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I recall being told in an algebra course in college that there exist groups with matching order tables and that are nonetheless not isomorphic. That is, if you list out the orders of all the elements in one group and all the orders of the elements in the other, the lists are "the same", and yet you can't match them up to yield an isomorphism. I vaguely recall being told that the smallest example of a pair of non-isomorphic groups like this consists of two groups of order 60. I thought also that there was a name for such groups. (i.e. an adjective that fills the gap in the sentence: "The fact that two groups are ***** is a necessary but not sufficient condition for them to be isomorphic," with the sneaky pair of groups I'm looking for being the smallest counter-example to the sufficiency.

Are my recollections correct? If so, can anyone give me the details of this pair of groups (and the elusive adjective, the non-recollection of which is preventing me from finding the info I want on the web, I suspect)?

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# Groups masquerading as isomorphic

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