# What are the finite groups with 8 or 16 conjugacy classes?

What are the list of finite groups with 8 or 16 conjugacy classes?

I learned that dihedral groups $D_{10}$ and $D_{13}$ have 8 conjugacy classes. (Here the order of these groups are $|D_{10}|=20$, $|D_{13}|=26$. Or some people may denote $D_{10}$ as $D_{20}$.) Of course we have trivial examples $Z_8$ and $Z_{16}$ have 8 or 16 conjugacy classes for each.

Are there other examples of non-Abelian groups with 8 or 16 conjugacy classes? I am mostly interested in the non-Abelian groups. Thank you. :o)

Add: Partial answers are fine. (Such as answering what Jack Schmidt points out there are 18 isomorphism classes of 8 conjugacy classes.)

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A computer search reveals that there are 53 nonabelian groups of order $\leq 100$ which have 8 or 16 conjugacy classes. – Dane Apr 17 '14 at 22:48
In addition to the dihedral group of order 20, which you mentioned, the dicyclic group of order 20 (see here) also has 8 conjugacy classes. – Dane Apr 17 '14 at 22:53
There are exactly 18 isomorphism classes of nonabelian groups with 8 conjugacy classes. I believe the groups with 16 conjugacy classes have not yet been classified. – Jack Schmidt Apr 17 '14 at 22:53
Thanks Jack and Dane, that is very helpful. Do you know the list of these 18 isomorphism classes of nonabelian groups with 8 conjugacy classes? This part of answer will be nice enough. :o) – annie heart Apr 18 '14 at 0:05

Here are the 101* finite nonabelian groups with 16 conjugacy classes whose order is less than 2000. (*I don't remember if there are any exceptional orders left out of the census.) The names are those given by GAP's structure description, and have not been cleaned up for this answer since there are so many groups.

• SmallGroup(40,1) = C5 : C8
• SmallGroup(40,5) = C4 x D10
• SmallGroup(40,7) = C2 x (C5 : C4)
• SmallGroup(40,13) = C2 x C2 x D10
• SmallGroup(48,31) = C4 x A4
• SmallGroup(48,49) = C2 x C2 x A4
• SmallGroup(52,1) = C13 : C4
• SmallGroup(52,4) = D52
• SmallGroup(58,1) = D58
• SmallGroup(64,41) = (C16 : C2) : C2
• SmallGroup(64,42) = (C16 : C2) : C2
• SmallGroup(64,43) = C2 . ((C8 x C2) : C2) = C8 . (C4 x C2)
• SmallGroup(64,46) = C16 : C4
• SmallGroup(64,134) = ((C4 x C4) : C2) : C2
• SmallGroup(64,135) = ((C4 x C4) : C2) : C2
• SmallGroup(64,136) = ((C4 x C4) : C2) : C2
• SmallGroup(64,137) = ((C4 x C4) : C2) : C2
• SmallGroup(64,138) = (((C4 x C2) : C2) : C2) : C2
• SmallGroup(64,139) = (((C4 x C2) : C2) : C2) : C2
• SmallGroup(64,149) = (C2 x (C8 : C2)) : C2
• SmallGroup(64,150) = (C2 x (C8 : C2)) : C2
• SmallGroup(64,151) = (C2 x Q16) : C2
• SmallGroup(64,152) = (C2 x QD16) : C2
• SmallGroup(64,153) = (C2 x D16) : C2
• SmallGroup(64,154) = (C2 x Q16) : C2
• SmallGroup(64,170) = (Q8 : C4) : C2
• SmallGroup(64,171) = ((C8 x C2) : C2) : C2
• SmallGroup(64,172) = (C2 x C2) . (C2 x D8) = (C4 x C2) . (C2 x C2 x C2)
• SmallGroup(64,177) = (C2 x D16) : C2
• SmallGroup(64,178) = (C2 x Q16) : C2
• SmallGroup(64,182) = C8 : Q8
• SmallGroup(64,190) = (C2 x D16) : C2
• SmallGroup(64,191) = (C2 x Q16) : C2
• SmallGroup(96,66) = SL(2,3) : C4
• SmallGroup(96,67) = SL(2,3) : C4
• SmallGroup(96,68) = C2 x ((C4 x C4) : C3)
• SmallGroup(96,188) = C2 x (C2 . S4 = SL(2,3) . C2)
• SmallGroup(96,189) = C2 x GL(2,3)
• SmallGroup(96,192) = (C2 . S4 = SL(2,3) . C2) : C2
• SmallGroup(96,229) = C2 x ((C2 x C2 x C2 x C2) : C3)
• SmallGroup(100,13) = D10 x D10
• SmallGroup(112,41) = C2 x ((C2 x C2 x C2) : C7)
• SmallGroup(120,39) = A4 x D10
• SmallGroup(136,3) = C17 : C8
• SmallGroup(136,13) = C2 x (C17 : C4)
• SmallGroup(144,184) = A4 x A4
• SmallGroup(156,1) = (C13 : C4) : C3
• SmallGroup(156,8) = C2 x ((C13 : C3) : C2)
• SmallGroup(160,235) = C2 x ((C2 x C2 x C2 x C2) : C5)
• SmallGroup(192,201) = (((C2 x D8) : C2) : C3) : C2
• SmallGroup(192,202) = ((((C4 x C2) : C2) : C2) : C2) : C3
• SmallGroup(196,8) = (C7 x C7) : C4
• SmallGroup(216,88) = ((C3 x C3) : C3) : Q8
• SmallGroup(216,96) = ((C18 x C2) : C3) : C2
• SmallGroup(216,99) = ((C6 x C6) : C3) : C2
• SmallGroup(240,191) = ((C2 x C2 x C2 x C2) : C5) : C3
• SmallGroup(312,51) = ((C26 x C2) : C3) : C2
• SmallGroup(336,210) = C2 x (((C2 x C2 x C2) : C7) : C3)
• SmallGroup(366,1) = (C61 : C3) : C2
• SmallGroup(384,5677) = ((((C4 x C4) : C3) : C2) : C2) : C2
• SmallGroup(384,5678) = ((((C2 x C2 x C2 x C2) : C3) : C2) : C2) : C2
• SmallGroup(384,5863) = ((C2 x ((C2 x C2 x C2 x C2) : C2)) : C2) : C3
• SmallGroup(384,5864) = (((C2 x C2 x Q8) : C2) : C2) : C3
• SmallGroup(384,5865) = ((C2 x C2 x C2) . (C2 x C2 x C2 x C2)) : C3
• SmallGroup(384,5866) = ((C2 x Q8) : Q8) : C3
• SmallGroup(384,18133) = ((((C4 x C4) : C2) : C2) : C3) : C2
• SmallGroup(400,134) = ((C5 x C5) : C4) : C4
• SmallGroup(400,207) = (((C5 x C5) : C4) : C2) : C2
• SmallGroup(400,212) = C2 x ((C5 x C5) : Q8)
• SmallGroup(406,1) = (C29 : C7) : C2
• SmallGroup(448,178) = (C4 x C4 x C4) : C7
• SmallGroup(448,1393) = (C2 x C2 x C2 x C2 x C2 x C2) : C7
• SmallGroup(448,1394) = (C2 x C2 x C2 x C2 x C2 x C2) : C7
• SmallGroup(576,8654) = ((A4 x A4) : C2) : C2
• SmallGroup(576,8661) = (C2 x C2 x C2 x C2 x C2 x C2) : C9
• SmallGroup(588,34) = ((C7 x C7) : C4) : C3
• SmallGroup(600,55) = ((C5 x C5) : C3) : C8
• SmallGroup(600,151) = (((C5 x C5) : C4) : C3) : C2
• SmallGroup(600,152) = C2 x (((C5 x C5) : C3) : C4)
• SmallGroup(610,1) = (C61 : C5) : C2
• SmallGroup(640,21454) = C2 . (((C2 x C2 x C2 x C2) : C5) : C4) = (((C2 x Q8) : C2) : C5) . C4
• SmallGroup(640,21455) = (((C2 x Q8) : C2) : C5) : C4
• SmallGroup(672,1044) = SL(2,7) : C2
• SmallGroup(672,1045) = C2 . (PSL(3,2) : C2) = SL(2,7) . C2
• SmallGroup(672,1257) = (C2 x C2 x ((C2 x C2 x C2) : C7)) : C3
• SmallGroup(864,2666) = ((C2 x ((C3 x C3) : C4)) : C4) : C3
• SmallGroup(864,4666) = (C2 x C2 x ((C3 x C3) : Q8)) : C3
• SmallGroup(1200,947) = (((C5 x C5) : Q8) : C3) : C2
• SmallGroup(1200,950) = C2 x (((C5 x C5) : Q8) : C3)
• SmallGroup(1320,134) = C2 x PSL(2,11)
• SmallGroup(1344,815) = ((C4 x C4 x C4) : C7) : C3
• SmallGroup(1344,11690) = ((C2 x C2 x C2 x C2 x C2 x C2) : C7) : C3
• SmallGroup(1344,11691) = ((C2 x C2 x C2 x C2 x C2 x C2) : C7) : C3
• SmallGroup(1440,4595) = A6 : C4
• SmallGroup(1440,5844) = C2 x (A6 . C2)
• SmallGroup(1452,20) = ((C11 x C11) : C3) : C4
• SmallGroup(1728,47862) = ((C2 x C2 x C2 x C2 x C2 x C2) : C9) : C3
• SmallGroup(1920,240998) = (C2 x C2 x C2 x C2 x C2) : A5
• SmallGroup(1920,240999) = C2 . ((C2 x C2 x C2 x C2) : A5)
• SmallGroup(1920,241004) = ((C2 x Q8) : C2) : A5
• SmallGroup(1944,803) = ((C3 x C3) . ((C3 x C3) : C3) = (C3 x C3 x C3) . (C3 x C3)) : C8
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Could both cases (8 and 16 conjugacy classes) not fit in one answer? Why two answers? – Jeppe Stig Nielsen Apr 18 '14 at 6:27
The answers are very different. The 8 class answer is correct, contains a bibliography, has been verified by both computer and theory by several people. The 16 class answer (this one) is incomplete, has no theory, has not been double-checked, and has no bibliography. By combining the two answers I would associate Vera-López–Vera-López with this shoddy answer. V-L^2 have written a series of papers on this topic, and their techniques should be used to more completely answer this question, but have not been (mostly because the complete answer is beyond us right now). – Jack Schmidt Apr 18 '14 at 11:26
Thanks Jack, two answers are fine. I am grateful to them. :o) – annie heart Apr 18 '14 at 19:54

Here are the 18 isomorphism classes of finite groups with 8 conjugacy classes:

• SmallGroup( 20, 1) = $\operatorname{AGL}(1,5)$
• SmallGroup( 20, 4) = $D_{10}$
• SmallGroup( 24, 13) = $C_2 \times A_4$
• SmallGroup( 26, 1) = $D_{13}$
• SmallGroup( 48, 3) = $C_3 \ltimes (C_4 \times C_4)$
• SmallGroup( 48, 28) = $\operatorname{SL}(2,3) \mathsf{Y} C_4$
• SmallGroup( 48, 29) = $\operatorname{GL}(2,3)$
• SmallGroup( 48, 50) = $C_3 \ltimes (C_2^4)$
• SmallGroup( 56, 11) = $\operatorname{AGL}(1,8)$
• SmallGroup( 68, 3) = $C_4 \ltimes C_{17}$
• SmallGroup( 78, 1) = $C_6 \ltimes C_{13}$
• SmallGroup( 80, 49) = $C_5 \ltimes C_2^4$
• SmallGroup(168, 43) = $\operatorname{A\Gamma L}(1,8)$
• SmallGroup(200, 44) = $Q_8 \ltimes (C_5 \times C_5)$
• SmallGroup(300, 23) = $C_4 \ltimes C_3 \ltimes C_5^2$
• SmallGroup(600,150) = $\operatorname{SL}(2,3) \ltimes C_5^2$
• SmallGroup(660, 13) = $\operatorname{PSL}(2,11)$
• SmallGroup(720,765) = $M_{10}$, the Mathieu group on 10 points

This list (with a computer-free proof of correctness) can be found on page 310 of (Vera-López–Vera-López, 1985).

• Vera López, Antonio; Vera López, Juan. “Classification of finite groups according to the number of conjugacy classes.” Israel J. Math. 51 (1985), no. 4, 305–338. MR804489 DOI:10.1007/BF02764723
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Dear Jack, thanks very much for the nice answer. Maybe you also know this simpler question? math.stackexchange.com/questions/759647/… – annie heart Apr 18 '14 at 20:17
– annie heart Apr 18 '14 at 20:38