What kind of fraction will have this property?

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List out i/19:

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They form a square, and the sum of each column is the same, 81.

i/7 also have such property:

142857
285714
428571
571428
714285
857142

Any other fractions have such property? What kind of fractions will have such property?

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You are after numbers that are the decimal period of so-called cyclic numbers.

$7$ and $19$ in this context are called full reptend primes (OEIS A001913):

Primes p such that the decimal expansion of 1/p has period p-1, which is the greatest period possible for any integer.

The first few are $7, 17, 19, 23, 29, 47, 59, 61, 97.$