A364783 Order of the symplectic group of 6 X 6 matrices over Z_n.

1, 1451520, 9170703360, 3044058071040, 457002000000000, 13311459341107200, 546914437209907200, 6383852471797678080, 95928796265538862080, 663347543040000000000, 7338585441586912128000, 27916153580121646694400, 245593958671812227742720, 793857243898924498944000

Offset: 1

Author

Robin Visser, Aug 07 2023

Keywords

• nonn
• mult

Comments

Let M be any fixed nonsingular skew-symmetric 6 X 6 matrix over the integers mod n. Then a(n) is the number of invertible 6 X 6 matrices A over the integers mod n such that A^T * M * A = M, where A^T denotes the transpose of A.

Links

• E. Artin, Geometric Algebra, Wiley Classics Library. John Wiley & Sons, Inc., New York, 1988. Reprint of the 1957 original, A Wiley-Interscience Publication.
• Larry C. Grove, Classical Groups and Geometric Algebra, Grad. Stud. Math., 39 American Mathematical Society, Providence, RI, 2002. x+169 pp.

Crossrefs

Cf. A364771, A364782.

Programs

Mathematica

f[p_, e_] := p^(21*e - 12)*(p^2 - 1)*(p^4 - 1)*(p^6 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 15] (* Amiram Eldar, Aug 08 2023 *)

Sage

def a(n):
    return product([p^(21*n.valuation(p) - 12)*(p^2 - 1)*(p^4 - 1)*(p^6 - 1)
        for p in n.prime_factors()])

Formula

a(n) = Product_{primes p dividing n} p^(21*v_p(n) - 12)*(p^2 - 1)*(p^4 - 1)*(p^6 - 1), where v_p(n) is the largest power k such that p^k divides n.

For primes p : a(p) = A003932(n), where A246655(n) = p.

Sum_{k=1..n} a(k) ~ c * n^22 / 22, where c = Product_{p prime} (1 - 1/p^3 - 1/p^5 + 1/p^9 + 1/p^11 - 1/p^13) = 0.8006965549... . - Amiram Eldar, Aug 08 2023