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A061064 Maximal number of zeros in the character table of a group with n elements.

%I A061064 #12 Oct 13 2024 00:17:36
%S A061064 0,0,0,0,0,1,0,3,0,2,0,4,0,3,0,12,0,9,0,8,4,5,0,27,0,6,16,12,0,25,0,
%T A061064 48,0,8,0,36,0,9,8,75,0,49,0,20,0,11,0,108,0,50,0,24,0,81,8,147,12,14,
%U A061064 0,100,0,15,36,192,0,121,0,32,0,98,0,243,0,18,16
%N A061064 Maximal number of zeros in the character table of a group with n elements.
%C A061064 A finite non-Abelian group G has an irreducible representation of degree >= 2 and the character of such representation always has a zero; so a(n) = 0 iff every group of order n is Abelian, i.e. n belongs to A051532.
%H A061064 Eric M. Schmidt, <a href="/A061064/b061064.txt">Table of n, a(n) for n = 1..1023</a>
%e A061064 a(6) = 1 because the character table of the symmetric group S_3 is / 1, 1, 1 / 1, 1, -1 / 2, -1, 0 /.
%o A061064 (GAP) A061064 := function(n) local max, i; max := 0; for i in [1..NumberSmallGroups(n)] do max := Maximum(max, Sum(Irr(SmallGroup(n,i)), chi->Number(chi, x->x=0))); od; return max; end; # _Eric M. Schmidt_, Aug 24 2012
%Y A061064 Cf. A051532.
%K A061064 nonn,nice
%O A061064 1,8
%A A061064 Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 05 2001
%E A061064 Added terms a(n) for n>=24, _Eric M. Schmidt_, Aug 24 2012.