User bof, in a comment, pointed to the history of The Monster. I think that's an excellent example, and worth expanding on. I quote from [Wikipedia][1]: 

The Monster was predicted by Bernd Fischer (unpublished, about 1973) and Robert Griess (1976) as a simple group containing a double cover of Fischer's Baby Monster group as a centralizer of an involution. Within a few months, the order of M was found by Griess using the Thompson order formula, and Fischer, Conway, Norton and Thompson discovered other groups as subquotients, including many of the known sporadic groups, and two new ones: the Thompson group and the Harada–Norton group. The character table of the Monster, a 194-by-194 array, was calculated in 1979 by Fischer and Donald Livingstone using computer programs written by Michael Thorne. **It was not clear in the 1970s whether the Monster actually exists.** [Emphasis mine] Griess (1982) constructed M as the automorphism group of the Griess algebra, a 196,884-dimensional commutative nonassociative algebra; he first announced his construction in Ann Arbor on January 14, 1980.... Griess's construction showed that the Monster exists. 


  [1]: https://en.wikipedia.org/wiki/Monster_group