# Sizes of Conjugacy Classes of a Group of Known Order

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.

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...