A003923 -id:A003923 - cp's OEIS frontend

A003053 Order of orthogonal group O(n, GF(2)).

1, 2, 6, 48, 720, 23040, 1451520, 185794560, 47377612800, 24257337753600, 24815256521932800, 50821645356918374400, 208114637736580743168000, 1704875112338069448032256000, 27930968965434591767112450048000, 915241991059360703024740763172864000

Offset: 1

N. J. A. Sloane

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..41
J. H. Conway et al., ATLAS of Finite Groups, Chapter 2.
F. J. MacWilliams, Orthogonal matrices over finite fields, Amer. Math. Monthly, 76 (1969), 152-164.

Bisections give A003923 and A090770.

h:=proc(n) local m;
if n mod 2 = 0 then m:=n/2;
2^(m^2)*mul( 4^i-1, i=1..m);
else m:=(n+1)/2;
2^(m^2)*mul( 4^i-1, i=1..m-1);
fi;
end;
# This produces a(n+1)

h[n_] := Module[{m}, If[EvenQ[n], m = n/2; 2^(m^2)*Product[4^i-1, {i, 1, m}], m = (n+1)/2; 2^(m^2)*Product[4^i-1, {i, 1, m-1}]]];
a[n_] := h[n-1];
Array[a, 16] (* Jean-François Alcover, Aug 18 2022, after Maple code *)

a(n) = n--; if (n % 2, m = (n+1)/2; 2^(m^2)*prod(k=1, m-1, 4^k-1), m = n/2; 2^(m^2)*prod(k=1, m, 4^k-1)); \\ Michel Marcus, Jul 13 2017

def size_binary_orthogonal_group(n):
    k = n-1
    if k%2==0:
        m=k//2
        p=2**(m**2)
        for i in range(1,m+1):
            p*=4**i-1
    else:
        m=(k+1)//2
        p=2**(m**2)
        for i in range(1,m):
            p*=4**i-1
    return p
#call and print output for a(n)
print([size_binary_orthogonal_group(n) for n in range(1, 10)])
# Nathan J. Russell, Nov 01 2017

from math import prod
def A003053(n): return (1 << (n//2)**2)*prod((1 << i)-1 for i in range(2,2*((n-1)//2)+1,2)) # Chai Wah Wu, Jun 20 2022

For formulas see Maple code.

Asymptotics: a(n) ~ c * 2^((n^2-n)/2), where c = (1/4; 1/4)infinity ~ 0.6885375... is expressed in terms of the Q-Pochhammer symbol. - _Cedric Lorand, Aug 07 2017

Edited by N. J. A. Sloane, Dec 30 2008

Edited by W. Edwin Clark et al., Jan 15 2015

A090770 a(n) = 2^(n^2 + 2n + 1)*Product_{j=1..n} (4^j - 1).

Original entry on oeis.org

2, 48, 23040, 185794560, 24257337753600, 50821645356918374400, 1704875112338069448032256000, 915241991059360703024740763172864000, 7861748876453505095791592854589753555681280000, 1080506416218846625176535970968094253434513802154475520000, 2376056471052200653607636735377527394627947719754523173734842368000000

Offset: 0

N. J. A. Sloane, Feb 10 2004

nonn

The order of the p-Clifford group for an odd prime p is a*p^(n^2+2n+1)*Product_{j=1..n} (p^(2*j)-1), where a = gcd(p+1,4). This is the sequence obtained by (illegally) setting p = 2.

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

Cf. A001309, A003956, A100221.
Cf. A092299 and A092301 (p=3), A092300 and A089989 (p=5), A090768 and A090769 (p=7).
A bisection of A003053, cf. A003923.

Table[2^(n^2+2n+1) Product[4^j-1,{j,n}],{n,0,10}] (* Harvey P. Dale, May 14 2022 *)

from math import prod
def A090770(n): return prod((1<Chai Wah Wu, Jun 20 2022

a(n) ~ c * 2^((n+1)*(2*n+1)), where c = A100221. - Amiram Eldar, Jul 06 2025

A001308 Order of Chevalley group D_n (2).

Original entry on oeis.org

1, 36, 20160, 174182400, 23499295948800, 50027557148216524800, 1691555775522928280469504000, 911666827031785075278550369566720000, 7846393898179181843651374899795632943267840000

Offset: 1

N. J. A. Sloane

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.

T. D. Noe, Table of n, a(n) for n = 1..20
Index entries for sequences related to groups.

Cf. A003923, A100221.

for m from 0 to 25 do N:=2^(m*(m-1))*mul( 4^i-1, i=1..m-1)*(2^m-1); lprint(N); od:

d[q_, n_] := q^(n*(n-1)) * (q^n-1) * Product[q^(2*k) - 1, {k, 1, n-1}]; Table[d[2, n], {n, 1, 9}] (* Amiram Eldar, Jul 07 2025 *)

a(n) = 2^(n*(n-1)) * (2^n - 1) * Product_{i=1..n-1} (2^(2*i) - 1).

a(n) ~ c * 2^(n*(2*n-1)), where c = A100221. - Amiram Eldar, Jul 07 2025

A144545 a(n) = 2^(n(n-1))(2^n + 1)*Product_{i=1..n-1} (4^i - 1).

Original entry on oeis.org

2, 3, 60, 25920, 197406720, 25015379558400, 51615733565620224000, 1718194449153210615595008000, 918817155086936330770931156779008000, 7877103854727828347931810809383874168094720000, 1081561598265935342583934931877242782978883444539392000000

Offset: 0

N. J. A. Sloane, Dec 30 2008

nonn

T. D. Noe, Table of n, a(n) for n = 0..20

Cf. A003923, A001308, A003053, A100221.

g:=m->2^(m*(m-1))*mul( 4^i-1, i=1..m-1)*(2^m+1);

a[n_] := 2^(n*(n-1))*(2^n + 1) * Product[4^i - 1, {i, 1, n-1}]; Array[a, 10, 0] (* Amiram Eldar, Jul 07 2025 *)

from math import prod
def A144545(n): return ((1<Chai Wah Wu, Jun 20 2022

a(n) ~ c * 2^(2*n^2-n), where c = A100221. - Amiram Eldar, Jul 07 2025

A346848 Number of conjugacy classes of the symplectic group Sp(2n, 2) over the field with 2 elements.

Original entry on oeis.org

1, 3, 11, 30, 81, 198, 477, 1089, 2451, 5358, 11567, 24537, 51577, 107205, 221378, 453900, 926395, 1882152, 3812232, 7699191, 15518112, 31220991, 62733296, 125911851, 252516626, 506082933, 1013780968, 2029989807, 4063678159, 8132877129, 16274093175

Offset: 0

Jan Kristian Haugland, Aug 06 2021

nonn

Sp(2n, 2) is isomorphic to the orthogonal group O(2n+1, 2) over the field with 2 elements, and is a simple and complete group for n>=3.

			a(2)=11, and Sp(4, 2) is isomorphic to the symmetric group S_6 which has 11 conjugacy classes.

Jan Kristian Haugland, Table of n, a(n) for n = 0..1000
G. E. Wall, On the conjugacy classes in the unitary, symplectic and orthogonal groups, J. Aust. Math. Soc., 3 (1963), 1-62.

Discrete convolution of A070933 and A098613. A003923 gives the order of the group.

cp's OEIS Frontend

A003053 Order of orthogonal group O(n, GF(2)).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Python

Formula

Extensions

A090770 a(n) = 2^(n^2 + 2n + 1)*Product_{j=1..n} (4^j - 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

A001308 Order of Chevalley group D_n (2).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A144545 a(n) = 2^(n*(n-1))*(2^n + 1)*Product_{i=1..n-1} (4^i - 1).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A346848 Number of conjugacy classes of the symplectic group Sp(2n, 2) over the field with 2 elements.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A144545 a(n) = 2^(n(n-1))(2^n + 1)*Product_{i=1..n-1} (4^i - 1).