5
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Find the cosets of $D_{2n}$ in $S_n$

I tried to take examples and see, I did it in the case when $n=4$, but for the case $n=6$, the computation is became huge.

Is there a simple way in which I can see the problem?

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1 Answer 1

1
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I used Python to get the $60$ (left) cosets in $S_6$; the first defined function was copied from here (I can be blamed for the remaining code).

Python Code

# Python function to print permutations of a given list 
def permutation(lst): 
  
    # If lst is empty then there are no permutations 
    if len(lst) == 0: 
        return [] 
  
    # If there is only one element in lst then, only 
    # one permuatation is possible 
    if len(lst) == 1: 
        return [lst] 
  
    # Find the permutations for lst if there are 
    # more than 1 characters 
  
    l = [] # empty list that will store current permutation 
  
    # Iterate the input(lst) and calculate the permutation 
    for i in range(len(lst)): 
       m = lst[i] 
  
       # Extract lst[i] or m from the list.  remLst is 
       # remaining list 
       remLst = lst[:i] + lst[i+1:] 
  
       # Generating all permutations where m is first 
       # element 
       for p in permutation(remLst): 
           l.append([m] + p) 
    return l 

def multiply_permutations(p,q):
    r = []
    for i in range(0,len(q)):
        #print(i)
        spot = int(q[i])
        r.append(p[spot-1])
    return r

def left_coset(p, C):
    make_coset = []
    for c in C:
        a = multiply_permutations(p, c)
        make_coset.append(a)
    return make_coset

Sym6_Partition = []

def expand_partition(Block):
    p = Block[0]
    for B in Sym6_Partition:
        if p in B:
            print('block already accounted for...')
            return
    Sym6_Partition.append(Block)
                  
    
    



  
# Driver program to test above function 
data = list('123456')
S6 = permutation(data)

H12c = [ ['1','2','3','4','5','6'], \
        ['2','3','4','5','6','1'], \
        ['3','4','5','6','1','2'], \
        ['4','5','6','1','2','3'], \
        ['5','6','1','2','3','4'], \
        ['6','1','2','3','4','5']  ]


while 1:
    H12_more = []
    for pitem in H12c:
        #print(pitem)
        prod = multiply_permutations(['2','1','6','5','4','3'], pitem)
        H12_more.append(prod) 
    #print(H12_more)
    break
        
H12 = H12c + H12_more
for h in H12:
    for k in H12:
        j = multiply_permutations(h, k)
        if j in H12:
            pass
        else:
            print('error', j, k, j)

print('H12 =', H12)
Sym6_Partition = []
expand_partition(H12) # add H12 (first item) to Sym6_Partition list

for p in S6:
    found_it = False        
    for B in Sym6_Partition:
        if p in B:
            found_it = True
            break
    if found_it:
        continue
    new_block = left_coset(p, H12)
    expand_partition(new_block)
    if len(Sym6_Partition) == 60:
        break

print()
print()
print("*** PARTITION ***")
for block in Sym6_Partition:
    print()
    print(block)
    print

$\text{***}$ OUTPUT $\text{***}$

H12 = [['1', '2', '3', '4', '5', '6'], ['2', '3', '4', '5', '6', '1'], ['3', '4', '5', '6', '1', '2'], ['4', '5', '6', '1', '2', '3'], ['5', '6', '1', '2', '3', '4'], ['6', '1', '2', '3', '4', '5'], ['2', '1', '6', '5', '4', '3'], ['1', '6', '5', '4', '3', '2'], ['6', '5', '4', '3', '2', '1'], ['5', '4', '3', '2', '1', '6'], ['4', '3', '2', '1', '6', '5'], ['3', '2', '1', '6', '5', '4']]


*** PARTITION ***

[['1', '2', '3', '4', '5', '6'], ['2', '3', '4', '5', '6', '1'], ['3', '4', '5', '6', '1', '2'], ['4', '5', '6', '1', '2', '3'], ['5', '6', '1', '2', '3', '4'], ['6', '1', '2', '3', '4', '5'], ['2', '1', '6', '5', '4', '3'], ['1', '6', '5', '4', '3', '2'], ['6', '5', '4', '3', '2', '1'], ['5', '4', '3', '2', '1', '6'], ['4', '3', '2', '1', '6', '5'], ['3', '2', '1', '6', '5', '4']]

[['1', '2', '3', '4', '6', '5'], ['2', '3', '4', '6', '5', '1'], ['3', '4', '6', '5', '1', '2'], ['4', '6', '5', '1', '2', '3'], ['6', '5', '1', '2', '3', '4'], ['5', '1', '2', '3', '4', '6'], ['2', '1', '5', '6', '4', '3'], ['1', '5', '6', '4', '3', '2'], ['5', '6', '4', '3', '2', '1'], ['6', '4', '3', '2', '1', '5'], ['4', '3', '2', '1', '5', '6'], ['3', '2', '1', '5', '6', '4']]

[['1', '2', '3', '5', '4', '6'], ['2', '3', '5', '4', '6', '1'], ['3', '5', '4', '6', '1', '2'], ['5', '4', '6', '1', '2', '3'], ['4', '6', '1', '2', '3', '5'], ['6', '1', '2', '3', '5', '4'], ['2', '1', '6', '4', '5', '3'], ['1', '6', '4', '5', '3', '2'], ['6', '4', '5', '3', '2', '1'], ['4', '5', '3', '2', '1', '6'], ['5', '3', '2', '1', '6', '4'], ['3', '2', '1', '6', '4', '5']]

[['1', '2', '3', '5', '6', '4'], ['2', '3', '5', '6', '4', '1'], ['3', '5', '6', '4', '1', '2'], ['5', '6', '4', '1', '2', '3'], ['6', '4', '1', '2', '3', '5'], ['4', '1', '2', '3', '5', '6'], ['2', '1', '4', '6', '5', '3'], ['1', '4', '6', '5', '3', '2'], ['4', '6', '5', '3', '2', '1'], ['6', '5', '3', '2', '1', '4'], ['5', '3', '2', '1', '4', '6'], ['3', '2', '1', '4', '6', '5']]

[['1', '2', '3', '6', '4', '5'], ['2', '3', '6', '4', '5', '1'], ['3', '6', '4', '5', '1', '2'], ['6', '4', '5', '1', '2', '3'], ['4', '5', '1', '2', '3', '6'], ['5', '1', '2', '3', '6', '4'], ['2', '1', '5', '4', '6', '3'], ['1', '5', '4', '6', '3', '2'], ['5', '4', '6', '3', '2', '1'], ['4', '6', '3', '2', '1', '5'], ['6', '3', '2', '1', '5', '4'], ['3', '2', '1', '5', '4', '6']]

[['1', '2', '3', '6', '5', '4'], ['2', '3', '6', '5', '4', '1'], ['3', '6', '5', '4', '1', '2'], ['6', '5', '4', '1', '2', '3'], ['5', '4', '1', '2', '3', '6'], ['4', '1', '2', '3', '6', '5'], ['2', '1', '4', '5', '6', '3'], ['1', '4', '5', '6', '3', '2'], ['4', '5', '6', '3', '2', '1'], ['5', '6', '3', '2', '1', '4'], ['6', '3', '2', '1', '4', '5'], ['3', '2', '1', '4', '5', '6']]

[['1', '2', '4', '3', '5', '6'], ['2', '4', '3', '5', '6', '1'], ['4', '3', '5', '6', '1', '2'], ['3', '5', '6', '1', '2', '4'], ['5', '6', '1', '2', '4', '3'], ['6', '1', '2', '4', '3', '5'], ['2', '1', '6', '5', '3', '4'], ['1', '6', '5', '3', '4', '2'], ['6', '5', '3', '4', '2', '1'], ['5', '3', '4', '2', '1', '6'], ['3', '4', '2', '1', '6', '5'], ['4', '2', '1', '6', '5', '3']]

[['1', '2', '4', '3', '6', '5'], ['2', '4', '3', '6', '5', '1'], ['4', '3', '6', '5', '1', '2'], ['3', '6', '5', '1', '2', '4'], ['6', '5', '1', '2', '4', '3'], ['5', '1', '2', '4', '3', '6'], ['2', '1', '5', '6', '3', '4'], ['1', '5', '6', '3', '4', '2'], ['5', '6', '3', '4', '2', '1'], ['6', '3', '4', '2', '1', '5'], ['3', '4', '2', '1', '5', '6'], ['4', '2', '1', '5', '6', '3']]

[['1', '2', '4', '5', '3', '6'], ['2', '4', '5', '3', '6', '1'], ['4', '5', '3', '6', '1', '2'], ['5', '3', '6', '1', '2', '4'], ['3', '6', '1', '2', '4', '5'], ['6', '1', '2', '4', '5', '3'], ['2', '1', '6', '3', '5', '4'], ['1', '6', '3', '5', '4', '2'], ['6', '3', '5', '4', '2', '1'], ['3', '5', '4', '2', '1', '6'], ['5', '4', '2', '1', '6', '3'], ['4', '2', '1', '6', '3', '5']]

[['1', '2', '4', '5', '6', '3'], ['2', '4', '5', '6', '3', '1'], ['4', '5', '6', '3', '1', '2'], ['5', '6', '3', '1', '2', '4'], ['6', '3', '1', '2', '4', '5'], ['3', '1', '2', '4', '5', '6'], ['2', '1', '3', '6', '5', '4'], ['1', '3', '6', '5', '4', '2'], ['3', '6', '5', '4', '2', '1'], ['6', '5', '4', '2', '1', '3'], ['5', '4', '2', '1', '3', '6'], ['4', '2', '1', '3', '6', '5']]

[['1', '2', '4', '6', '3', '5'], ['2', '4', '6', '3', '5', '1'], ['4', '6', '3', '5', '1', '2'], ['6', '3', '5', '1', '2', '4'], ['3', '5', '1', '2', '4', '6'], ['5', '1', '2', '4', '6', '3'], ['2', '1', '5', '3', '6', '4'], ['1', '5', '3', '6', '4', '2'], ['5', '3', '6', '4', '2', '1'], ['3', '6', '4', '2', '1', '5'], ['6', '4', '2', '1', '5', '3'], ['4', '2', '1', '5', '3', '6']]

[['1', '2', '4', '6', '5', '3'], ['2', '4', '6', '5', '3', '1'], ['4', '6', '5', '3', '1', '2'], ['6', '5', '3', '1', '2', '4'], ['5', '3', '1', '2', '4', '6'], ['3', '1', '2', '4', '6', '5'], ['2', '1', '3', '5', '6', '4'], ['1', '3', '5', '6', '4', '2'], ['3', '5', '6', '4', '2', '1'], ['5', '6', '4', '2', '1', '3'], ['6', '4', '2', '1', '3', '5'], ['4', '2', '1', '3', '5', '6']]

[['1', '2', '5', '3', '4', '6'], ['2', '5', '3', '4', '6', '1'], ['5', '3', '4', '6', '1', '2'], ['3', '4', '6', '1', '2', '5'], ['4', '6', '1', '2', '5', '3'], ['6', '1', '2', '5', '3', '4'], ['2', '1', '6', '4', '3', '5'], ['1', '6', '4', '3', '5', '2'], ['6', '4', '3', '5', '2', '1'], ['4', '3', '5', '2', '1', '6'], ['3', '5', '2', '1', '6', '4'], ['5', '2', '1', '6', '4', '3']]

[['1', '2', '5', '3', '6', '4'], ['2', '5', '3', '6', '4', '1'], ['5', '3', '6', '4', '1', '2'], ['3', '6', '4', '1', '2', '5'], ['6', '4', '1', '2', '5', '3'], ['4', '1', '2', '5', '3', '6'], ['2', '1', '4', '6', '3', '5'], ['1', '4', '6', '3', '5', '2'], ['4', '6', '3', '5', '2', '1'], ['6', '3', '5', '2', '1', '4'], ['3', '5', '2', '1', '4', '6'], ['5', '2', '1', '4', '6', '3']]

[['1', '2', '5', '4', '3', '6'], ['2', '5', '4', '3', '6', '1'], ['5', '4', '3', '6', '1', '2'], ['4', '3', '6', '1', '2', '5'], ['3', '6', '1', '2', '5', '4'], ['6', '1', '2', '5', '4', '3'], ['2', '1', '6', '3', '4', '5'], ['1', '6', '3', '4', '5', '2'], ['6', '3', '4', '5', '2', '1'], ['3', '4', '5', '2', '1', '6'], ['4', '5', '2', '1', '6', '3'], ['5', '2', '1', '6', '3', '4']]

[['1', '2', '5', '4', '6', '3'], ['2', '5', '4', '6', '3', '1'], ['5', '4', '6', '3', '1', '2'], ['4', '6', '3', '1', '2', '5'], ['6', '3', '1', '2', '5', '4'], ['3', '1', '2', '5', '4', '6'], ['2', '1', '3', '6', '4', '5'], ['1', '3', '6', '4', '5', '2'], ['3', '6', '4', '5', '2', '1'], ['6', '4', '5', '2', '1', '3'], ['4', '5', '2', '1', '3', '6'], ['5', '2', '1', '3', '6', '4']]

[['1', '2', '5', '6', '3', '4'], ['2', '5', '6', '3', '4', '1'], ['5', '6', '3', '4', '1', '2'], ['6', '3', '4', '1', '2', '5'], ['3', '4', '1', '2', '5', '6'], ['4', '1', '2', '5', '6', '3'], ['2', '1', '4', '3', '6', '5'], ['1', '4', '3', '6', '5', '2'], ['4', '3', '6', '5', '2', '1'], ['3', '6', '5', '2', '1', '4'], ['6', '5', '2', '1', '4', '3'], ['5', '2', '1', '4', '3', '6']]

[['1', '2', '5', '6', '4', '3'], ['2', '5', '6', '4', '3', '1'], ['5', '6', '4', '3', '1', '2'], ['6', '4', '3', '1', '2', '5'], ['4', '3', '1', '2', '5', '6'], ['3', '1', '2', '5', '6', '4'], ['2', '1', '3', '4', '6', '5'], ['1', '3', '4', '6', '5', '2'], ['3', '4', '6', '5', '2', '1'], ['4', '6', '5', '2', '1', '3'], ['6', '5', '2', '1', '3', '4'], ['5', '2', '1', '3', '4', '6']]

[['1', '2', '6', '3', '4', '5'], ['2', '6', '3', '4', '5', '1'], ['6', '3', '4', '5', '1', '2'], ['3', '4', '5', '1', '2', '6'], ['4', '5', '1', '2', '6', '3'], ['5', '1', '2', '6', '3', '4'], ['2', '1', '5', '4', '3', '6'], ['1', '5', '4', '3', '6', '2'], ['5', '4', '3', '6', '2', '1'], ['4', '3', '6', '2', '1', '5'], ['3', '6', '2', '1', '5', '4'], ['6', '2', '1', '5', '4', '3']]

[['1', '2', '6', '3', '5', '4'], ['2', '6', '3', '5', '4', '1'], ['6', '3', '5', '4', '1', '2'], ['3', '5', '4', '1', '2', '6'], ['5', '4', '1', '2', '6', '3'], ['4', '1', '2', '6', '3', '5'], ['2', '1', '4', '5', '3', '6'], ['1', '4', '5', '3', '6', '2'], ['4', '5', '3', '6', '2', '1'], ['5', '3', '6', '2', '1', '4'], ['3', '6', '2', '1', '4', '5'], ['6', '2', '1', '4', '5', '3']]

[['1', '2', '6', '4', '3', '5'], ['2', '6', '4', '3', '5', '1'], ['6', '4', '3', '5', '1', '2'], ['4', '3', '5', '1', '2', '6'], ['3', '5', '1', '2', '6', '4'], ['5', '1', '2', '6', '4', '3'], ['2', '1', '5', '3', '4', '6'], ['1', '5', '3', '4', '6', '2'], ['5', '3', '4', '6', '2', '1'], ['3', '4', '6', '2', '1', '5'], ['4', '6', '2', '1', '5', '3'], ['6', '2', '1', '5', '3', '4']]

[['1', '2', '6', '4', '5', '3'], ['2', '6', '4', '5', '3', '1'], ['6', '4', '5', '3', '1', '2'], ['4', '5', '3', '1', '2', '6'], ['5', '3', '1', '2', '6', '4'], ['3', '1', '2', '6', '4', '5'], ['2', '1', '3', '5', '4', '6'], ['1', '3', '5', '4', '6', '2'], ['3', '5', '4', '6', '2', '1'], ['5', '4', '6', '2', '1', '3'], ['4', '6', '2', '1', '3', '5'], ['6', '2', '1', '3', '5', '4']]

[['1', '2', '6', '5', '3', '4'], ['2', '6', '5', '3', '4', '1'], ['6', '5', '3', '4', '1', '2'], ['5', '3', '4', '1', '2', '6'], ['3', '4', '1', '2', '6', '5'], ['4', '1', '2', '6', '5', '3'], ['2', '1', '4', '3', '5', '6'], ['1', '4', '3', '5', '6', '2'], ['4', '3', '5', '6', '2', '1'], ['3', '5', '6', '2', '1', '4'], ['5', '6', '2', '1', '4', '3'], ['6', '2', '1', '4', '3', '5']]

[['1', '2', '6', '5', '4', '3'], ['2', '6', '5', '4', '3', '1'], ['6', '5', '4', '3', '1', '2'], ['5', '4', '3', '1', '2', '6'], ['4', '3', '1', '2', '6', '5'], ['3', '1', '2', '6', '5', '4'], ['2', '1', '3', '4', '5', '6'], ['1', '3', '4', '5', '6', '2'], ['3', '4', '5', '6', '2', '1'], ['4', '5', '6', '2', '1', '3'], ['5', '6', '2', '1', '3', '4'], ['6', '2', '1', '3', '4', '5']]

[['1', '3', '2', '4', '5', '6'], ['3', '2', '4', '5', '6', '1'], ['2', '4', '5', '6', '1', '3'], ['4', '5', '6', '1', '3', '2'], ['5', '6', '1', '3', '2', '4'], ['6', '1', '3', '2', '4', '5'], ['3', '1', '6', '5', '4', '2'], ['1', '6', '5', '4', '2', '3'], ['6', '5', '4', '2', '3', '1'], ['5', '4', '2', '3', '1', '6'], ['4', '2', '3', '1', '6', '5'], ['2', '3', '1', '6', '5', '4']]

[['1', '3', '2', '4', '6', '5'], ['3', '2', '4', '6', '5', '1'], ['2', '4', '6', '5', '1', '3'], ['4', '6', '5', '1', '3', '2'], ['6', '5', '1', '3', '2', '4'], ['5', '1', '3', '2', '4', '6'], ['3', '1', '5', '6', '4', '2'], ['1', '5', '6', '4', '2', '3'], ['5', '6', '4', '2', '3', '1'], ['6', '4', '2', '3', '1', '5'], ['4', '2', '3', '1', '5', '6'], ['2', '3', '1', '5', '6', '4']]

[['1', '3', '2', '5', '4', '6'], ['3', '2', '5', '4', '6', '1'], ['2', '5', '4', '6', '1', '3'], ['5', '4', '6', '1', '3', '2'], ['4', '6', '1', '3', '2', '5'], ['6', '1', '3', '2', '5', '4'], ['3', '1', '6', '4', '5', '2'], ['1', '6', '4', '5', '2', '3'], ['6', '4', '5', '2', '3', '1'], ['4', '5', '2', '3', '1', '6'], ['5', '2', '3', '1', '6', '4'], ['2', '3', '1', '6', '4', '5']]

[['1', '3', '2', '5', '6', '4'], ['3', '2', '5', '6', '4', '1'], ['2', '5', '6', '4', '1', '3'], ['5', '6', '4', '1', '3', '2'], ['6', '4', '1', '3', '2', '5'], ['4', '1', '3', '2', '5', '6'], ['3', '1', '4', '6', '5', '2'], ['1', '4', '6', '5', '2', '3'], ['4', '6', '5', '2', '3', '1'], ['6', '5', '2', '3', '1', '4'], ['5', '2', '3', '1', '4', '6'], ['2', '3', '1', '4', '6', '5']]

[['1', '3', '2', '6', '4', '5'], ['3', '2', '6', '4', '5', '1'], ['2', '6', '4', '5', '1', '3'], ['6', '4', '5', '1', '3', '2'], ['4', '5', '1', '3', '2', '6'], ['5', '1', '3', '2', '6', '4'], ['3', '1', '5', '4', '6', '2'], ['1', '5', '4', '6', '2', '3'], ['5', '4', '6', '2', '3', '1'], ['4', '6', '2', '3', '1', '5'], ['6', '2', '3', '1', '5', '4'], ['2', '3', '1', '5', '4', '6']]

[['1', '3', '2', '6', '5', '4'], ['3', '2', '6', '5', '4', '1'], ['2', '6', '5', '4', '1', '3'], ['6', '5', '4', '1', '3', '2'], ['5', '4', '1', '3', '2', '6'], ['4', '1', '3', '2', '6', '5'], ['3', '1', '4', '5', '6', '2'], ['1', '4', '5', '6', '2', '3'], ['4', '5', '6', '2', '3', '1'], ['5', '6', '2', '3', '1', '4'], ['6', '2', '3', '1', '4', '5'], ['2', '3', '1', '4', '5', '6']]

[['1', '3', '4', '2', '5', '6'], ['3', '4', '2', '5', '6', '1'], ['4', '2', '5', '6', '1', '3'], ['2', '5', '6', '1', '3', '4'], ['5', '6', '1', '3', '4', '2'], ['6', '1', '3', '4', '2', '5'], ['3', '1', '6', '5', '2', '4'], ['1', '6', '5', '2', '4', '3'], ['6', '5', '2', '4', '3', '1'], ['5', '2', '4', '3', '1', '6'], ['2', '4', '3', '1', '6', '5'], ['4', '3', '1', '6', '5', '2']]

[['1', '3', '4', '2', '6', '5'], ['3', '4', '2', '6', '5', '1'], ['4', '2', '6', '5', '1', '3'], ['2', '6', '5', '1', '3', '4'], ['6', '5', '1', '3', '4', '2'], ['5', '1', '3', '4', '2', '6'], ['3', '1', '5', '6', '2', '4'], ['1', '5', '6', '2', '4', '3'], ['5', '6', '2', '4', '3', '1'], ['6', '2', '4', '3', '1', '5'], ['2', '4', '3', '1', '5', '6'], ['4', '3', '1', '5', '6', '2']]

[['1', '3', '4', '5', '2', '6'], ['3', '4', '5', '2', '6', '1'], ['4', '5', '2', '6', '1', '3'], ['5', '2', '6', '1', '3', '4'], ['2', '6', '1', '3', '4', '5'], ['6', '1', '3', '4', '5', '2'], ['3', '1', '6', '2', '5', '4'], ['1', '6', '2', '5', '4', '3'], ['6', '2', '5', '4', '3', '1'], ['2', '5', '4', '3', '1', '6'], ['5', '4', '3', '1', '6', '2'], ['4', '3', '1', '6', '2', '5']]

[['1', '3', '4', '6', '2', '5'], ['3', '4', '6', '2', '5', '1'], ['4', '6', '2', '5', '1', '3'], ['6', '2', '5', '1', '3', '4'], ['2', '5', '1', '3', '4', '6'], ['5', '1', '3', '4', '6', '2'], ['3', '1', '5', '2', '6', '4'], ['1', '5', '2', '6', '4', '3'], ['5', '2', '6', '4', '3', '1'], ['2', '6', '4', '3', '1', '5'], ['6', '4', '3', '1', '5', '2'], ['4', '3', '1', '5', '2', '6']]

[['1', '3', '5', '2', '4', '6'], ['3', '5', '2', '4', '6', '1'], ['5', '2', '4', '6', '1', '3'], ['2', '4', '6', '1', '3', '5'], ['4', '6', '1', '3', '5', '2'], ['6', '1', '3', '5', '2', '4'], ['3', '1', '6', '4', '2', '5'], ['1', '6', '4', '2', '5', '3'], ['6', '4', '2', '5', '3', '1'], ['4', '2', '5', '3', '1', '6'], ['2', '5', '3', '1', '6', '4'], ['5', '3', '1', '6', '4', '2']]

[['1', '3', '5', '2', '6', '4'], ['3', '5', '2', '6', '4', '1'], ['5', '2', '6', '4', '1', '3'], ['2', '6', '4', '1', '3', '5'], ['6', '4', '1', '3', '5', '2'], ['4', '1', '3', '5', '2', '6'], ['3', '1', '4', '6', '2', '5'], ['1', '4', '6', '2', '5', '3'], ['4', '6', '2', '5', '3', '1'], ['6', '2', '5', '3', '1', '4'], ['2', '5', '3', '1', '4', '6'], ['5', '3', '1', '4', '6', '2']]

[['1', '3', '5', '4', '2', '6'], ['3', '5', '4', '2', '6', '1'], ['5', '4', '2', '6', '1', '3'], ['4', '2', '6', '1', '3', '5'], ['2', '6', '1', '3', '5', '4'], ['6', '1', '3', '5', '4', '2'], ['3', '1', '6', '2', '4', '5'], ['1', '6', '2', '4', '5', '3'], ['6', '2', '4', '5', '3', '1'], ['2', '4', '5', '3', '1', '6'], ['4', '5', '3', '1', '6', '2'], ['5', '3', '1', '6', '2', '4']]

[['1', '3', '5', '6', '2', '4'], ['3', '5', '6', '2', '4', '1'], ['5', '6', '2', '4', '1', '3'], ['6', '2', '4', '1', '3', '5'], ['2', '4', '1', '3', '5', '6'], ['4', '1', '3', '5', '6', '2'], ['3', '1', '4', '2', '6', '5'], ['1', '4', '2', '6', '5', '3'], ['4', '2', '6', '5', '3', '1'], ['2', '6', '5', '3', '1', '4'], ['6', '5', '3', '1', '4', '2'], ['5', '3', '1', '4', '2', '6']]

[['1', '3', '6', '2', '4', '5'], ['3', '6', '2', '4', '5', '1'], ['6', '2', '4', '5', '1', '3'], ['2', '4', '5', '1', '3', '6'], ['4', '5', '1', '3', '6', '2'], ['5', '1', '3', '6', '2', '4'], ['3', '1', '5', '4', '2', '6'], ['1', '5', '4', '2', '6', '3'], ['5', '4', '2', '6', '3', '1'], ['4', '2', '6', '3', '1', '5'], ['2', '6', '3', '1', '5', '4'], ['6', '3', '1', '5', '4', '2']]

[['1', '3', '6', '2', '5', '4'], ['3', '6', '2', '5', '4', '1'], ['6', '2', '5', '4', '1', '3'], ['2', '5', '4', '1', '3', '6'], ['5', '4', '1', '3', '6', '2'], ['4', '1', '3', '6', '2', '5'], ['3', '1', '4', '5', '2', '6'], ['1', '4', '5', '2', '6', '3'], ['4', '5', '2', '6', '3', '1'], ['5', '2', '6', '3', '1', '4'], ['2', '6', '3', '1', '4', '5'], ['6', '3', '1', '4', '5', '2']]

[['1', '3', '6', '4', '2', '5'], ['3', '6', '4', '2', '5', '1'], ['6', '4', '2', '5', '1', '3'], ['4', '2', '5', '1', '3', '6'], ['2', '5', '1', '3', '6', '4'], ['5', '1', '3', '6', '4', '2'], ['3', '1', '5', '2', '4', '6'], ['1', '5', '2', '4', '6', '3'], ['5', '2', '4', '6', '3', '1'], ['2', '4', '6', '3', '1', '5'], ['4', '6', '3', '1', '5', '2'], ['6', '3', '1', '5', '2', '4']]

[['1', '3', '6', '5', '2', '4'], ['3', '6', '5', '2', '4', '1'], ['6', '5', '2', '4', '1', '3'], ['5', '2', '4', '1', '3', '6'], ['2', '4', '1', '3', '6', '5'], ['4', '1', '3', '6', '5', '2'], ['3', '1', '4', '2', '5', '6'], ['1', '4', '2', '5', '6', '3'], ['4', '2', '5', '6', '3', '1'], ['2', '5', '6', '3', '1', '4'], ['5', '6', '3', '1', '4', '2'], ['6', '3', '1', '4', '2', '5']]

[['1', '4', '2', '3', '5', '6'], ['4', '2', '3', '5', '6', '1'], ['2', '3', '5', '6', '1', '4'], ['3', '5', '6', '1', '4', '2'], ['5', '6', '1', '4', '2', '3'], ['6', '1', '4', '2', '3', '5'], ['4', '1', '6', '5', '3', '2'], ['1', '6', '5', '3', '2', '4'], ['6', '5', '3', '2', '4', '1'], ['5', '3', '2', '4', '1', '6'], ['3', '2', '4', '1', '6', '5'], ['2', '4', '1', '6', '5', '3']]

[['1', '4', '2', '3', '6', '5'], ['4', '2', '3', '6', '5', '1'], ['2', '3', '6', '5', '1', '4'], ['3', '6', '5', '1', '4', '2'], ['6', '5', '1', '4', '2', '3'], ['5', '1', '4', '2', '3', '6'], ['4', '1', '5', '6', '3', '2'], ['1', '5', '6', '3', '2', '4'], ['5', '6', '3', '2', '4', '1'], ['6', '3', '2', '4', '1', '5'], ['3', '2', '4', '1', '5', '6'], ['2', '4', '1', '5', '6', '3']]

[['1', '4', '2', '5', '3', '6'], ['4', '2', '5', '3', '6', '1'], ['2', '5', '3', '6', '1', '4'], ['5', '3', '6', '1', '4', '2'], ['3', '6', '1', '4', '2', '5'], ['6', '1', '4', '2', '5', '3'], ['4', '1', '6', '3', '5', '2'], ['1', '6', '3', '5', '2', '4'], ['6', '3', '5', '2', '4', '1'], ['3', '5', '2', '4', '1', '6'], ['5', '2', '4', '1', '6', '3'], ['2', '4', '1', '6', '3', '5']]

[['1', '4', '2', '6', '3', '5'], ['4', '2', '6', '3', '5', '1'], ['2', '6', '3', '5', '1', '4'], ['6', '3', '5', '1', '4', '2'], ['3', '5', '1', '4', '2', '6'], ['5', '1', '4', '2', '6', '3'], ['4', '1', '5', '3', '6', '2'], ['1', '5', '3', '6', '2', '4'], ['5', '3', '6', '2', '4', '1'], ['3', '6', '2', '4', '1', '5'], ['6', '2', '4', '1', '5', '3'], ['2', '4', '1', '5', '3', '6']]

[['1', '4', '3', '2', '5', '6'], ['4', '3', '2', '5', '6', '1'], ['3', '2', '5', '6', '1', '4'], ['2', '5', '6', '1', '4', '3'], ['5', '6', '1', '4', '3', '2'], ['6', '1', '4', '3', '2', '5'], ['4', '1', '6', '5', '2', '3'], ['1', '6', '5', '2', '3', '4'], ['6', '5', '2', '3', '4', '1'], ['5', '2', '3', '4', '1', '6'], ['2', '3', '4', '1', '6', '5'], ['3', '4', '1', '6', '5', '2']]

[['1', '4', '3', '2', '6', '5'], ['4', '3', '2', '6', '5', '1'], ['3', '2', '6', '5', '1', '4'], ['2', '6', '5', '1', '4', '3'], ['6', '5', '1', '4', '3', '2'], ['5', '1', '4', '3', '2', '6'], ['4', '1', '5', '6', '2', '3'], ['1', '5', '6', '2', '3', '4'], ['5', '6', '2', '3', '4', '1'], ['6', '2', '3', '4', '1', '5'], ['2', '3', '4', '1', '5', '6'], ['3', '4', '1', '5', '6', '2']]

[['1', '4', '3', '5', '2', '6'], ['4', '3', '5', '2', '6', '1'], ['3', '5', '2', '6', '1', '4'], ['5', '2', '6', '1', '4', '3'], ['2', '6', '1', '4', '3', '5'], ['6', '1', '4', '3', '5', '2'], ['4', '1', '6', '2', '5', '3'], ['1', '6', '2', '5', '3', '4'], ['6', '2', '5', '3', '4', '1'], ['2', '5', '3', '4', '1', '6'], ['5', '3', '4', '1', '6', '2'], ['3', '4', '1', '6', '2', '5']]

[['1', '4', '3', '6', '2', '5'], ['4', '3', '6', '2', '5', '1'], ['3', '6', '2', '5', '1', '4'], ['6', '2', '5', '1', '4', '3'], ['2', '5', '1', '4', '3', '6'], ['5', '1', '4', '3', '6', '2'], ['4', '1', '5', '2', '6', '3'], ['1', '5', '2', '6', '3', '4'], ['5', '2', '6', '3', '4', '1'], ['2', '6', '3', '4', '1', '5'], ['6', '3', '4', '1', '5', '2'], ['3', '4', '1', '5', '2', '6']]

[['1', '4', '5', '2', '3', '6'], ['4', '5', '2', '3', '6', '1'], ['5', '2', '3', '6', '1', '4'], ['2', '3', '6', '1', '4', '5'], ['3', '6', '1', '4', '5', '2'], ['6', '1', '4', '5', '2', '3'], ['4', '1', '6', '3', '2', '5'], ['1', '6', '3', '2', '5', '4'], ['6', '3', '2', '5', '4', '1'], ['3', '2', '5', '4', '1', '6'], ['2', '5', '4', '1', '6', '3'], ['5', '4', '1', '6', '3', '2']]

[['1', '4', '5', '3', '2', '6'], ['4', '5', '3', '2', '6', '1'], ['5', '3', '2', '6', '1', '4'], ['3', '2', '6', '1', '4', '5'], ['2', '6', '1', '4', '5', '3'], ['6', '1', '4', '5', '3', '2'], ['4', '1', '6', '2', '3', '5'], ['1', '6', '2', '3', '5', '4'], ['6', '2', '3', '5', '4', '1'], ['2', '3', '5', '4', '1', '6'], ['3', '5', '4', '1', '6', '2'], ['5', '4', '1', '6', '2', '3']]

[['1', '4', '6', '2', '3', '5'], ['4', '6', '2', '3', '5', '1'], ['6', '2', '3', '5', '1', '4'], ['2', '3', '5', '1', '4', '6'], ['3', '5', '1', '4', '6', '2'], ['5', '1', '4', '6', '2', '3'], ['4', '1', '5', '3', '2', '6'], ['1', '5', '3', '2', '6', '4'], ['5', '3', '2', '6', '4', '1'], ['3', '2', '6', '4', '1', '5'], ['2', '6', '4', '1', '5', '3'], ['6', '4', '1', '5', '3', '2']]

[['1', '4', '6', '3', '2', '5'], ['4', '6', '3', '2', '5', '1'], ['6', '3', '2', '5', '1', '4'], ['3', '2', '5', '1', '4', '6'], ['2', '5', '1', '4', '6', '3'], ['5', '1', '4', '6', '3', '2'], ['4', '1', '5', '2', '3', '6'], ['1', '5', '2', '3', '6', '4'], ['5', '2', '3', '6', '4', '1'], ['2', '3', '6', '4', '1', '5'], ['3', '6', '4', '1', '5', '2'], ['6', '4', '1', '5', '2', '3']]

[['1', '5', '2', '3', '4', '6'], ['5', '2', '3', '4', '6', '1'], ['2', '3', '4', '6', '1', '5'], ['3', '4', '6', '1', '5', '2'], ['4', '6', '1', '5', '2', '3'], ['6', '1', '5', '2', '3', '4'], ['5', '1', '6', '4', '3', '2'], ['1', '6', '4', '3', '2', '5'], ['6', '4', '3', '2', '5', '1'], ['4', '3', '2', '5', '1', '6'], ['3', '2', '5', '1', '6', '4'], ['2', '5', '1', '6', '4', '3']]

[['1', '5', '2', '4', '3', '6'], ['5', '2', '4', '3', '6', '1'], ['2', '4', '3', '6', '1', '5'], ['4', '3', '6', '1', '5', '2'], ['3', '6', '1', '5', '2', '4'], ['6', '1', '5', '2', '4', '3'], ['5', '1', '6', '3', '4', '2'], ['1', '6', '3', '4', '2', '5'], ['6', '3', '4', '2', '5', '1'], ['3', '4', '2', '5', '1', '6'], ['4', '2', '5', '1', '6', '3'], ['2', '5', '1', '6', '3', '4']]

[['1', '5', '3', '2', '4', '6'], ['5', '3', '2', '4', '6', '1'], ['3', '2', '4', '6', '1', '5'], ['2', '4', '6', '1', '5', '3'], ['4', '6', '1', '5', '3', '2'], ['6', '1', '5', '3', '2', '4'], ['5', '1', '6', '4', '2', '3'], ['1', '6', '4', '2', '3', '5'], ['6', '4', '2', '3', '5', '1'], ['4', '2', '3', '5', '1', '6'], ['2', '3', '5', '1', '6', '4'], ['3', '5', '1', '6', '4', '2']]

[['1', '5', '3', '4', '2', '6'], ['5', '3', '4', '2', '6', '1'], ['3', '4', '2', '6', '1', '5'], ['4', '2', '6', '1', '5', '3'], ['2', '6', '1', '5', '3', '4'], ['6', '1', '5', '3', '4', '2'], ['5', '1', '6', '2', '4', '3'], ['1', '6', '2', '4', '3', '5'], ['6', '2', '4', '3', '5', '1'], ['2', '4', '3', '5', '1', '6'], ['4', '3', '5', '1', '6', '2'], ['3', '5', '1', '6', '2', '4']]

[['1', '5', '4', '2', '3', '6'], ['5', '4', '2', '3', '6', '1'], ['4', '2', '3', '6', '1', '5'], ['2', '3', '6', '1', '5', '4'], ['3', '6', '1', '5', '4', '2'], ['6', '1', '5', '4', '2', '3'], ['5', '1', '6', '3', '2', '4'], ['1', '6', '3', '2', '4', '5'], ['6', '3', '2', '4', '5', '1'], ['3', '2', '4', '5', '1', '6'], ['2', '4', '5', '1', '6', '3'], ['4', '5', '1', '6', '3', '2']]

[['1', '5', '4', '3', '2', '6'], ['5', '4', '3', '2', '6', '1'], ['4', '3', '2', '6', '1', '5'], ['3', '2', '6', '1', '5', '4'], ['2', '6', '1', '5', '4', '3'], ['6', '1', '5', '4', '3', '2'], ['5', '1', '6', '2', '3', '4'], ['1', '6', '2', '3', '4', '5'], ['6', '2', '3', '4', '5', '1'], ['2', '3', '4', '5', '1', '6'], ['3', '4', '5', '1', '6', '2'], ['4', '5', '1', '6', '2', '3']]
$\endgroup$
4
  • $\begingroup$ Where do you learn to write such programs in general.We have been taught $C$ and Java in our school and they dont teach us to write such programs? How do I learn coding such mathematical problems? $\endgroup$
    – Antimony
    Oct 25, 2020 at 19:25
  • $\begingroup$ I have some practice using Python to express math concepts. The language is expressive and you can even construct sets using it. If you have a questions about the language I've found stackoverflow to be a good resource. $\endgroup$ Oct 25, 2020 at 19:35
  • $\begingroup$ @CopyPastelt .Thankyou so much for the help.Just one more thing I wanted to know how do I figure out the disjoint cycles ("(21)(65)(43)" from "(216543)") . They all seem a single 6-cycle to me. $\endgroup$
    – Antimony
    Oct 31, 2020 at 12:27
  • $\begingroup$ @smita The geekforgeeks code was using the one-line_notation to represent a permutation and so that is what my answer is using. $\endgroup$ Oct 31, 2020 at 14:14

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