[Python] How to find Lyndon words among necklaces? (Or how to find the periodic necklaces?)

There are A003239(n) binary necklaces of length 2n.
A022553(n) among them are primitive, i.e. not periodic. They correspond to Lyndon words.
For n=6 there are 80 necklaces. 75 of them are primitive, leaving 5 that must be periodic.

It should be trivial to show examples with Python, but something is wrong with my code.
For n=6 it finds only 4 instead of 5 periodic necklaces.
(For n=7 it does not find the single periodic necklace. For n=8 all 10 are found.)

from itertools import combinations
from math import log10, ceil


def necklace_representative(necklace):

def rotate(key):
return necklace[key:] + necklace[:key]

rotations = [rotate(i) for i in range(len(necklace))]

return min(rotations)


n = 6
length = 2 * n
necklaces = set()

for places in combinations(range(length), n):
word = [int(i in places) for i in range(length)]
necklaces.add(tuple(necklace_representative(word)))

decimal_length = ceil(log10(len(necklaces) + 1))

for i, word in enumerate(sorted(necklaces)):

print(
str(i+1).zfill(decimal_length),
''.join([str(_) for _ in word]),
'periodic' if word[0:n] == word[n:length] else ''
)


01 000000111111
02 000001011111
03 000001101111
04 000001110111
05 000001111011
06 000001111101
07 000010011111
08 000010101111
09 000010110111
10 000010111011
11 000010111101
12 000011001111
13 000011010111
14 000011011011
15 000011011101
16 000011100111
17 000011101011
18 000011101101
19 000011110011
20 000011110101
21 000011111001
22 000100011111
23 000100101111
24 000100110111
25 000100111011
26 000100111101
27 000101001111
28 000101010111
29 000101011011
30 000101011101
31 000101100111
32 000101101011
33 000101101101
34 000101110011
35 000101110101
36 000101111001
37 000110001111
38 000110010111
39 000110011011
40 000110011101
41 000110100111
42 000110101011
43 000110101101
44 000110110011
45 000110110101
46 000110111001
47 000111000111 periodic
48 000111001011
49 000111001101
50 000111010011
51 000111010101
52 000111011001
53 000111100101
54 000111101001
55 001001001111
56 001001010111
57 001001011011
58 001001011101
59 001001100111
60 001001101011
61 001001101101
62 001001110011
63 001001110101
64 001010010111
65 001010011011
66 001010011101
67 001010100111
68 001010101011
69 001010101101
70 001010110011
71 001010110101
72 001011001011 periodic
73 001011001101
74 001011010011
75 001011010101
76 001100110011
77 001100110101
78 001101001101 periodic
79 001101010101
80 010101010101 periodic

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