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Order of the symplectic group of 4 X 4 matrices over Z_n.
1

%I #13 Aug 07 2023 02:12:37

%S 1,720,51840,737280,9360000,37324800,276595200,754974720,3061100160,

%T 6739200000,25721308800,38220595200,137037962880,199148544000,

%U 485222400000,773094113280,2008994088960,2203992115200,6114035779200,6900940800000,14338695168000,18519342336000,41348052472320

%N Order of the symplectic group of 4 X 4 matrices over Z_n.

%C Let M be any fixed nonsingular skew-symmetric 4 X 4 matrix over the integers mod n. Then a(n) is the number of invertible 4 X 4 matrices A over the integers mod n such that A^T * M * A = M, where A^T denotes the transpose of A.

%H E. Artin, <a href="https://doi.org/10.1002/9781118164518">Geometric Algebra</a>, Wiley Classics Library. John Wiley & Sons, Inc., New York, 1988. Reprint of the 1957 original, A Wiley-Interscience Publication.

%H Larry C. Grove, <a href="https://doi.org/10.1090/gsm/039">Classical Groups and Geometric Algebra</a>, Grad. Stud. Math., 39 American Mathematical Society, Providence, RI, 2002. x+169 pp.

%F a(n) = Product_{primes p dividing n} p^(10*v_p(n) - 6)*(p^2 - 1)*(p^4 - 1), where v_p(n) is the largest power k such that p^k divides n.

%F For primes p : a(p) = A003931(n), where A246655(n) = p.

%F Sum_{k=1..n} a(k) ~ c * n^11 / 11, where c = Product_{p prime} (1 - (p^4+p^2-1)/p^7) = 0.8062434963... . - _Amiram Eldar_, Aug 07 2023

%t f[p_, e_] := p^(10*e - 6)*(p^2 - 1)*(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* _Amiram Eldar_, Aug 07 2023 *)

%o (Sage)

%o def a(n):

%o return product([p^(10*n.valuation(p)-6)*(p^2 - 1)*(p^4 - 1)

%o for p in n.prime_factors()])

%Y Cf. A011786, A305186.

%K nonn,mult

%O 1,2

%A _Robin Visser_, Aug 06 2023