Polytope of Type {40,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {40,6}*1920d
if this polytope has a name.
Group : SmallGroup(1920,238620)
Rank : 3
Schlafli Type : {40,6}
Number of vertices, edges, etc : 160, 480, 24
Order of s0s1s2 : 120
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {20,6}*960e
   4-fold quotients : {40,6}*480, {20,6}*480c
   5-fold quotients : {8,6}*384g
   8-fold quotients : {20,6}*240a, {20,6}*240b
   10-fold quotients : {4,6}*192b
   12-fold quotients : {40,2}*160
   16-fold quotients : {10,6}*120
   20-fold quotients : {8,6}*96, {4,6}*96
   24-fold quotients : {20,2}*80
   40-fold quotients : {4,6}*48a, {4,3}*48, {4,6}*48b, {4,6}*48c
   48-fold quotients : {10,2}*40
   60-fold quotients : {8,2}*32
   80-fold quotients : {4,3}*24, {2,6}*24
   96-fold quotients : {5,2}*20
   120-fold quotients : {4,2}*16
   160-fold quotients : {2,3}*12
   240-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
      12 facets:
         12 of {40}*80
      80 vertex figures:
         80 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2> of order 2.
      12 facets:
         12 of {40}*80
      80 vertex figures:
         80 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      12 facets:
         12 of {40}*80
      80 vertex figures:
         80 of {6}*12
   P/N, where N=<s1*s2*s1*s2> of order 3.
      8 facets:
         8 of {40}*80
      80 vertex figures:
         40 of {2}*4
         40 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1> of order 4.
      6 facets:
         6 of {40}*80
      40 vertex figures:
         40 of {6}*12
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 4.
      6 facets:
         6 of {40}*80
      40 vertex figures:
         40 of {6}*12
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 4.
      6 facets:
         6 of {40}*80
      40 vertex figures:
         40 of {6}*12

Permutation Representation (GAP) : 
s0 := (  1,  3)(  2,  4)(  5, 19)(  6, 20)(  7, 17)(  8, 18)(  9, 15)( 10, 16)( 11, 13)( 12, 14)( 21, 23)( 22, 24)( 25, 39)( 26, 40)( 27, 37)( 28, 38)( 29, 35)( 30, 36)( 31, 33)( 32, 34)( 41, 43)( 42, 44)( 45, 59)( 46, 60)( 47, 57)( 48, 58)( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 61, 63)( 62, 64)( 65, 79)( 66, 80)( 67, 77)( 68, 78)( 69, 75)( 70, 76)( 71, 73)( 72, 74)( 81, 83)( 82, 84)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89, 95)( 90, 96)( 91, 93)( 92, 94)(101,103)(102,104)(105,119)(106,120)(107,117)(108,118)(109,115)(110,116)(111,113)(112,114)(121,183)(122,184)(123,181)(124,182)(125,199)(126,200)(127,197)(128,198)(129,195)(130,196)(131,193)(132,194)(133,191)(134,192)(135,189)(136,190)(137,187)(138,188)(139,185)(140,186)(141,203)(142,204)(143,201)(144,202)(145,219)(146,220)(147,217)(148,218)(149,215)(150,216)(151,213)(152,214)(153,211)(154,212)(155,209)(156,210)(157,207)(158,208)(159,205)(160,206)(161,223)(162,224)(163,221)(164,222)(165,239)(166,240)(167,237)(168,238)(169,235)(170,236)(171,233)(172,234)(173,231)(174,232)(175,229)(176,230)(177,227)(178,228)(179,225)(180,226)(241,363)(242,364)(243,361)(244,362)(245,379)(246,380)(247,377)(248,378)(249,375)(250,376)(251,373)(252,374)(253,371)(254,372)(255,369)(256,370)(257,367)(258,368)(259,365)(260,366)(261,383)(262,384)(263,381)(264,382)(265,399)(266,400)(267,397)(268,398)(269,395)(270,396)(271,393)(272,394)(273,391)(274,392)(275,389)(276,390)(277,387)(278,388)(279,385)(280,386)(281,403)(282,404)(283,401)(284,402)(285,419)(286,420)(287,417)(288,418)(289,415)(290,416)(291,413)(292,414)(293,411)(294,412)(295,409)(296,410)(297,407)(298,408)(299,405)(300,406)(301,423)(302,424)(303,421)(304,422)(305,439)(306,440)(307,437)(308,438)(309,435)(310,436)(311,433)(312,434)(313,431)(314,432)(315,429)(316,430)(317,427)(318,428)(319,425)(320,426)(321,443)(322,444)(323,441)(324,442)(325,459)(326,460)(327,457)(328,458)(329,455)(330,456)(331,453)(332,454)(333,451)(334,452)(335,449)(336,450)(337,447)(338,448)(339,445)(340,446)(341,463)(342,464)(343,461)(344,462)(345,479)(346,480)(347,477)(348,478)(349,475)(350,476)(351,473)(352,474)(353,471)(354,472)(355,469)(356,470)(357,467)(358,468)(359,465)(360,466);;
s1 := (  1,245)(  2,246)(  3,248)(  4,247)(  5,241)(  6,242)(  7,244)(  8,243)(  9,257)( 10,258)( 11,260)( 12,259)( 13,253)( 14,254)( 15,256)( 16,255)( 17,249)( 18,250)( 19,252)( 20,251)( 21,285)( 22,286)( 23,288)( 24,287)( 25,281)( 26,282)( 27,284)( 28,283)( 29,297)( 30,298)( 31,300)( 32,299)( 33,293)( 34,294)( 35,296)( 36,295)( 37,289)( 38,290)( 39,292)( 40,291)( 41,265)( 42,266)( 43,268)( 44,267)( 45,261)( 46,262)( 47,264)( 48,263)( 49,277)( 50,278)( 51,280)( 52,279)( 53,273)( 54,274)( 55,276)( 56,275)( 57,269)( 58,270)( 59,272)( 60,271)( 61,305)( 62,306)( 63,308)( 64,307)( 65,301)( 66,302)( 67,304)( 68,303)( 69,317)( 70,318)( 71,320)( 72,319)( 73,313)( 74,314)( 75,316)( 76,315)( 77,309)( 78,310)( 79,312)( 80,311)( 81,345)( 82,346)( 83,348)( 84,347)( 85,341)( 86,342)( 87,344)( 88,343)( 89,357)( 90,358)( 91,360)( 92,359)( 93,353)( 94,354)( 95,356)( 96,355)( 97,349)( 98,350)( 99,352)(100,351)(101,325)(102,326)(103,328)(104,327)(105,321)(106,322)(107,324)(108,323)(109,337)(110,338)(111,340)(112,339)(113,333)(114,334)(115,336)(116,335)(117,329)(118,330)(119,332)(120,331)(121,425)(122,426)(123,428)(124,427)(125,421)(126,422)(127,424)(128,423)(129,437)(130,438)(131,440)(132,439)(133,433)(134,434)(135,436)(136,435)(137,429)(138,430)(139,432)(140,431)(141,465)(142,466)(143,468)(144,467)(145,461)(146,462)(147,464)(148,463)(149,477)(150,478)(151,480)(152,479)(153,473)(154,474)(155,476)(156,475)(157,469)(158,470)(159,472)(160,471)(161,445)(162,446)(163,448)(164,447)(165,441)(166,442)(167,444)(168,443)(169,457)(170,458)(171,460)(172,459)(173,453)(174,454)(175,456)(176,455)(177,449)(178,450)(179,452)(180,451)(181,365)(182,366)(183,368)(184,367)(185,361)(186,362)(187,364)(188,363)(189,377)(190,378)(191,380)(192,379)(193,373)(194,374)(195,376)(196,375)(197,369)(198,370)(199,372)(200,371)(201,405)(202,406)(203,408)(204,407)(205,401)(206,402)(207,404)(208,403)(209,417)(210,418)(211,420)(212,419)(213,413)(214,414)(215,416)(216,415)(217,409)(218,410)(219,412)(220,411)(221,385)(222,386)(223,388)(224,387)(225,381)(226,382)(227,384)(228,383)(229,397)(230,398)(231,400)(232,399)(233,393)(234,394)(235,396)(236,395)(237,389)(238,390)(239,392)(240,391);;
s2 := (  1, 21)(  2, 24)(  3, 23)(  4, 22)(  5, 25)(  6, 28)(  7, 27)(  8, 26)(  9, 29)( 10, 32)( 11, 31)( 12, 30)( 13, 33)( 14, 36)( 15, 35)( 16, 34)( 17, 37)( 18, 40)( 19, 39)( 20, 38)( 42, 44)( 46, 48)( 50, 52)( 54, 56)( 58, 60)( 61, 81)( 62, 84)( 63, 83)( 64, 82)( 65, 85)( 66, 88)( 67, 87)( 68, 86)( 69, 89)( 70, 92)( 71, 91)( 72, 90)( 73, 93)( 74, 96)( 75, 95)( 76, 94)( 77, 97)( 78,100)( 79, 99)( 80, 98)(102,104)(106,108)(110,112)(114,116)(118,120)(121,141)(122,144)(123,143)(124,142)(125,145)(126,148)(127,147)(128,146)(129,149)(130,152)(131,151)(132,150)(133,153)(134,156)(135,155)(136,154)(137,157)(138,160)(139,159)(140,158)(162,164)(166,168)(170,172)(174,176)(178,180)(181,201)(182,204)(183,203)(184,202)(185,205)(186,208)(187,207)(188,206)(189,209)(190,212)(191,211)(192,210)(193,213)(194,216)(195,215)(196,214)(197,217)(198,220)(199,219)(200,218)(222,224)(226,228)(230,232)(234,236)(238,240)(241,261)(242,264)(243,263)(244,262)(245,265)(246,268)(247,267)(248,266)(249,269)(250,272)(251,271)(252,270)(253,273)(254,276)(255,275)(256,274)(257,277)(258,280)(259,279)(260,278)(282,284)(286,288)(290,292)(294,296)(298,300)(301,321)(302,324)(303,323)(304,322)(305,325)(306,328)(307,327)(308,326)(309,329)(310,332)(311,331)(312,330)(313,333)(314,336)(315,335)(316,334)(317,337)(318,340)(319,339)(320,338)(342,344)(346,348)(350,352)(354,356)(358,360)(361,381)(362,384)(363,383)(364,382)(365,385)(366,388)(367,387)(368,386)(369,389)(370,392)(371,391)(372,390)(373,393)(374,396)(375,395)(376,394)(377,397)(378,400)(379,399)(380,398)(402,404)(406,408)(410,412)(414,416)(418,420)(421,441)(422,444)(423,443)(424,442)(425,445)(426,448)(427,447)(428,446)(429,449)(430,452)(431,451)(432,450)(433,453)(434,456)(435,455)(436,454)(437,457)(438,460)(439,459)(440,458)(462,464)(466,468)(470,472)(474,476)(478,480);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) : 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(480)!(  1,  3)(  2,  4)(  5, 19)(  6, 20)(  7, 17)(  8, 18)(  9, 15)( 10, 16)( 11, 13)( 12, 14)( 21, 23)( 22, 24)( 25, 39)( 26, 40)( 27, 37)( 28, 38)( 29, 35)( 30, 36)( 31, 33)( 32, 34)( 41, 43)( 42, 44)( 45, 59)( 46, 60)( 47, 57)( 48, 58)( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 61, 63)( 62, 64)( 65, 79)( 66, 80)( 67, 77)( 68, 78)( 69, 75)( 70, 76)( 71, 73)( 72, 74)( 81, 83)( 82, 84)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89, 95)( 90, 96)( 91, 93)( 92, 94)(101,103)(102,104)(105,119)(106,120)(107,117)(108,118)(109,115)(110,116)(111,113)(112,114)(121,183)(122,184)(123,181)(124,182)(125,199)(126,200)(127,197)(128,198)(129,195)(130,196)(131,193)(132,194)(133,191)(134,192)(135,189)(136,190)(137,187)(138,188)(139,185)(140,186)(141,203)(142,204)(143,201)(144,202)(145,219)(146,220)(147,217)(148,218)(149,215)(150,216)(151,213)(152,214)(153,211)(154,212)(155,209)(156,210)(157,207)(158,208)(159,205)(160,206)(161,223)(162,224)(163,221)(164,222)(165,239)(166,240)(167,237)(168,238)(169,235)(170,236)(171,233)(172,234)(173,231)(174,232)(175,229)(176,230)(177,227)(178,228)(179,225)(180,226)(241,363)(242,364)(243,361)(244,362)(245,379)(246,380)(247,377)(248,378)(249,375)(250,376)(251,373)(252,374)(253,371)(254,372)(255,369)(256,370)(257,367)(258,368)(259,365)(260,366)(261,383)(262,384)(263,381)(264,382)(265,399)(266,400)(267,397)(268,398)(269,395)(270,396)(271,393)(272,394)(273,391)(274,392)(275,389)(276,390)(277,387)(278,388)(279,385)(280,386)(281,403)(282,404)(283,401)(284,402)(285,419)(286,420)(287,417)(288,418)(289,415)(290,416)(291,413)(292,414)(293,411)(294,412)(295,409)(296,410)(297,407)(298,408)(299,405)(300,406)(301,423)(302,424)(303,421)(304,422)(305,439)(306,440)(307,437)(308,438)(309,435)(310,436)(311,433)(312,434)(313,431)(314,432)(315,429)(316,430)(317,427)(318,428)(319,425)(320,426)(321,443)(322,444)(323,441)(324,442)(325,459)(326,460)(327,457)(328,458)(329,455)(330,456)(331,453)(332,454)(333,451)(334,452)(335,449)(336,450)(337,447)(338,448)(339,445)(340,446)(341,463)(342,464)(343,461)(344,462)(345,479)(346,480)(347,477)(348,478)(349,475)(350,476)(351,473)(352,474)(353,471)(354,472)(355,469)(356,470)(357,467)(358,468)(359,465)(360,466);
s1 := Sym(480)!(  1,245)(  2,246)(  3,248)(  4,247)(  5,241)(  6,242)(  7,244)(  8,243)(  9,257)( 10,258)( 11,260)( 12,259)( 13,253)( 14,254)( 15,256)( 16,255)( 17,249)( 18,250)( 19,252)( 20,251)( 21,285)( 22,286)( 23,288)( 24,287)( 25,281)( 26,282)( 27,284)( 28,283)( 29,297)( 30,298)( 31,300)( 32,299)( 33,293)( 34,294)( 35,296)( 36,295)( 37,289)( 38,290)( 39,292)( 40,291)( 41,265)( 42,266)( 43,268)( 44,267)( 45,261)( 46,262)( 47,264)( 48,263)( 49,277)( 50,278)( 51,280)( 52,279)( 53,273)( 54,274)( 55,276)( 56,275)( 57,269)( 58,270)( 59,272)( 60,271)( 61,305)( 62,306)( 63,308)( 64,307)( 65,301)( 66,302)( 67,304)( 68,303)( 69,317)( 70,318)( 71,320)( 72,319)( 73,313)( 74,314)( 75,316)( 76,315)( 77,309)( 78,310)( 79,312)( 80,311)( 81,345)( 82,346)( 83,348)( 84,347)( 85,341)( 86,342)( 87,344)( 88,343)( 89,357)( 90,358)( 91,360)( 92,359)( 93,353)( 94,354)( 95,356)( 96,355)( 97,349)( 98,350)( 99,352)(100,351)(101,325)(102,326)(103,328)(104,327)(105,321)(106,322)(107,324)(108,323)(109,337)(110,338)(111,340)(112,339)(113,333)(114,334)(115,336)(116,335)(117,329)(118,330)(119,332)(120,331)(121,425)(122,426)(123,428)(124,427)(125,421)(126,422)(127,424)(128,423)(129,437)(130,438)(131,440)(132,439)(133,433)(134,434)(135,436)(136,435)(137,429)(138,430)(139,432)(140,431)(141,465)(142,466)(143,468)(144,467)(145,461)(146,462)(147,464)(148,463)(149,477)(150,478)(151,480)(152,479)(153,473)(154,474)(155,476)(156,475)(157,469)(158,470)(159,472)(160,471)(161,445)(162,446)(163,448)(164,447)(165,441)(166,442)(167,444)(168,443)(169,457)(170,458)(171,460)(172,459)(173,453)(174,454)(175,456)(176,455)(177,449)(178,450)(179,452)(180,451)(181,365)(182,366)(183,368)(184,367)(185,361)(186,362)(187,364)(188,363)(189,377)(190,378)(191,380)(192,379)(193,373)(194,374)(195,376)(196,375)(197,369)(198,370)(199,372)(200,371)(201,405)(202,406)(203,408)(204,407)(205,401)(206,402)(207,404)(208,403)(209,417)(210,418)(211,420)(212,419)(213,413)(214,414)(215,416)(216,415)(217,409)(218,410)(219,412)(220,411)(221,385)(222,386)(223,388)(224,387)(225,381)(226,382)(227,384)(228,383)(229,397)(230,398)(231,400)(232,399)(233,393)(234,394)(235,396)(236,395)(237,389)(238,390)(239,392)(240,391);
s2 := Sym(480)!(  1, 21)(  2, 24)(  3, 23)(  4, 22)(  5, 25)(  6, 28)(  7, 27)(  8, 26)(  9, 29)( 10, 32)( 11, 31)( 12, 30)( 13, 33)( 14, 36)( 15, 35)( 16, 34)( 17, 37)( 18, 40)( 19, 39)( 20, 38)( 42, 44)( 46, 48)( 50, 52)( 54, 56)( 58, 60)( 61, 81)( 62, 84)( 63, 83)( 64, 82)( 65, 85)( 66, 88)( 67, 87)( 68, 86)( 69, 89)( 70, 92)( 71, 91)( 72, 90)( 73, 93)( 74, 96)( 75, 95)( 76, 94)( 77, 97)( 78,100)( 79, 99)( 80, 98)(102,104)(106,108)(110,112)(114,116)(118,120)(121,141)(122,144)(123,143)(124,142)(125,145)(126,148)(127,147)(128,146)(129,149)(130,152)(131,151)(132,150)(133,153)(134,156)(135,155)(136,154)(137,157)(138,160)(139,159)(140,158)(162,164)(166,168)(170,172)(174,176)(178,180)(181,201)(182,204)(183,203)(184,202)(185,205)(186,208)(187,207)(188,206)(189,209)(190,212)(191,211)(192,210)(193,213)(194,216)(195,215)(196,214)(197,217)(198,220)(199,219)(200,218)(222,224)(226,228)(230,232)(234,236)(238,240)(241,261)(242,264)(243,263)(244,262)(245,265)(246,268)(247,267)(248,266)(249,269)(250,272)(251,271)(252,270)(253,273)(254,276)(255,275)(256,274)(257,277)(258,280)(259,279)(260,278)(282,284)(286,288)(290,292)(294,296)(298,300)(301,321)(302,324)(303,323)(304,322)(305,325)(306,328)(307,327)(308,326)(309,329)(310,332)(311,331)(312,330)(313,333)(314,336)(315,335)(316,334)(317,337)(318,340)(319,339)(320,338)(342,344)(346,348)(350,352)(354,356)(358,360)(361,381)(362,384)(363,383)(364,382)(365,385)(366,388)(367,387)(368,386)(369,389)(370,392)(371,391)(372,390)(373,393)(374,396)(375,395)(376,394)(377,397)(378,400)(379,399)(380,398)(402,404)(406,408)(410,412)(414,416)(418,420)(421,441)(422,444)(423,443)(424,442)(425,445)(426,448)(427,447)(428,446)(429,449)(430,452)(431,451)(432,450)(433,453)(434,456)(435,455)(436,454)(437,457)(438,460)(439,459)(440,458)(462,464)(466,468)(470,472)(474,476)(478,480);
poly := sub<Sym(480)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) : 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope

Twisty Puzzle