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A346008 Order of the full automorphism group of an n^2 X n^2 Sudoku puzzle.

%I A346008 #34 Oct 27 2021 14:33:01
%S A346008 1,128,3359232,126806761930752,17832200896512000000000000,
%T A346008 20122639448358307421277388800000000000000,
%U A346008 346671850578027965617950152200042758191185920000000000000000,158635147791426908154211087484339310324630213259159597497553256448000000000000000000,3135383389315524601627656266493367412334920325664589642523187933340624422000766361791574835200000000000000000000
%N A346008 Order of the full automorphism group of an n^2 X n^2 Sudoku puzzle.
%C A346008 a(n) is the order of the automorphism group of the n^2 X n^2 Sudoku graph (see A182866).
%F A346008 a(n) = 2*(n!)^(2n+2) for n > 1.
%e A346008 For n=2, a(2) = 128 is the number of symmetries of a Shidoku puzzle.
%e A346008 For n=3, a(3) = 3359232 is the number of symmetries of standard 9 X 9 Sudoku puzzle.
%t A346008 Join[{1},Table[2*n!^(2*n+2),{n,2,9}]] (* _Stefano Spezia_, Jul 27 2021 *)
%o A346008 (SageMath)
%o A346008 M = matrix(n^4,n^4)
%o A346008 for i in [0..n^4-1]:
%o A346008     for j in [0..n^4-1]:
%o A346008         if i!=j:
%o A346008             if i%n^2==j%n^2:
%o A346008                 M[i,j]=1
%o A346008             if floor(i/n^2)==floor(j/n^2):
%o A346008                 M[i,j]=1
%o A346008 D = Graph(M, format='adjacency_matrix')
%o A346008 for col in [0..n-1]:
%o A346008     for row in [0..n-1]:
%o A346008         tl = n*col + n^3*row
%o A346008         s = []
%o A346008         for i in [0..n-1]:
%o A346008             for j in [0..n-1]:
%o A346008                 s.append(tl + i + n^2*j)
%o A346008         D.add_clique(s)
%o A346008 print(D.automorphism_group().order())
%Y A346008 Cf. A159299.
%K A346008 nonn
%O A346008 1,2
%A A346008 _Carl-Fredrik Nyberg Brodda_, _Shay Jordan_, Jul 27 2021