# # +--------------+ # | 1 2 | # # | u | # # | 3 4 | # +--------------+--------------+--------------+--------------+ # | 5 6 | 9 10 | 13 14 | 17 18 | # # | l | f | r | b | # # | 7 8 | 11 12 | 15 16 | 19 20 | # +--------------+--------------+--------------+--------------+ # | 21 22 | # # | d | # # | 23 24 | # +--------------+ U := ( 1, 2, 4, 3)( 6, 18, 14, 10)( 5, 17, 13, 9); L := ( 5, 6, 8, 7)( 1, 9, 21, 20)( 3, 11, 23, 18); F := (9, 10, 12, 11)( 3, 13, 22, 8)( 4, 15, 21, 6); R := (13, 14, 16, 15)( 10, 2, 19, 22)( 12, 4, 17, 24); B := (17, 18, 20, 19)( 2, 5, 23, 16)( 1,7, 24, 14); D := (21, 22, 24, 23)(7, 11, 15, 19)(8, 12, 16, 20); cube := Group(U,L,F,R,B,D);; #gap> Size(cube); #88179840 square_group := Group(U^2,L^2,F^2,R^2,B^2,D^2);; #gap> Size(square_group); #96 #gap> IsAbelian(square_group); #false square_two_faces := Group(U^2,R^2);; #gap> Size(square_two_faces); #6 #gap> IsAbelian(square_two_faces); #false two_faces:= Group(U,R); ## 2-faced cube #gap> Size(two_faces); #29160 LoadPackage("grape"); Gamma := CayleyGraph( two_faces, [U,U^(-1), R, R^(-1)] );; Diameter( Gamma ); #17 N1:= Size( cube ); Collected( Factors(N1) ); N2:=Size(square_group); Collected( Factors(N2) ); N3:=Size(two_faces); Collected( Factors(N3) ); Hgens:=[]; for g in G do if (4^g in [4,13,10]) and (2^g in [2,14,17]) then Hgens:=Concatenation([g],Hgens); fi; od; #gap> H:=Group(Hgens); # #gap> Size(H); #972 Kgens:=[]; for g in G do if (4^g in [4,13,10]) and (2^g in [2,14,17]) and (3^g in [3,6,9]) then Kgens:=Concatenation([g],Kgens); fi; od; K:=Group(Kgens); # #gap> IsAbelian(H); #true #gap> IsNormal(two_faces,K); #true #gap> C := Complementclasses(two_faces,K);; #gap> Length(C); #3 #gap> IdGroup(C[1]); IdGroup(H); #[ 120, 34 ] #[ 120, 34 ] gap.eval('U := ( 1, 2, 4, 3)( 6, 18, 14, 10)( 5, 17, 13, 9);') gap.eval('L := ( 5, 6, 8, 7)( 1, 9, 21, 20)( 3, 11, 23, 18);') gap.eval('F := (9, 10, 12, 11)( 3, 13, 22, 8)( 4, 15, 21, 6);') gap.eval('R := (13, 14, 16, 15)( 10, 2, 19, 22)( 12, 4, 17, 24);') gap.eval('B := (17, 18, 20, 19)( 2, 5, 23, 16)( 1,7, 24, 14);') gap.eval('D := (21, 22, 24, 23)(7, 11, 15, 19)(8, 12, 16, 20);') gap.eval('two_faces:= Group(U,R);') ## 2-faced cube gap.eval('RR := UnivariatePolynomialRing(Rationals,"x")') gap.eval('x := Indeterminate(RR,"x")') gap.eval('repns := IrreducibleRepresentations(two_faces);') gap.eval('Ri:=R^(-1); Ui:=U^(-1)') polys = gap.eval('Q:=List(repns,rho->CharacteristicPolynomial((R^rho+Ri^rho+U^rho+Ui^rho)/4));') spolys = polys.split(",") sage: spolys ['[ x-1', ' x+1', ' x^4-1/2*x^2-1/8*x', ' x^4-1/2*x^2+1/8*x', ' \n x^5+x^4-1/8*x^3-3/8*x^2-31/256*x-3/256', ' \n x^5-x^4-1/8*x^3+3/8*x^2-31/256*x+3/256', ' x^6-5/8*x^4+9/256*x^2', ' \n x^6-2*x^5+17/16*x^4+1/8*x^3-55/256*x^2+5/128*x', ' \n x^6-2*x^5+17/16*x^4+1/8*x^3-55/256*x^2+5/128*x', ' \n x^6+2*x^5+17/16*x^4-1/8*x^3-55/256*x^2-5/128*x', ' \n x^6+2*x^5+17/16*x^4-1/8*x^3-55/256*x^2-5/128*x', ' x^6-7/16*x^4+9/256*x^2', ' \n x^6-7/16*x^4+9/256*x^2', ' x^6-7/16*x^4+9/256*x^2', ' x^6-7/16*x^4+9/256*x^2', ' \n x^10-5/4*x^8+29/64*x^6-11/256*x^4+1/1024*x^2', ' \n x^10-13/8*x^8+227/256*x^6-379/2048*x^4+865/65536*x^2-9/262144', ' \n x^10-13/8*x^8+227/256*x^6-379/2048*x^4+865/65536*x^2-9/262144', ' \n x^15+x^14-7/4*x^13-15/8*x^12+67/64*x^11+161/128*x^10-269/1024*x^9-791/2048*x^8+373/16384*x^7+1815/32768*x^6+257/262144*x^5-1655/524288*x^4-209/1048576*x^3+33/1048576*x^2', ' \n x^15+x^14-7/4*x^13-15/8*x^12+67/64*x^11+161/128*x^10-269/1024*x^9-791/2048*x^8+373/16384*x^7+1815/32768*x^6+257/262144*x^5-1655/524288*x^4-209/1048576*x^3+33/1048576*x^2', ' \n x^15-x^14-7/4*x^13+15/8*x^12+67/64*x^11-161/128*x^10-269/1024*x^9+791/2048*x^8+373/16384*x^7-1815/32768*x^6+257/262144*x^5+1655/524288*x^4-209/1048576*x^3-33/1048576*x^2', ' \n x^15-x^14-7/4*x^13+15/8*x^12+67/64*x^11-161/128*x^10-269/1024*x^9+791/2048*x^8+373/16384*x^7-1815/32768*x^6+257/262144*x^5+1655/524288*x^4-209/1048576*x^3-33/1048576*x^2', ' \n x^15+x^14-5/4*x^13-11/8*x^12+15/32*x^11+41/64*x^10-57/1024*x^9-17/128*x^8-25/8192*x^7+409/32768*x^6+267/262144*x^5-225/524288*x^4-27/524288*x^3', ' \n x^15+x^14-5/4*x^13-11/8*x^12+15/32*x^11+41/64*x^10-57/1024*x^9-17/128*x^8-25/8192*x^7+409/32768*x^6+267/262144*x^5-225/524288*x^4-27/524288*x^3', ' \n x^15-x^14-5/4*x^13+11/8*x^12+15/32*x^11-41/64*x^10-57/1024*x^9+17/128*x^8-25/8192*x^7-409/32768*x^6+267/262144*x^5+225/524288*x^4-27/524288*x^3', ' \n x^15-x^14-5/4*x^13+11/8*x^12+15/32*x^11-41/64*x^10-57/1024*x^9+17/128*x^8-25/8192*x^7-409/32768*x^6+267/262144*x^5+225/524288*x^4-27/524288*x^3', ' \n x^20-15/8*x^18+343/256*x^16-947/2048*x^14+5385/65536*x^12-1975/262144*x^10+171/524288*x^8-81/16777216*x^6', ' \n x^20-23/8*x^18+863/256*x^16-4263/2048*x^14+47969/65536*x^12-39015/262144*x^10+17873/1048576*x^8-17065/16777216*x^6+1675/67108864*x^4-9/268435456*x^2', ' \n x^20-11/4*x^18+1/4*x^17+383/128*x^16-33/64*x^15-1665/1024*x^14+409/1024*x^13+30113/65536*x^12-2389/16384*x^11-8339/131072*x^10+3429/131072*x^9+6733/2097152*x^8-4493/2097152*x^7+255/4194304*x^6+1035/16777216*x^5-27/4194304*x^4', ' \n x^20-11/4*x^18-1/4*x^17+383/128*x^16+33/64*x^15-1665/1024*x^14-409/1024*x^13+30113/65536*x^12+2389/16384*x^11-8339/131072*x^10-3429/131072*x^9+6733/2097152*x^8+4493/2097152*x^7+255/4194304*x^6-1035/16777216*x^5-27/4194304*x^4', ' \n x^20-19/8*x^18+1/8*x^17+583/256*x^16-15/64*x^15-2307/2048*x^14+173/1024*x^13+19985/65536*x^12-1939/32768*x^11-11599/262144*x^10+1383/131072*x^9+12941/4194304*x^8-3827/4194304*x^7-4545/67108864*x^6+4203/134217728*x^5-1053/1073741824*x^4-81/536870912*x^3', ' \n x^20-19/8*x^18+1/8*x^17+583/256*x^16-15/64*x^15-2307/2048*x^14+173/1024*x^13+19985/65536*x^12-1939/32768*x^11-11599/262144*x^10+1383/131072*x^9+12941/4194304*x^8-3827/4194304*x^7-4545/67108864*x^6+4203/134217728*x^5-1053/1073741824*x^4-81/536870912*x^3', ' \n x^20-19/8*x^18-1/8*x^17+583/256*x^16+15/64*x^15-2307/2048*x^14-173/1024*x^13+19985/65536*x^12+1939/32768*x^11-11599/262144*x^10-1383/131072*x^9+12941/4194304*x^8+3827/4194304*x^7-4545/67108864*x^6-4203/134217728*x^5-1053/1073741824*x^4+81/536870912*x^3', ' \n x^20-19/8*x^18-1/8*x^17+583/256*x^16+15/64*x^15-2307/2048*x^14-173/1024*x^13+19985/65536*x^12+1939/32768*x^11-11599/262144*x^10-1383/131072*x^9+12941/4194304*x^8+3827/4194304*x^7-4545/67108864*x^6-4203/134217728*x^5-1053/1073741824*x^4+81/536870912*x^3', ' \n x^24-49/16*x^22+251/64*x^20-11193/4096*x^18+74081/65536*x^16-74655/262144*x^14+181589/4194304*x^12-256711/67108864*x^10+24485/134217728*x^8-17325/4294967296*x^6+2025/68719476736*x^4', ' \n x^24-49/16*x^22+251/64*x^20-11193/4096*x^18+74081/65536*x^16-74655/262144*x^14+181589/4194304*x^12-256711/67108864*x^10+24485/134217728*x^8-17325/4294967296*x^6+2025/68719476736*x^4', ' \n x^30-2*x^29-2*x^28+25/4*x^27+29/64*x^26-131/16*x^25+4273/2048*x^24+5985/1024*x^23-85449/32768*x^22-40391/16384*x^21+196577/131072*x^20+161635/262144*x^19-8419227/16777216*x^18-700219/8388608*x^17+13914393/134217728*x^16+114345/33554432*x^15-57307779/4294967296*x^14+1306667/2147483648*x^13+1124601/1073741824*x^12-1676057/17179869184*x^11-12978199/274877906944*x^10+798365/137438953472*x^9+1158747/1099511627776*x^8-172467/1099511627776*x^7-29349/4398046511104*x^6+891/549755813888*x^5-243/4398046511104*x^4', ' \n x^30+2*x^29-2*x^28-25/4*x^27+29/64*x^26+131/16*x^25+4273/2048*x^24-5985/1024*x^23-85449/32768*x^22+40391/16384*x^21+196577/131072*x^20-161635/262144*x^19-8419227/16777216*x^18+700219/8388608*x^17+13914393/134217728*x^16-114345/33554432*x^15-57307779/4294967296*x^14-1306667/2147483648*x^13+1124601/1073741824*x^12+1676057/17179869184*x^11-12978199/274877906944*x^10-798365/137438953472*x^9+1158747/1099511627776*x^8+172467/1099511627776*x^7-29349/4398046511104*x^6-891/549755813888*x^5-243/4398046511104*x^4', ' \n x^30-7/2*x^28+675/128*x^26-2305/512*x^24+158051/65536*x^22-222215/262144*x^20+1664023/8388608*x^18-2057103/67108864*x^16+12998729/4294967296*x^14-3054553/17179869184*x^12+1467909/274877906944*x^10-61479/1099511627776*x^8+729/4398046511104*x^6', ' \n x^30-7/2*x^28+339/64*x^26-2329/512*x^24+40073/16384*x^22-56183/65536*x^20+103711/524288*x^18-497769/16777216*x^16+46817/16777216*x^14-659845/4294967296*x^12+74655/17179869184*x^10-6561/137438953472*x^8+729/4398046511104*x^6', ' \n x^30-7/2*x^28+339/64*x^26-2329/512*x^24+40073/16384*x^22-56183/65536*x^20+103711/524288*x^18-497769/16777216*x^16+46817/16777216*x^14-659845/4294967296*x^12+74655/17179869184*x^10-6561/137438953472*x^8+729/4398046511104*x^6', ' \n x^30-7/2*x^28+339/64*x^26-9331/2048*x^24+80767/32768*x^22-28741/32768*x^20+3500405/16777216*x^18-4430727/134217728*x^16+14520893/4294967296*x^14-1813721/8589934592*x^12+1955097/274877906944*x^10-106191/1099511627776*x^8+729/4398046511104*x^6', ' \n x^30-7/2*x^28+339/64*x^26-9331/2048*x^24+80767/32768*x^22-28741/32768*x^20+3500405/16777216*x^18-4430727/134217728*x^16+14520893/4294967296*x^14-1813721/8589934592*x^12+1955097/274877906944*x^10-106191/1099511627776*x^8+729/4398046511104*x^6', ' \n x^40-41/8*x^38+3033/256*x^36-16751/1024*x^34+493399/32768*x^32-160035/16384*x^30+19312727/4194304*x^28-215228887/134217728*x^26+1784864733/4294967296*x^24-2753833459/34359738368*x^22+12569084507/1099511627776*x^20-10470446433/8796093022208*x^18+24934513837/281474976710656*x^16-5130498001/1125899906842624*x^14+2766623697/18014398509481984*x^12-222843393/72057594037927936*x^10+1114641/36028797018963968*x^8-6561/72057594037927936*x^6', ' \n x^60-8*x^58+1917/64*x^56-142919/2048*x^54+1859019/16384*x^52-35882357/262144*x^50+2133413375/16777216*x^48-6254811143/67108864*x^46+58825379151/1073741824*x^44-897178954303/34359738368*x^42+698432843951/68719476736*x^40-445910069851/137438953472*x^38+239496223521569/281474976710656*x^36-103150625244453/562949953421312*x^34+72772931095165/2251799813685248*x^32-1339176458034443/288230376151711744*x^30+1246480992575401/2305843009213693952*x^28-232395616366671/4611686018427387904*x^26+1096007854049473/295147905179352825856*x^24-1003257629124963/4722366482869645213696*x^22+696048784264073/75557863725914323419136*x^20-2768494214987/9444732965739290427392*x^18+31680943305507/4835703278458516698824704*x^16-1875293183943/19342813113834066795298816*x^14+271514764773/309485009821345068724781056*x^12-2666213253/618970019642690137449562112*x^10+43046721/4951760157141521099596496896*x^8', ' \n x^60-8*x^58+1917/64*x^56-142919/2048*x^54+1859019/16384*x^52-35882357/262144*x^50+2133413375/16777216*x^48-6254811143/67108864*x^46+58825379151/1073741824*x^44-897178954303/34359738368*x^42+698432843951/68719476736*x^40-445910069851/137438953472*x^38+239496223521569/281474976710656*x^36-103150625244453/562949953421312*x^34+72772931095165/2251799813685248*x^32-1339176458034443/288230376151711744*x^30+1246480992575401/2305843009213693952*x^28-232395616366671/4611686018427387904*x^26+1096007854049473/295147905179352825856*x^24-1003257629124963/4722366482869645213696*x^22+696048784264073/75557863725914323419136*x^20-2768494214987/9444732965739290427392*x^18+31680943305507/4835703278458516698824704*x^16-1875293183943/19342813113834066795298816*x^14+271514764773/309485009821345068724781056*x^12-2666213253/618970019642690137449562112*x^10+43046721/4951760157141521099596496896*x^8', ' \n x^60-7*x^58+1459/64*x^56-94007/2048*x^54+1049481/16384*x^52-17255247/262144*x^50+866781007/16777216*x^48-2127985441/67108864*x^46+16596067621/1073741824*x^44-207663710791/34359738368*x^42+262153929971/137438953472*x^40-535767211867/1099511627776*x^38+28366877796193/281474976710656*x^36-9476540381583/562949953421312*x^34+10175363591367/4503599627370496*x^32-69667878672835/288230376151711744*x^30+46971503582069/2305843009213693952*x^28-3067758560429/2305843009213693952*x^26+19421945265877/295147905179352825856*x^24-11262593325699/4722366482869645213696*x^22+4555713849417/75557863725914323419136*x^20-148415655645/151115727451828646838272*x^18+43072328583/4835703278458516698824704*x^16-612042885/19342813113834066795298816*x^14+531441/309485009821345068724781056*x^12', ' \n x^60-7*x^58+1459/64*x^56-94007/2048*x^54+1049481/16384*x^52-17255247/262144*x^50+866781007/16777216*x^48-2127985441/67108864*x^46+16596067621/1073741824*x^44-207663710791/34359738368*x^42+262153929971/137438953472*x^40-535767211867/1099511627776*x^38+28366877796193/281474976710656*x^36-9476540381583/562949953421312*x^34+10175363591367/4503599627370496*x^32-69667878672835/288230376151711744*x^30+46971503582069/2305843009213693952*x^28-3067758560429/2305843009213693952*x^26+19421945265877/295147905179352825856*x^24-11262593325699/4722366482869645213696*x^22+4555713849417/75557863725914323419136*x^20-148415655645/151115727451828646838272*x^18+43072328583/4835703278458516698824704*x^16-612042885/19342813113834066795298816*x^14+531441/309485009821345068724781056*x^12 ]'] p = sage_eval(spolys[10].replace("\n ","")) sage: PR = PolynomialRing(RR,"x") sage: PR(p) 1.0000000000000000*x^6 + 2.0000000000000000*x^5 + 1.0625000000000000*x^4 - 0.12500000000000000*x^3 - 0.21484375000000000*x^2 - 0.039062500000000000*x sage: PR(p).roots() [-0.90450849718747373, -0.78627567280010557, -0.38099409927046651, -0.34549150281252627, 0, 0.41726977207057209]