In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less well-documented than the main narrative of discovery and confirmation. Specifically, I am referring to the question (posted 11 years ago) [Groups that do not exist][1], which asked whether there were finite simple groups conjectured at some point that turned out not to exist.

To build upon that discussion, I am looking for a very specific kind of artifact from the history of group theory: explicit character tables for such conjectured but non-existent finite simple groups, if they ever were constructed. My understanding is that character tables were computed for many groups as part of the classification effort. 

**Question**: Could anyone provide references to or copies of explicit (complete) character tables developed for finite simple groups that were later shown not to exist?

If possible, please elucidate the reasoning behind the exclusion.


  [1]: https://mathoverflow.net/questions/115735/groups-that-do-not-exist

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Regarding the conjecture mentioned in Dave Benson's answer, which states that for any two distinct odd primes $p$ and $q$, the integer $\frac{q^p - 1}{q - 1}$ never divides $\frac{p^q - 1}{p - 1}$, I am unsure to what extent this has been verified. Below is a Python script that confirms the conjecture holds for $p, q < 5000$ within 2 minutes. It also identifies the pair $(p, q) = (3313,17)$ as noted in JoshuaZ's comment, for the sole non-coprime example in this range,

**Computation**  

    ...$time python3 check_primes.py
    Pair not relatively prime: p=3313, q=17, gcd=112643
    
    real	2m5.001s
    user	2m4.981s
    sys	0m0.012s

**Code**  

    # check_primes.py
    from sympy import primerange, gcd
    
    def check_relative_prime_and_divisibility(p, q):
        # Calculate the expressions (p^q - 1)/(p - 1) and (q^p - 1)/(q - 1)
        num1 = (pow(p, q) - 1) // (p - 1)
        num2 = (pow(q, p) - 1) // (q - 1)
        
        # Check if num1 and num2 are relatively prime
        d = gcd(num1, num2)
        if d == 1:
            return False
        
        # If they are not relatively prime, check divisibility
        print(f"Pair not relatively prime: p={p}, q={q}, gcd={d}")
        return num2 % num1 == 0
    
    def find_prime_pairs(max_prime):
        # Generate a list of odd primes up to max_prime
        primes = list(primerange(3, max_prime))
        
        # Iterate over all pairs of odd primes p, q where p > q
        for i in range(len(primes)):
            p = primes[i]
            #print(p)
            for j in range(i):
                q = primes[j]
                if check_relative_prime_and_divisibility(p, q):
                    print(f"Found divisible pair: p={p}, q={q}")
    
    # Set a maximum prime number limit according to your laptop's computational power
    max_prime = 5000 # You can adjust this value
    find_prime_pairs(max_prime)