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Number of pointed groups of order n: that is, Sum_{G = group of order n} Number of conjugacy classes in G.
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%I #9 Nov 28 2023 16:51:54

%S 1,2,3,8,5,9,7,34,18,14,11,40,13,19,15,161,17,57,19,61,26,29,23,197,

%T 50,34,103,76,29,66,31,912,33,44,35,267,37,49,46,293,41,107,43,116,90,

%U 59,47,1096,98,148,51,143,53,371,62,377,66,74,59,325,61,79,156,7068

%N Number of pointed groups of order n: that is, Sum_{G = group of order n} Number of conjugacy classes in G.

%C Number of pairs (G, g in G) for G a group of order n.

%C This has the same relation to A000001 (groups) as A000081 (pointed trees, also called rooted trees) does to trees (A000055).

%H Klaus Brockhaus, <a href="/A126102/b126102.txt">Table of n, a(n) for n=1..255</a>

%o (Magma) SmallGroupDatabase();

%o for o in [1..64] do

%o t1:=0;

%o t2:=NumberOfSmallGroups(o);

%o for n in [1..t2] do

%o G:=Group(o,n);

%o t1:=t1 + #ConjugacyClasses(G);

%o end for;

%o print(t1);

%o end for;

%o (Magma) D:=SmallGroupDatabase(); [ &+[ #ConjugacyClasses(Group(D, o, n)): n in [1..NumberOfSmallGroups(D, o)] ]: o in [1..64] ]; /* Klaus Brockhaus, Mar 06 2007 */

%Y Cf. A000001 (groups). See A126103 for a different and better version.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Mar 06 2007