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 A064527 Numbers n such that there exists a finite group G of order n such that all entries in its character table are integers. 1
 1, 2, 4, 6, 8, 12, 16, 18, 24, 32, 36, 48, 54, 64, 72, 96, 108, 120, 128, 144, 162, 192, 200, 216, 240, 256, 288, 324, 384, 400, 432, 480, 486, 512, 576, 648, 720, 768, 800, 864, 960, 972, 1024, 1152, 1200, 1296, 1440, 1458, 1536, 1600, 1728, 1920, 1944 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The list contains all numbers of the form 2^w*3^u for w> 0, u>=0. But it also contains 120, 200, 240 and 400. It contains n! for all n because the symmetric groups have integral character tables. By taking direct products, we get all numbers of the form n! * 2^w * 3^u, w > 0, u >= 0. The 200 comes from a semidirect product of an elementary group of order 25 with a quaternion group of order 8, with fixed-point-free action (a Frobenius group). - Derek Holt From Eric M. Schmidt, Feb 22 2013: (Start) A group of order n has integral character table iff g^m is conjugate to g for all group elements g and all m coprime to n. A necessary condition for a group G to have an integral character table is for G/G' to be an elementary Abelian 2-group. Therefore, by the Feit-Thompson theorem, the only odd term in this sequence is 1. R. Gow proved (see reference) that no prime greater than 5 can divide the order of a solvable group with integral character table. (End) REFERENCES Roderick Gow, Groups whose characters are rational-valued, J. Algebra 40 (1976) 280-299. Hegedus Pal, Structure of Solvable Rational Groups, Proc. London Math. Soc. (2005) 90 (2): 439-471. LINKS PROG (GAP) HasIntegerCharTable := function(G) local cc, ccr, e; cc := ConjugacyClasses(G); ccr := List(cc, Representative); e := Exponent(G); return ForAll([2..e-1], m->(not (IsPrimeInt(m) and GcdInt(m, e)=1)) or ForAll([1..Length(cc)], j->ccr[j]^m in cc[j])); end; A064527 := function(max) local res, i, j; res := [1]; for i in [2, 4..max-(max mod 2)] do if ForAny(res, j->i/j in res) then Add(res, i); continue; fi; for j in [1..NumberSmallGroups(i)] do if HasIntegerCharTable(SmallGroup(i, j)) then Add(res, i); continue; fi; od; od; return res; end; # Eric M. Schmidt, Feb 22 2013 CROSSREFS Contains A000142 and A007694. Sequence in context: A067946 A227270 A145853 * A007694 A219653 A050622 Adjacent sequences:  A064524 A064525 A064526 * A064528 A064529 A064530 KEYWORD nonn,nice AUTHOR Tim Brooks (tim_brooks(AT)my-deja.com), Oct 07 2001 EXTENSIONS More terms from Derek Holt (mareg(AT)csv.warwick.ac.uk), Oct 07, 2001 More terms from Eric M. Schmidt, Feb 22 2013 STATUS approved

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