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I have a interesting question,

Let G be a finite group with exactly 2 conjugacy class of the same order. when degrees of exactly 2 irreducible characters are equivalent?

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Changed the tags, as the question has nothing to do with characteristic classes. – Mark Grant Aug 8 '11 at 10:43
Seems to work for A_5 and PSL(2,7). But it does not work for A_4. The condition is quite restrictive and I am not sure if there is a clean general statement. – Geoff Robinson Aug 8 '11 at 10:45
Doesn't work for dihedral group of order 10; conjugacy classes have orders 1, 2, 2, 5, irreducible characters have degrees 1, 1, 2, 2. – Gerry Myerson Aug 8 '11 at 10:53
Also doesn't work for $S_4$ and $S_5$ (class lengths 1,3,6,6,8 vs degrees 1,1,2,3,3 and 1,10,15,20,20,24,30 vs 1,1,4,4,5,5,6). – Frieder Ladisch Aug 8 '11 at 12:26
What's the motivation for this question? – j.c. Aug 8 '11 at 12:38

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