Polytopes for The Higman-Sims Group HS


The Higman-Sims Group HS has :

  • 465 rank 3 polytopes,
  • 111 rank 4 polytopes,
  • 4 rank 5 polytopes,
  • and no higher rank polytopes.


Polytopes of rank 5

indexTypeFacet GroupVF GroupFacet(Facet)Facet(VF)VF(VF)Dual
hs[577]{3,8,5,3}Order 252000Order 7680Order 720Order 480Simple, Order 60hs[579]
hs[578]{5,8,5,3}Order 252000Order 7680Order 720Order 480Simple, Order 60hs[580]
hs[579]{3,5,8,3}Order 7680Order 252000Simple, Order 60Order 480Order 720hs[577]
hs[580]{3,5,8,5}Order 7680Order 252000Simple, Order 60Order 480Order 720hs[578]

Polytopes of rank 4

indexTypeFacet GroupVF GroupDual
hs[466]{4,6,6}Order 336Order 40320hs[572]
hs[467]{4,10,4}Order 252000Order 1440hs[492]
hs[468]{4,10,3}Order 252000Order 360hs[565]
hs[469]{4,8,4}Order 252000Order 1440hs[493]
hs[470]{6,6,4}Order 1920Order 1440hs[517]
hs[471]{3,8,6}Order 336Order 40320hs[494]
hs[472]{3,8,6}Order 336Order 252000hs[495]
hs[473]{8,12,4}Order 252000Solvable, Order 1536hs[521]
hs[474]{8,6,4}Order 40320Order 2880hs[514]
hs[475]{8,4,6}Order 336Order 2880hs[571]
hs[476]{8,6,6}Order 336Order 7680hs[520]
hs[477]{8,6,5}Order 252000Order 120hs[527]
hs[478]{8,8,3}Order 252000Solvable, Order 192hs[564]
hs[479]{8,10,3}Order 252000Order 120hs[574]
hs[480]{8,8,3}Order 252000Order 1440hs[562]
hs[481]{8,3,6}Order 336Solvable, Order 192hs[544]
hs[482]{8,3,5}Order 336Order 120hs[529]
hs[483]{8,8,4}Order 252000Order 1440hs[569]
hs[484]{10,6,6}Order 5040Order 7680hs[518]
hs[485]{10,12,4}Order 252000Solvable, Order 1536hs[522]
hs[486]{10,10,4}Order 252000Order 1440hs[525]
hs[487]{10,10,3}Order 252000Order 120hs[573]
hs[488]{10,6,3}Order 252000Solvable, Order 192hs[566]
hs[489]{10,5,6}Order 5040Order 600hs[531]
hs[490]{10,8,6}Order 252000Solvable, Order 192hs[545]
hs[491]{10,3,6}Order 360Solvable, Order 192hs[543]
hs[492]{4,10,4}Order 1440Order 252000hs[467]
hs[493]{4,8,4}Order 1440Order 252000hs[469]
hs[494]{6,8,3}Order 40320Order 336hs[471]
hs[495]{6,8,3}Order 252000Order 336hs[472]
hs[496]{6,10,5}Order 252000Solvable, Order 320hs[528]
hs[497]{6,5,5}Order 480Solvable, Order 320hs[530]
hs[498]{8,3,8}Order 336Order 336hs[498]
hs[499]{8,3,10}Order 336Order 5040hs[506]
hs[500]{8,5,10}Order 40320Solvable, Order 500hs[503]
hs[501]{8,6,8}Order 336Order 252000hs[533]
hs[502]{8,4,4}Order 252000Solvable, Order 128hs[508]
hs[503]{10,5,8}Solvable, Order 500Order 40320hs[500]
hs[504]{10,4,4}Order 252000Solvable, Order 128hs[516]
hs[505]{10,10,8}Solvable, Order 500Order 252000hs[535]
hs[506]{10,3,8}Order 5040Order 336hs[499]
hs[507]{10,8,4}Order 252000Order 336hs[554]
hs[508]{4,4,8}Solvable, Order 128Order 252000hs[502]
hs[509]{4,5,4}Order 1920Order 1920hs[509]
hs[510]{4,12,4}Solvable, Order 1536Order 40320hs[570]
hs[511]{4,12,4}Solvable, Order 1536Order 252000hs[560]
hs[512]{4,12,5}Solvable, Order 1536Order 252000hs[550]
hs[513]{4,6,8}Order 120Order 252000hs[534]
hs[514]{4,6,8}Order 2880Order 40320hs[474]
hs[515]{4,8,8}Order 1440Order 252000hs[536]
hs[516]{4,4,10}Solvable, Order 128Order 252000hs[504]
hs[517]{4,6,6}Order 1440Order 1920hs[470]
hs[518]{6,6,10}Order 7680Order 5040hs[484]
hs[519]{6,6,4}Order 7680Order 1440hs[555]
hs[520]{6,6,8}Order 7680Order 336hs[476]
hs[521]{4,12,8}Solvable, Order 1536Order 252000hs[473]
hs[522]{4,12,10}Solvable, Order 1536Order 252000hs[485]
hs[523]{4,12,8}Solvable, Order 1536Order 252000hs[538]
hs[524]{4,12,4}Solvable, Order 1536Order 252000hs[561]
hs[525]{4,10,10}Order 1440Order 252000hs[486]
hs[526]{4,8,4}Order 2880Order 336hs[553]
hs[527]{5,6,8}Order 120Order 252000hs[477]
hs[528]{5,10,6}Solvable, Order 320Order 252000hs[496]
hs[529]{5,3,8}Order 120Order 336hs[482]
hs[530]{5,5,6}Solvable, Order 320Order 480hs[497]
hs[531]{6,5,10}Order 600Order 5040hs[489]
hs[532]{6,5,4}Order 600Order 720hs[558]
hs[533]{8,6,8}Order 252000Order 336hs[501]
hs[534]{8,6,4}Order 252000Order 120hs[513]
hs[535]{8,10,10}Order 252000Solvable, Order 500hs[505]
hs[536]{8,8,4}Order 252000Order 1440hs[515]
hs[537]{8,8,3}Order 252000Solvable, Order 192hs[563]
hs[538]{8,12,4}Order 252000Solvable, Order 1536hs[523]
hs[539]{8,5,5}Order 252000Order 600hs[546]
hs[540]{8,6,3}Order 252000Solvable, Order 192hs[567]
hs[541]{8,6,5}Order 252000Order 600hs[551]
hs[542]{8,10,3}Order 252000Order 120hs[576]
hs[543]{6,3,10}Solvable, Order 192Order 360hs[491]
hs[544]{6,3,8}Solvable, Order 192Order 336hs[481]
hs[545]{6,8,10}Solvable, Order 192Order 252000hs[490]
hs[546]{5,5,8}Order 600Order 252000hs[539]
hs[547]{5,5,4}Order 600Order 1440hs[568]
hs[548]{5,5,5}Order 600Order 600hs[548]
hs[549]{5,8,4}Order 252000Order 1440hs[556]
hs[550]{5,12,4}Order 252000Solvable, Order 1536hs[512]
hs[551]{5,6,8}Order 600Order 252000hs[541]
hs[552]{5,6,4}Order 600Order 2880hs[557]
hs[553]{4,8,4}Order 336Order 2880hs[526]
hs[554]{4,8,10}Order 336Order 252000hs[507]
hs[555]{4,6,6}Order 1440Order 7680hs[519]
hs[556]{4,8,5}Order 1440Order 252000hs[549]
hs[557]{4,6,5}Order 2880Order 600hs[552]
hs[558]{4,5,6}Order 720Order 600hs[532]
hs[559]{4,10,3}Order 252000Order 120hs[575]
hs[560]{4,12,4}Order 252000Solvable, Order 1536hs[511]
hs[561]{4,12,4}Order 252000Solvable, Order 1536hs[524]
hs[562]{3,8,8}Order 1440Order 252000hs[480]
hs[563]{3,8,8}Solvable, Order 192Order 252000hs[537]
hs[564]{3,8,8}Solvable, Order 192Order 252000hs[478]
hs[565]{3,10,4}Order 360Order 252000hs[468]
hs[566]{3,6,10}Solvable, Order 192Order 252000hs[488]
hs[567]{3,6,8}Solvable, Order 192Order 252000hs[540]
hs[568]{4,5,5}Order 1440Order 600hs[547]
hs[569]{4,8,8}Order 1440Order 252000hs[483]
hs[570]{4,12,4}Order 40320Solvable, Order 1536hs[510]
hs[571]{6,4,8}Order 2880Order 336hs[475]
hs[572]{6,6,4}Order 40320Order 336hs[466]
hs[573]{3,10,10}Order 120Order 252000hs[487]
hs[574]{3,10,8}Order 120Order 252000hs[479]
hs[575]{3,10,4}Order 120Order 252000hs[559]
hs[576]{3,10,8}Order 120Order 252000hs[542]

Polytopes of rank 3

indexTypeDual
hs[1]{5,12}hs[143]
hs[2]{5,8}hs[111]
hs[3]{5,8}hs[114]
hs[4]{5,12}hs[141]
hs[5]{6,8}hs[116]
hs[6]{6,8}hs[153]
hs[7]{6,12}hs[133]
hs[8]{6,8}hs[104]
hs[9]{6,8}hs[152]
hs[10]{6,8}hs[154]
hs[11]{6,12}hs[135]
hs[12]{8,12}hs[50]
hs[13]{8,12}hs[48]
hs[14]{8,10}hs[41]
hs[15]{8,8}hs[86]
hs[16]{8,12}hs[51]
hs[17]{8,8}hs[17]
hs[18]{8,10}hs[406]
hs[19]{8,8}hs[332]
hs[20]{8,15}hs[225]
hs[21]{8,15}hs[177]
hs[22]{8,15}hs[255]
hs[23]{8,15}hs[178]
hs[24]{8,8}hs[373]
hs[25]{8,5}hs[264]
hs[26]{8,10}hs[379]
hs[27]{8,8}hs[330]
hs[28]{8,15}hs[183]
hs[29]{8,8}hs[363]
hs[30]{8,10}hs[433]
hs[31]{8,7}hs[289]
hs[32]{8,7}hs[319]
hs[33]{8,6}hs[451]
hs[34]{8,10}hs[430]
hs[35]{8,10}hs[397]
hs[36]{8,15}hs[231]
hs[37]{8,10}hs[417]
hs[38]{8,15}hs[251]
hs[39]{8,15}hs[180]
hs[40]{10,12}hs[46]
hs[41]{10,8}hs[14]
hs[42]{10,8}hs[331]
hs[43]{10,15}hs[218]
hs[44]{12,12}hs[44]
hs[45]{12,8}hs[87]
hs[46]{12,10}hs[40]
hs[47]{12,12}hs[49]
hs[48]{12,8}hs[13]
hs[49]{12,12}hs[47]
hs[50]{12,8}hs[12]
hs[51]{12,8}hs[16]
hs[52]{12,15}hs[208]
hs[53]{12,10}hs[435]
hs[54]{12,7}hs[290]
hs[55]{12,7}hs[271]
hs[56]{12,15}hs[182]
hs[57]{12,10}hs[377]
hs[58]{12,15}hs[176]
hs[59]{12,10}hs[407]
hs[60]{12,6}hs[444]
hs[61]{12,15}hs[202]
hs[62]{12,15}hs[181]
hs[63]{12,8}hs[351]
hs[64]{12,8}hs[354]
hs[65]{12,10}hs[402]
hs[66]{12,7}hs[318]
hs[67]{12,7}hs[296]
hs[68]{12,10}hs[418]
hs[69]{12,10}hs[416]
hs[70]{12,15}hs[256]
hs[71]{12,7}hs[269]
hs[72]{12,8}hs[346]
hs[73]{12,10}hs[408]
hs[74]{12,15}hs[179]
hs[75]{12,15}hs[250]
hs[76]{4,15}hs[260]
hs[77]{4,7}hs[270]
hs[78]{4,15}hs[210]
hs[79]{6,10}hs[403]
hs[80]{6,10}hs[376]
hs[81]{6,15}hs[175]
hs[82]{6,7}hs[291]
hs[83]{6,15}hs[219]
hs[84]{6,10}hs[398]
hs[85]{6,10}hs[439]
hs[86]{8,8}hs[15]
hs[87]{8,12}hs[45]
hs[88]{8,15}hs[247]
hs[89]{8,7}hs[295]
hs[90]{8,10}hs[381]
hs[91]{8,8}hs[375]
hs[92]{8,15}hs[220]
hs[93]{8,15}hs[226]
hs[94]{8,10}hs[378]
hs[95]{8,10}hs[400]
hs[96]{8,7}hs[268]
hs[97]{8,15}hs[224]
hs[98]{8,7}hs[272]
hs[99]{8,8}hs[333]
hs[100]{8,10}hs[380]
hs[101]{8,15}hs[252]
hs[102]{4,15}hs[174]
hs[103]{4,15}hs[212]
hs[104]{8,6}hs[8]
hs[105]{8,12}hs[138]
hs[106]{8,6}hs[173]
hs[107]{8,8}hs[156]
hs[108]{8,12}hs[128]
hs[109]{8,12}hs[130]
hs[110]{8,6}hs[148]
hs[111]{8,5}hs[2]
hs[112]{8,8}hs[112]
hs[113]{8,8}hs[113]
hs[114]{8,5}hs[3]
hs[115]{8,10}hs[166]
hs[116]{8,6}hs[5]
hs[117]{8,8}hs[159]
hs[118]{8,12}hs[140]
hs[119]{8,8}hs[119]
hs[120]{8,12}hs[126]
hs[121]{8,10}hs[169]
hs[122]{8,8}hs[165]
hs[123]{8,8}hs[123]
hs[124]{12,12}hs[136]
hs[125]{12,6}hs[172]
hs[126]{12,8}hs[120]
hs[127]{12,12}hs[127]
hs[128]{12,8}hs[108]
hs[129]{12,6}hs[146]
hs[130]{12,8}hs[109]
hs[131]{12,8}hs[162]
hs[132]{12,10}hs[168]
hs[133]{12,6}hs[7]
hs[134]{12,8}hs[157]
hs[135]{12,6}hs[11]
hs[136]{12,12}hs[124]
hs[137]{12,10}hs[167]
hs[138]{12,8}hs[105]
hs[139]{12,8}hs[158]
hs[140]{12,8}hs[118]
hs[141]{12,5}hs[4]
hs[142]{12,12}hs[142]
hs[143]{12,5}hs[1]
hs[144]{12,6}hs[150]
hs[145]{12,8}hs[164]
hs[146]{6,12}hs[129]
hs[147]{6,8}hs[163]
hs[148]{6,8}hs[110]
hs[149]{6,6}hs[149]
hs[150]{6,12}hs[144]
hs[151]{6,6}hs[151]
hs[152]{8,6}hs[9]
hs[153]{8,6}hs[6]
hs[154]{8,6}hs[10]
hs[155]{8,8}hs[155]
hs[156]{8,8}hs[107]
hs[157]{8,12}hs[134]
hs[158]{8,12}hs[139]
hs[159]{8,8}hs[117]
hs[160]{8,10}hs[171]
hs[161]{8,10}hs[170]
hs[162]{8,12}hs[131]
hs[163]{8,6}hs[147]
hs[164]{8,12}hs[145]
hs[165]{8,8}hs[122]
hs[166]{10,8}hs[115]
hs[167]{10,12}hs[137]
hs[168]{10,12}hs[132]
hs[169]{10,8}hs[121]
hs[170]{10,8}hs[161]
hs[171]{10,8}hs[160]
hs[172]{6,12}hs[125]
hs[173]{6,8}hs[106]
hs[174]{15,4}hs[102]
hs[175]{15,6}hs[81]
hs[176]{15,12}hs[58]
hs[177]{15,8}hs[21]
hs[178]{15,8}hs[23]
hs[179]{15,12}hs[74]
hs[180]{15,8}hs[39]
hs[181]{15,12}hs[62]
hs[182]{15,12}hs[56]
hs[183]{15,8}hs[28]
hs[184]{15,6}hs[452]
hs[185]{15,8}hs[347]
hs[186]{15,15}hs[186]
hs[187]{15,15}hs[223]
hs[188]{15,15}hs[188]
hs[189]{15,7}hs[280]
hs[190]{15,10}hs[385]
hs[191]{15,10}hs[401]
hs[192]{15,8}hs[336]
hs[193]{15,5}hs[267]
hs[194]{15,15}hs[209]
hs[195]{15,15}hs[195]
hs[196]{15,10}hs[442]
hs[197]{15,15}hs[197]
hs[198]{15,7}hs[311]
hs[199]{15,10}hs[384]
hs[200]{15,6}hs[462]
hs[201]{15,8}hs[372]
hs[202]{15,12}hs[61]
hs[203]{15,10}hs[411]
hs[204]{15,7}hs[314]
hs[205]{15,8}hs[367]
hs[206]{15,5}hs[455]
hs[207]{15,7}hs[312]
hs[208]{15,12}hs[52]
hs[209]{15,15}hs[194]
hs[210]{15,4}hs[78]
hs[211]{15,10}hs[386]
hs[212]{15,4}hs[103]
hs[213]{15,6}hs[450]
hs[214]{15,3}hs[458]
hs[215]{15,4}hs[465]
hs[216]{15,4}hs[459]
hs[217]{15,15}hs[217]
hs[218]{15,10}hs[43]
hs[219]{15,6}hs[83]
hs[220]{15,8}hs[92]
hs[221]{15,10}hs[421]
hs[222]{15,8}hs[365]
hs[223]{15,15}hs[187]
hs[224]{15,8}hs[97]
hs[225]{15,8}hs[20]
hs[226]{15,8}hs[93]
hs[227]{15,7}hs[288]
hs[228]{15,10}hs[412]
hs[229]{15,6}hs[453]
hs[230]{15,10}hs[405]
hs[231]{15,8}hs[36]
hs[232]{15,8}hs[337]
hs[233]{15,10}hs[434]
hs[234]{15,15}hs[234]
hs[235]{15,15}hs[235]
hs[236]{15,7}hs[304]
hs[237]{15,10}hs[391]
hs[238]{15,10}hs[390]
hs[239]{15,15}hs[239]
hs[240]{15,10}hs[422]
hs[241]{15,15}hs[241]
hs[242]{15,7}hs[273]
hs[243]{15,15}hs[243]
hs[244]{15,8}hs[356]
hs[245]{15,7}hs[286]
hs[246]{15,15}hs[246]
hs[247]{15,8}hs[88]
hs[248]{15,15}hs[248]
hs[249]{15,15}hs[249]
hs[250]{15,12}hs[75]
hs[251]{15,8}hs[38]
hs[252]{15,8}hs[101]
hs[253]{15,8}hs[374]
hs[254]{15,15}hs[254]
hs[255]{15,8}hs[22]
hs[256]{15,12}hs[70]
hs[257]{15,4}hs[457]
hs[258]{15,8}hs[357]
hs[259]{15,7}hs[283]
hs[260]{15,4}hs[76]
hs[261]{15,8}hs[343]
hs[262]{15,7}hs[315]
hs[263]{15,7}hs[279]
hs[264]{5,8}hs[25]
hs[265]{5,7}hs[325]
hs[266]{5,8}hs[359]
hs[267]{5,15}hs[193]
hs[268]{7,8}hs[96]
hs[269]{7,12}hs[71]
hs[270]{7,4}hs[77]
hs[271]{7,12}hs[55]
hs[272]{7,8}hs[98]
hs[273]{7,15}hs[242]
hs[274]{7,4}hs[463]
hs[275]{7,10}hs[441]
hs[276]{7,8}hs[366]
hs[277]{7,10}hs[415]
hs[278]{7,8}hs[350]
hs[279]{7,15}hs[263]
hs[280]{7,15}hs[189]
hs[281]{7,10}hs[382]
hs[282]{7,7}hs[306]
hs[283]{7,15}hs[259]
hs[284]{7,7}hs[284]
hs[285]{7,6}hs[447]
hs[286]{7,15}hs[245]
hs[287]{7,8}hs[349]
hs[288]{7,15}hs[227]
hs[289]{7,8}hs[31]
hs[290]{7,12}hs[54]
hs[291]{7,6}hs[82]
hs[292]{7,6}hs[461]
hs[293]{7,5}hs[454]
hs[294]{7,7}hs[327]
hs[295]{7,8}hs[89]
hs[296]{7,12}hs[67]
hs[297]{7,8}hs[341]
hs[298]{7,6}hs[449]
hs[299]{7,10}hs[431]
hs[300]{7,7}hs[300]
hs[301]{7,8}hs[342]
hs[302]{7,7}hs[302]
hs[303]{7,10}hs[392]
hs[304]{7,15}hs[236]
hs[305]{7,7}hs[305]
hs[306]{7,7}hs[282]
hs[307]{7,8}hs[369]
hs[308]{7,8}hs[370]
hs[309]{7,8}hs[364]
hs[310]{7,10}hs[396]
hs[311]{7,15}hs[198]
hs[312]{7,15}hs[207]
hs[313]{7,10}hs[429]
hs[314]{7,15}hs[204]
hs[315]{7,15}hs[262]
hs[316]{7,10}hs[410]
hs[317]{7,10}hs[414]
hs[318]{7,12}hs[66]
hs[319]{7,8}hs[32]
hs[320]{7,10}hs[413]
hs[321]{7,10}hs[428]
hs[322]{7,7}hs[322]
hs[323]{7,8}hs[361]
hs[324]{7,6}hs[329]
hs[325]{7,5}hs[265]
hs[326]{7,8}hs[339]
hs[327]{7,7}hs[294]
hs[328]{6,8}hs[358]
hs[329]{6,7}hs[324]
hs[330]{8,8}hs[27]
hs[331]{8,10}hs[42]
hs[332]{8,8}hs[19]
hs[333]{8,8}hs[99]
hs[334]{8,6}hs[448]
hs[335]{8,10}hs[388]
hs[336]{8,15}hs[192]
hs[337]{8,15}hs[232]
hs[338]{8,8}hs[338]
hs[339]{8,7}hs[326]
hs[340]{8,10}hs[437]
hs[341]{8,7}hs[297]
hs[342]{8,7}hs[301]
hs[343]{8,15}hs[261]
hs[344]{8,10}hs[420]
hs[345]{8,8}hs[355]
hs[346]{8,12}hs[72]
hs[347]{8,15}hs[185]
hs[348]{8,8}hs[348]
hs[349]{8,7}hs[287]
hs[350]{8,7}hs[278]
hs[351]{8,12}hs[63]
hs[352]{8,10}hs[389]
hs[353]{8,10}hs[426]
hs[354]{8,12}hs[64]
hs[355]{8,8}hs[345]
hs[356]{8,15}hs[244]
hs[357]{8,15}hs[258]
hs[358]{8,6}hs[328]
hs[359]{8,5}hs[266]
hs[360]{8,8}hs[360]
hs[361]{8,7}hs[323]
hs[362]{8,10}hs[436]
hs[363]{8,8}hs[29]
hs[364]{8,7}hs[309]
hs[365]{8,15}hs[222]
hs[366]{8,7}hs[276]
hs[367]{8,15}hs[205]
hs[368]{8,10}hs[404]
hs[369]{8,7}hs[307]
hs[370]{8,7}hs[308]
hs[371]{8,10}hs[438]
hs[372]{8,15}hs[201]
hs[373]{8,8}hs[24]
hs[374]{8,15}hs[253]
hs[375]{8,8}hs[91]
hs[376]{10,6}hs[80]
hs[377]{10,12}hs[57]
hs[378]{10,8}hs[94]
hs[379]{10,8}hs[26]
hs[380]{10,8}hs[100]
hs[381]{10,8}hs[90]
hs[382]{10,7}hs[281]
hs[383]{10,6}hs[460]
hs[384]{10,15}hs[199]
hs[385]{10,15}hs[190]
hs[386]{10,15}hs[211]
hs[387]{10,6}hs[445]
hs[388]{10,8}hs[335]
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hs[464]{4,10}hs[425]
hs[465]{4,15}hs[215]

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