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 A226193 The number of medial quasigroups of order n, up to isomorphism. 2
 1, 1, 5, 13, 19, 5, 41, 73, 116, 19, 109, 65, 155, 41, 95, 669, 271, 116, 341, 247, 205, 109, 505, 365, 1084, 155, 1574, 533, 811, 95, 929, 4193, 545, 271, 779, 1508, 1331, 341, 775, 1387, 1639, 205, 1805, 1417, 2204, 505, 2161, 3345, 4388, 1084, 1355, 2015, 2755, 1574, 2071, 2993, 1705, 811, 3421, 1235, 3659, 929, 4756 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS See the Wikipedia link for "Medial" for definitions. This article also contains the Bruck-Toyoda theorem which characterizes medial quasigroups in terms of abelian groups. LINKS David Stanovsky, Table of n, a(n) for n = 1..63 David Stanovský, Petr Vojtechovský, Central and medial quasigroups of small order, arxiv preprint arXiv:1511.03534 [math.GR], 2015. Wikipedia, Medial MAPLE a:=proc(n) if n = 1 then      return 1;   else return MAGMA:-Enumerate(n, 'medial', 'quasigroup'); end if; end proc; PROG # GAP, gives the number of medial quasigroups over SmallGroup(n, k) LoadPackage("loops"); MQ := function( n, k )     local G, ct, elms, inv, A, f_reps, count, f, Cf, O, g_reps, g, Cfg, W, unused, c, Wc;     G := SmallGroup( n, k );     G := IntoLoop( G );     ct := CayleyTable( G );     elms := Elements( G );     inv := List( List( [1..n], i -> elms[i]^(-1) ), x -> x![1] );     A := AutomorphismGroup( G );     f_reps := List( ConjugacyClasses( A ), Representative );     count := 0;     for f in f_reps do         Cf := Centralizer( A, f );         O := OrbitsDomain( Cf, A );         g_reps := List( O, x -> x[1] );         for g in g_reps do             Cfg := Intersection( Cf, Centralizer( A, g ) );             W := Set( [1..n], w -> ct[w][ inv[ ct[w^f][w^g] ] ] );             unused := [1..n];             while not IsEmpty( unused ) do                 c := unused[1];                 if f*g=g*f then count := count + 1; fi;                 if Size(W) = Length(unused) then                     unused := [];                 else                     Wc := Set( W, w -> ct[w][c] );                     Wc := Union( Orbits( Cfg, Wc ) );                     unused := Difference( unused, Wc );                 fi;             od;         od;     od;     return count; end; # David Stanovsky, Nov 12 2015 CROSSREFS Sequence in context: A190432 A197563 A022138 * A028274 A272723 A245177 Adjacent sequences:  A226190 A226191 A226192 * A226194 A226195 A226196 KEYWORD nonn,hard,mult AUTHOR W. Edwin Clark, May 30 2013 EXTENSIONS a(9)-a(63) from David Stanovsky, Nov 12 2015 STATUS approved

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Last modified December 9 16:42 EST 2019. Contains 329879 sequences. (Running on oeis4.)