Today’s date, US style, is 9/26/2023, and there is only one group, up to isomorphism, of size 9262023. You could verify this in Mathematica with the command

FiniteGroupCount[9262023]

which returns 1.

For a given *n*, when is there only one group of size *n*?

There are two requirements. First, *n* has to be the product of distinct primes, i.e. no prime appears in the factorization with a power greater than 1. Second, no prime divides one less than another prime.

Now

9262023 = 3 × 41 × 257 ×293

and you can check that 3 does not divide 40, 256, or 292, nor does 41 divide 2, 252, or 292, etc.

A more compact way to state the criteria above is to say

gcd(*n*, φ(*n*)) = 1

where φ(*n*) is Euler’s totient function, the number of positive numbers less than *n* and relatively prime to *n*.

Why are these criteria equivalent? If

*n* = *pqr*…

then

φ(*n*) = (*p* − 1)(*q* − 1)(*r* − 1)…

If *n* and φ(*n*) have a nontrivial common factor, it has to be one of the prime factors of *n*, and none of these divide any term of φ(*n*).

Source: Dieter Jungnickel. On the Uniqueness of the Cyclic Group of Order *n. *The American Mathematical Monthly, Vol. 99, No. 6. (Jun. – Jul., 1992), pp. 545–547.