of elements in group =  [1, 5, 5, 5, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 2, 10, 10]
corrected title by suggestion of SamHopkins.
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mathoverflowUser
mathoverflowUser
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Why do these finite group Dedekind matrices seem to have integer spectrum when specialized to the order of group elements?

corrected typo
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mathoverflowUser
mathoverflowUser
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$$\langle f_p,f_p\rangle = \langle f_p,f_p| f_p^2 = 1, f_q^2 = 1, (f_p \circ f_q)^n = 1\rangle = D_n$$$$\langle f_p,f_q\rangle = \langle f_p,f_q| f_p^2 = 1, f_q^2 = 1, (f_p \circ f_q)^n = 1\rangle = D_n$$ is the Dihedral group of degree $n(p,q)=:n$ for some natural number $n$, which depends on $p,q$.

This is because $\frac{1}{\frac{1}{x}} = x \bmod (p)$, so $f_p^2 = 1$ and if we look at the product $f_p \circ f_q$ which has finite order $n:=n(p,q)$, we see that this is just the presentation of the dihedral group of degree $n$ and order $2n$: $$\langle f_p,f_p| f_p^2 = 1, f_q^2 = 1, (f_p \circ f_q)^n = 1\rangle = D_n$$$$\langle f_p,f_q| f_p^2 = 1, f_q^2 = 1, (f_p \circ f_q)^n = 1\rangle = D_n$$

$$\langle f_p,f_p\rangle = \langle f_p,f_p| f_p^2 = 1, f_q^2 = 1, (f_p \circ f_q)^n = 1\rangle = D_n$$ is the Dihedral group of degree $n(p,q)=:n$ for some natural number $n$, which depends on $p,q$.

This is because $\frac{1}{\frac{1}{x}} = x \bmod (p)$, so $f_p^2 = 1$ and if we look at the product $f_p \circ f_q$ which has finite order $n:=n(p,q)$, we see that this is just the presentation of the dihedral group of degree $n$ and order $2n$: $$\langle f_p,f_p| f_p^2 = 1, f_q^2 = 1, (f_p \circ f_q)^n = 1\rangle = D_n$$

$$\langle f_p,f_q\rangle = \langle f_p,f_q| f_p^2 = 1, f_q^2 = 1, (f_p \circ f_q)^n = 1\rangle = D_n$$ is the Dihedral group of degree $n(p,q)=:n$ for some natural number $n$, which depends on $p,q$.

This is because $\frac{1}{\frac{1}{x}} = x \bmod (p)$, so $f_p^2 = 1$ and if we look at the product $f_p \circ f_q$ which has finite order $n:=n(p,q)$, we see that this is just the presentation of the dihedral group of degree $n$ and order $2n$: $$\langle f_p,f_q| f_p^2 = 1, f_q^2 = 1, (f_p \circ f_q)^n = 1\rangle = D_n$$

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