A178795 -id:A178795 - cp's OEIS frontend

A008868 Order of simple Chevalley group E_8 (q), q = prime power.

337804753143634806261388190614085595079991692242467651576160959909068800000, 18830052912953932311099032439972660332140886784940152038522449391826616580150109878711243949982163694448626420940800000, 191797292142671717754639757897512906421357507604216557533558287598236977154127870984484770345340348298409697395609822849492217656441474908160000000000

Offset: 1

N. J. A. Sloane

Coxeter and Moser p. 131 remark that the first term in this sequence is comparable to Eddington's estimate of the number of protons in the universe. See also the comment in A004231!

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.

Cf. A003133, A004231, A178795.

q^120*(q^2-1)*(q^8-1)*(q^12-1)*(q^14-1)*(q^18-1)*(q^20-1)*(q^24-1)*(q^30-1);

Table[q^120*(q^2-1)*(q^8-1)*(q^12-1)*(q^14-1)*(q^18-1)*(q^20-1)*(q^24-1)*(q^30-1), {q, Select[Range[4], PrimePowerQ]}] (* Jean-François Alcover, Aug 24 2022 *)

a(3) added by N. J. A. Sloane, Sep 16 2008

A178779 Expansion of the polynomial x^12(x^6-1)(x^4-1)(x^3-1)(x-1) in increasing powers of x.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, -1, 0, 1, -1, 2, -1, 1, 0, -1, 0, -1, 1

Offset: 0

N. J. A. Sloane, Dec 26 2010

q^24*(q^12-1)*(q^8-1)*(q^6-1)*(q^2-1) is the order of the simple group F_4(q), if q is a prime power.

The x-polynomial f(x) and the q-polynomial g(q) are such that g(q) = f(q^2) - Jean-François Alcover, Aug 25 2022

R. L. Griess, Jr., Twelve Sporadic Groups, Springer, 1998; see p. 169.

Cf. A008913, A178795.

Vecrev(x^12*(x^6-1)*(x^4-1)*(x^3-1)*(x-1)) \\ Michel Marcus, Aug 25 2022

A178781 Expansion of the polynomial (x^9-1)(x^7-1)(x^6-1)(x^5-1)(x^4-1)(x^3-1)(x-1) in increasing powers of x.

Original entry on oeis.org

-1, 1, 0, 1, 0, 0, 0, -1, -1, 0, -1, 0, 1, 0, 2, 0, 1, 0, 0, -1, 0, -2, 0, -1, 0, 1, 0, 1, 1, 0, 0, 0, -1, 0, -1, 1

Offset: 0

N. J. A. Sloane, Dec 26 2010

q^63*(q^18-1)*(q^14-1)*(q^12-1)*(q^10-1)*(q^8-1)*(q^6-1)*(q^2-1) is the order of the simple group E_7(q), if q is a prime power.

The x-polynomial f(x) and the q-polynomial g(q) are such that g(q) = q^63*f(q^2) - Jean-François Alcover, Aug 25 2022

R. L. Griess, Jr., Twelve Sporadic Groups, Springer, 1998; see p. 169.

Cf. A008869, A178795.

Vecrev((x^9-1)*(x^7-1)*(x^6-1)*(x^5-1)*(x^4-1)*(x^3-1)*(x-1)) \\ Michel Marcus, Aug 25 2022

A178780 Expansion of the polynomial x^36(x^12-1)(x^9-1)(x^8-1)(x^6-1)(x^5-1)(x^2-1) in increasing powers of x.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, -1, -1, 1, 0, -1, 1, 2, -1, 0, 3, 0, -2, 1, 1, -3, -1, 2, -1, -3, 1, 1, -2, 0, 3, 0, -1, 2, 1, -1, 0, 1, -1, -1, 0, 0, -1, 0, 1

Offset: 0

N. J. A. Sloane, Dec 26 2010

q^36*(q^12-1)*(q^9-1)*(q^8-1)*(q^6-1)*(q^5-1)*(q^2-1) is the order of the simple group E_6(q), if q is a prime power.

R. L. Griess, Jr., Twelve Sporadic Groups, Springer, 1998; see p. 169.

Cf. A008871, A178779, A178781, A178795.

A178798 Expansion of the polynomial x^36(x^12-1)(x^9+1)(x^8-1)(x^6-1)(x^5-1)(x^2-1) in increasing powers of x.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 1, 1, -1, 0, -1, -1, 0, 1, 0, -1, 2, 0, -1, -1, 1, -1, 0, 1, -1, 1, 1, 0, -2, 1, 0, -1, 0, 1, 1, 0, 1, -1, -1, 0, 0, -1, 0, 1

Offset: 0

N. J. A. Sloane, Dec 26 2010

x^36*(x^12-1)*(x^9+1)*(x^8-1)*(x^6-1)*(x^5-1)*(x^2-1) is the order of the twisted Chevalley group 2_E_6 (q), q = prime power.

R. L. Griess, Jr., Twelve Sporadic Groups, Springer, 1998; see p. 169.

Cf. A008915, A008868, A178795.

cp's OEIS Frontend

A008868 Order of simple Chevalley group E_8 (q), q = prime power.

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A178779 Expansion of the polynomial x^12*(x^6-1)*(x^4-1)*(x^3-1)*(x-1) in increasing powers of x.

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A178781 Expansion of the polynomial (x^9-1)*(x^7-1)*(x^6-1)*(x^5-1)*(x^4-1)*(x^3-1)*(x-1) in increasing powers of x.

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A178780 Expansion of the polynomial x^36*(x^12-1)*(x^9-1)*(x^8-1)*(x^6-1)*(x^5-1)*(x^2-1) in increasing powers of x.

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A178798 Expansion of the polynomial x^36*(x^12-1)*(x^9+1)*(x^8-1)*(x^6-1)*(x^5-1)*(x^2-1) in increasing powers of x.

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A178779 Expansion of the polynomial x^12(x^6-1)(x^4-1)(x^3-1)(x-1) in increasing powers of x.

A178781 Expansion of the polynomial (x^9-1)(x^7-1)(x^6-1)(x^5-1)(x^4-1)(x^3-1)(x-1) in increasing powers of x.

A178780 Expansion of the polynomial x^36(x^12-1)(x^9-1)(x^8-1)(x^6-1)(x^5-1)(x^2-1) in increasing powers of x.

A178798 Expansion of the polynomial x^36(x^12-1)(x^9+1)(x^8-1)(x^6-1)(x^5-1)(x^2-1) in increasing powers of x.