Which finite groups have exactly 2 conjugacy classes of the same order and exactly 2 irreducible characters of the same degree?

Dears

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?

edited Aug 8 2011 at 12:40

jc
3,885●9●23

asked Aug 8 2011 at 9:55

Hesam Sharifi
11●1

Changed the tags, as the question has nothing to do with characteristic classes. – Mark Grant Aug 8 2011 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 2011 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 2011 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). – F. Ladisch Aug 8 2011 at 12:26

What's the motivation for this question? mathoverflow.net/howtoask#motivation – jc Aug 8 2011 at 12:38

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