A003950 -id:A003950 - cp's OEIS frontend

A163347 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

1, 8, 56, 392, 2744, 19180, 134064, 937104, 6550320, 45786384, 320044452, 2237094216, 15637173048, 109303031880, 764022547512, 5340478146444, 37329666414768, 260932440209616, 1823904280240560, 12748996716570576

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

nonn

The initial terms coincide with those of A003950, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, 6, 6, 6, -21).

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^5)/(1-7*x+27*x^5-21*x^6) )); // G. C. Greubel, May 12 2019

CoefficientList[Series[(1+x)*(1-x^5)/(1-7*x+27*x^5-21*x^6), {x, 0, 30}], x] (* or *) LinearRecurrence[{6,6,6,6,-21}, {1,8,56,392,2744,19180}, 30] (* G. C. Greubel, Dec 19 2016 *)
coxG[{5, 21, -6}] (* The coxG program is at A169452 *) (* G. C. Greubel, May 12 2019 *)

my(x='x+O('x^30)); Vec((1+x)*(1-x^5)/(1-7*x+27*x^5-21*x^6)) \\ G. C. Greubel, Dec 19 2016

((1+x)*(1-x^5)/(1-7*x+27*x^5-21*x^6)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 12 2019

G.f.: (t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(21*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).

a(n) = 6*a(n-1)+6*a(n-2)+6*a(n-3)+6*a(n-4)-21*a(n-5). - Wesley Ivan Hurt, May 10 2021

A163924 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

Original entry on oeis.org

1, 8, 56, 392, 2744, 19208, 134428, 940800, 6584256, 46080384, 322496832, 2257016832, 15795891636, 110548662840, 773682621768, 5414672451384, 37894967433288, 265210605012024, 1856095143363468, 12990012903371952

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

nonn

The initial terms coincide with those of A003950, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,6,6,6,6,-21).

a:=[8,56,392,2744,19208,134428];; for n in [7..30] do a[n]:=6*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]) -21*a[n-6]; od; Concatenation([1], a); # G. C. Greubel, Aug 10 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^6)/(1-7*t+27*t^6-21*t^7) )); // G. C. Greubel, Aug 10 2019

seq(coeff(series((1+t)*(1-t^6)/(1-7*t+27*t^6-21*t^7), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 10 2019

coxG[{6,21,-6}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jan 24 2016 *)
CoefficientList[Series[(1+t)*(1-t^6)/(1-7*t+27*t^6-21*t^7), {t,0,30}], t] (* G. C. Greubel, Aug 08 2017 *)

my(t='t+O('t^30)); Vec((1+t)*(1-t^6)/(1-7*t+27*t^6-21*t^7)) \\ G. C. Greubel, Aug 08 2017

def A163924_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^6)/(1-7*t+27*t^6-21*t^7)).list()
A163924_list(30) # G. C. Greubel, Aug 10 2019

G.f.: (t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(21*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).

a(n) = -21*a(n-6) + 6*Sum_{k=1..5} a(n-k). - Wesley Ivan Hurt, May 11 2021

A164373 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

Original entry on oeis.org

1, 8, 56, 392, 2744, 19208, 134456, 941164, 6587952, 46114320, 322790832, 2259469968, 15815828784, 110707574544, 774930433956, 5424354927432, 37969377752376, 265777897314888, 1860391054122552, 13022357800350024

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

nonn

The initial terms coincide with those of A003950, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6, 6, 6, 6, 6, 6, -21).

a:=[8, 56, 392, 2744, 19208, 134456, 941164];; for n in [8..30] do a[n]:=6*(a[n-1] +a[n-2]+a[n-3]+a[n-4]+a[n-5]+a[n-6]) -21*a[n-7]; od; Concatenation([1], a); # G. C. Greubel, Aug 28 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8) )); // G. C. Greubel, Aug 28 2019

seq(coeff(series((1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Aug 28 2019

CoefficientList[Series[(1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8), {t, 0, 30}], t] (* G. C. Greubel, Sep 17 2017 *)
coxG[{7, 21, -6}] (* The coxG program is at A169452 *) (* G. C. Greubel, Aug 28 2019 *)

my(t='t+O('t^30)); Vec((1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8)) \\ G. C. Greubel, Sep 17 2017

def A164373_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+t)*(1-t^7)/(1-7*t+27*t^7-21*t^8)).list()
A164373_list(30) # G. C. Greubel, Aug 28 2019

G.f.: (t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(21*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).

a(n) = -21*a(n-7) + 6*Sum_{k=1..6} a(n-k). - Wesley Ivan Hurt, May 11 2021

A165786 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

Original entry on oeis.org

1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 46118408, 322828828, 2259801600, 15818609856, 110730259584, 775111751232, 5425781797632, 37980469356480, 265863262906752, 1861042682227008, 13027297668747264

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

nonn

The initial terms coincide with those of A003950, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6,6,6,6,6,6,6,6,6,-21).

a:=[8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 46118408, 322828828];; for n in [11..20] do a[n]:=6*Sum([1..9], j-> a[n-j]) -21*a[n-10]; od; Concatenation([1], a); # G. C. Greubel, Sep 22 2019

R:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+t)*(1-t^10)/(1-7*t+27*t^10-21*t^11) )); // G. C. Greubel, Sep 22 2019

seq(coeff(series((1+t)*(1-t^10)/(1-7*t+27*t^10-21*t^11), t, n+1), t, n), n = 0..20); # G. C. Greubel, Sep 22 2019

With[{num=Total[2t^Range[9]]+t^10+1,den=Total[-6 t^Range[9]]+21t^10+1}, CoefficientList[Series[num/den,{t,0,30}],t]] (* Harvey P. Dale, Oct 20 2011 *)
CoefficientList[Series[(1+t)*(1-t^10)/(1-7*t+27*t^10-21*t^11), {t,0,20}], t] (* or *) coxG[{10, 21, -6}] (* The coxG program is at A169452 *) (* G. C. Greubel, Sep 22 2019 *)

my(t='t+O('t^20)); Vec((1+t)*(1-t^10)/(1-7*t+27*t^10-21*t^11)) \\ G. C. Greubel, Sep 22 2019

def A165786_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^10)/(1-7*t+27*t^10-21*t^11) ).list()
A165786_list(30) # G. C. Greubel, Sep 22 2019

G.f.: (t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(21*t^10 - 6*t^9 - 6*t^8 - 6*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).

A166366 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

Original entry on oeis.org

1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 46118408, 322828856, 2259801964, 15818613552, 110730293520, 775112045232, 5425784250768, 37980489294384, 265863421833744, 1861043930247600, 13027307353612944

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

nonn

The initial terms coincide with those of A003950, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6,6,6,6,6,6,6,6,6,6,-21).

seq(coeff(series((1+t)*(1-t^11)/(1-7*t+27*t^11-21*t^12), t, n+1), t, n), n = 0 .. 30); # G. C. Greubel, Mar 13 2020

CoefficientList[Series[(1+t)*(1-t^11)/(1-7*t+27*t^11-21*t^12), {t,0,30}], t] (* G. C. Greubel, May 10 2016 *)
coxG[{11, 21, -6}] (* The coxG program is in A169452 *) (* G. C. Greubel, Mar 13 2020 *)

def A166366_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+t)*(1-t^11)/(1-7*t+27*t^11-21*t^12) ).list()
A166366_list(30) # G. C. Greubel, Mar 13 2020

G.f.: (t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(21*t^11 - 6*t^10 - 6*t^9 - 6*t^8 - 6*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).

A166538 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

Original entry on oeis.org

1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 46118408, 322828856, 2259801992, 15818613916, 110730297216, 775112079168, 5425784544768, 37980491747520, 265863441771648, 1861044089174592, 13027308601633536

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

nonn

The initial terms coincide with those of A003950, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6,6,6,6,6,6,6,6,6,6,6,-21).

Cf. A003950, A154638, A169452.

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x)*(1-x^12)/(1-7*x+27*x^12-21*x^13) )); // G. C. Greubel, Aug 23 2024

CoefficientList[Series[(1+t)*(1-t^12)/(1-7*t+27*t^12-21*t^13), {t, 0, 50}], t] (* G. C. Greubel, May 16 2016; Aug 23 2024 *)
coxG[{12,21,-6}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Oct 24 2016 *)

def A166538_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x)*(1-x^12)/(1-7*x+27*x^12-21*x^13) ).list()
A166538_list(30) # G. C. Greubel, Aug 23 2024

G.f.: (t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(21*t^12 - 6*t^11 - 6*t^10 - 6*t^9 - 6*t^8 - 6*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).
From G. C. Greubel, Aug 23 2024: (Start)
a(n) = 6*Sum_{j=1..11} a(n-j) - 21*a(n-12).
G.f.: (1 + x)*(1 - x^12)/(1 - 7*x + 27*x^12 - 21*x^13). (End)

A166910 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^13 = I.

Original entry on oeis.org

1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 46118408, 322828856, 2259801992, 15818613944, 110730297580, 775112082864, 5425784578704, 37980492041520, 265863444224784, 1861044109112496, 13027308760560528

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

nonn

The initial terms coincide with those of A003950, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, -21).

CoefficientList[Series[(t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(21*t^13 - 6*t^12 - 6*t^11 - 6*t^10 - 6*t^9 - 6*t^8 - 6*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1), {t, 0, 50}], t] (* G. C. Greubel, May 28 2016 *)
coxG[{13,21,-6}] (* The coxG program is at A169452 *) (* Harvey P. Dale, Jul 05 2021 *)

G.f.: (t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(21*t^13 - 6*t^12 - 6*t^11 - 6*t^10 - 6*t^9 - 6*t^8 - 6*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).

A167109 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^14 = I.

Original entry on oeis.org

1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 46118408, 322828856, 2259801992, 15818613944, 110730297608, 775112083228, 5425784582400, 37980492075456, 265863444518784, 1861044111565632, 13027308780498432

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

nonn

The initial terms coincide with those of A003950, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, -21).

With[{num=Total[2t^Range[13]]+t^14+1,den=Total[-6 t^Range[13]]+ 21t^14+1}, CoefficientList[Series[num/den,{t,0,30}],t]] (* Harvey P. Dale, Jul 15 2011 *)
CoefficientList[Series[(t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/ (21*t^14 - 6*t^13 - 6*t^12 - 6*t^11 - 6*t^10 - 6*t^9 - 6*t^8 - 6*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1), {t, 0, 50}], t] (* G. C. Greubel, Jun 03 2016 *)

G.f.: (t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(21*t^14 - 6*t^13 - 6*t^12 - 6*t^11 - 6*t^10 - 6*t^9 - 6*t^8 - 6*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).

A167653 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^15 = I.

Original entry on oeis.org

1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 46118408, 322828856, 2259801992, 15818613944, 110730297608, 775112083256, 5425784582764, 37980492079152, 265863444552720, 1861044111859632, 13027308782951568

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

nonn

The initial terms coincide with those of A003950, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, -21).

CoefficientList[Series[(t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(21*t^15 - 6*t^14 - 6*t^13 - 6*t^12 - 6*t^11 - 6*t^10 - 6*t^9 - 6*t^8 - 6*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1), {t, 0, 50}], t] (* G. C. Greubel, Jun 18 2016 *)

G.f.: (t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(21*t^15 - 6*t^14 - 6*t^13 - 6*t^12 - 6*t^11 - 6*t^10 - 6*t^9 - 6*t^8 - 6*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).

A167899 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^16 = I.

Original entry on oeis.org

1, 8, 56, 392, 2744, 19208, 134456, 941192, 6588344, 46118408, 322828856, 2259801992, 15818613944, 110730297608, 775112083256, 5425784582792, 37980492079516, 265863444556416, 1861044111893568, 13027308783245568

Offset: 0

John Cannon and N. J. A. Sloane, Dec 03 2009

nonn

The initial terms coincide with those of A003950, although the two sequences are eventually different.

Computed with MAGMA using commands similar to those used to compute A154638.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, -21).

CoefficientList[Series[(t^16 + 2*t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/(21*t^16 - 6*t^15 - 6*t^14 - 6*t^13 - 6*t^12 - 6*t^11 - 6*t^10 - 6*t^9 - 6*t^8 - 6*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1), {t, 0, 50}], t] (* G. C. Greubel, Jul 01 2016 *)

G.f.: (t^16 + 2*t^15 + 2*t^14 + 2*t^13 + 2*t^12 + 2*t^11 + 2*t^10 + 2*t^9 + 2*t^8 + 2*t^7 + 2*t^6 + 2*t^5 + 2*t^4 + 2*t^3 + 2*t^2 + 2*t + 1)/( 21*t^16 - 6*t^15 - 6*t^14 - 6*t^13 - 6*t^12 - 6*t^11 - 6*t^10 - 6*t^9 - 6*t^8 - 6*t^7 - 6*t^6 - 6*t^5 - 6*t^4 - 6*t^3 - 6*t^2 - 6*t + 1).

cp's OEIS Frontend

A163347 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^5 = I.

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A163924 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^6 = I.

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A164373 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^7 = I.

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A165786 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^10 = I.

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A166366 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^11 = I.

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A166538 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^12 = I.

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A166910 Number of reduced words of length n in Coxeter group on 8 generators S_i with relations (S_i)^2 = (S_i S_j)^13 = I.

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