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Ultraworld
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An example in my book on character theory gives an arbitrary non-abelian group G of order 27 and claims that |G'| = 3 (order of the commutator subgroup equals 3) and that there are 11 conjugation-classes. I really don't know why. I only manage to deduce that |Z(G)| = 3 (the order of the center of the group equals 3).
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