2 added 494 characters in body; added 9 characters in body; deleted 2 characters in body; added 9 characters in body

In principle, we can compute the character table of a finite group algorithmically. For example, [McKay, J. K. S. A method for computing the character table of a finite group. 1968 Computers in Mathematical Research pp. 140--148 North-Holland, Amsterdam MR0236278 (38 #4575)] and 4575)], [McKay, J. K. S. Algorithm 307. Comm. ACM 10, 7, (July 1967) 450-451.]450-451.], Dixon, John D. High speed computation of group characters. Numer. Math. 10 1967 446--450. MR0224726 (37 #325)] and others; the ideas go back to Burnside, at least, it seems. McKay's program was used to compute the characters of $J_1$ and $J_3$ in all of 84 minutes at the time, with a whole 16K store (which is 12 times smaller than the size of the background image I am using as background to my desktop computer!)

Using a bit of character theory, as in Noah's answer, we can then tell which characters are afforded by real representations, and what are the real faithful representations.