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Suppose G is a group of order 48 (centre consisting identity only). Show it has a conjugacy class of order 3.

I know that the size of the conjugacy classes are limited to divisors of 48: 1,2,3,4,6,8,12,16,24, and 48. These classes also partition G so their sizes must sum to the order of G (so I cannot use 48 and have to include 1). I am not sure how to continue from here.

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We know that $G$ partitions into conjugacy classes and the only possible orders of these classes are those that you mention. There is only one conjugacy class of order $1$ since the center is trivial. Now the sum of the occurring orders is $48$, which is even, so there is at least one conjugacy class of odd order bigger than one...

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