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Consider the next decision problem:
Given two finite groups represented by their multiplicity table, determine if they are isomorphic or not.
Clearly, this problem belongs to NP since given a witness in the form of an isomorphism between the groups it can be verified in polynomial time (to the input size). But what more can be said about this problem in terms of complexity?

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Interesting. Thank you. – Ido Aug 12 '12 at 21:06
Good question. I'd be interested to know more about the hardest case: p-groups. – user641 Aug 12 '12 at 22:53
Also, are things easier when the groups are permutation groups? – user641 Aug 12 '12 at 22:54
up vote 4 down vote accepted

As Colin McQuillan mentions above, this is an excellent survey of the state of the art:

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When posting (almost verbatim) something that has already been posted as a comment by someone else, the common courtesy would be to at least mention the author of the comment and post as community wiki.. – tomasz Aug 12 '12 at 21:05
@tomasz To be quite fair, the comment entirely escaped my attention while posting. I was quite surprised to see it had an earlier timestamp. – Erick Wong Aug 12 '12 at 21:15

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