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Polytope of Type {3,6,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,20}*720
if this polytope has a name.
Group : SmallGroup(720,500)
Rank : 4
Schlafli Type : {3,6,20}
Number of vertices, edges, etc : 3, 9, 60, 20
Order of s0s1s2s3 : 60
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,6,20,2} of size 1440
Vertex Figure Of :
   {2,3,6,20} of size 1440
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,6,10}*360
   3-fold quotients : {3,2,20}*240
   5-fold quotients : {3,6,4}*144
   6-fold quotients : {3,2,10}*120
   10-fold quotients : {3,6,2}*72
   12-fold quotients : {3,2,5}*60
   15-fold quotients : {3,2,4}*48
   30-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,6,40}*1440, {6,6,20}*1440c
Permutation Representation (GAP) :
s0 := (  1,271)(  2,272)(  3,273)(  4,274)(  5,275)(  6,281)(  7,282)(  8,283)
(  9,284)( 10,285)( 11,276)( 12,277)( 13,278)( 14,279)( 15,280)( 16,301)
( 17,302)( 18,303)( 19,304)( 20,305)( 21,311)( 22,312)( 23,313)( 24,314)
( 25,315)( 26,306)( 27,307)( 28,308)( 29,309)( 30,310)( 31,286)( 32,287)
( 33,288)( 34,289)( 35,290)( 36,296)( 37,297)( 38,298)( 39,299)( 40,300)
( 41,291)( 42,292)( 43,293)( 44,294)( 45,295)( 46,316)( 47,317)( 48,318)
( 49,319)( 50,320)( 51,326)( 52,327)( 53,328)( 54,329)( 55,330)( 56,321)
( 57,322)( 58,323)( 59,324)( 60,325)( 61,346)( 62,347)( 63,348)( 64,349)
( 65,350)( 66,356)( 67,357)( 68,358)( 69,359)( 70,360)( 71,351)( 72,352)
( 73,353)( 74,354)( 75,355)( 76,331)( 77,332)( 78,333)( 79,334)( 80,335)
( 81,341)( 82,342)( 83,343)( 84,344)( 85,345)( 86,336)( 87,337)( 88,338)
( 89,339)( 90,340)( 91,226)( 92,227)( 93,228)( 94,229)( 95,230)( 96,236)
( 97,237)( 98,238)( 99,239)(100,240)(101,231)(102,232)(103,233)(104,234)
(105,235)(106,256)(107,257)(108,258)(109,259)(110,260)(111,266)(112,267)
(113,268)(114,269)(115,270)(116,261)(117,262)(118,263)(119,264)(120,265)
(121,241)(122,242)(123,243)(124,244)(125,245)(126,251)(127,252)(128,253)
(129,254)(130,255)(131,246)(132,247)(133,248)(134,249)(135,250)(136,181)
(137,182)(138,183)(139,184)(140,185)(141,191)(142,192)(143,193)(144,194)
(145,195)(146,186)(147,187)(148,188)(149,189)(150,190)(151,211)(152,212)
(153,213)(154,214)(155,215)(156,221)(157,222)(158,223)(159,224)(160,225)
(161,216)(162,217)(163,218)(164,219)(165,220)(166,196)(167,197)(168,198)
(169,199)(170,200)(171,206)(172,207)(173,208)(174,209)(175,210)(176,201)
(177,202)(178,203)(179,204)(180,205);;
s1 := (  1,291)(  2,292)(  3,293)(  4,294)(  5,295)(  6,286)(  7,287)(  8,288)
(  9,289)( 10,290)( 11,296)( 12,297)( 13,298)( 14,299)( 15,300)( 16,276)
( 17,277)( 18,278)( 19,279)( 20,280)( 21,271)( 22,272)( 23,273)( 24,274)
( 25,275)( 26,281)( 27,282)( 28,283)( 29,284)( 30,285)( 31,306)( 32,307)
( 33,308)( 34,309)( 35,310)( 36,301)( 37,302)( 38,303)( 39,304)( 40,305)
( 41,311)( 42,312)( 43,313)( 44,314)( 45,315)( 46,336)( 47,337)( 48,338)
( 49,339)( 50,340)( 51,331)( 52,332)( 53,333)( 54,334)( 55,335)( 56,341)
( 57,342)( 58,343)( 59,344)( 60,345)( 61,321)( 62,322)( 63,323)( 64,324)
( 65,325)( 66,316)( 67,317)( 68,318)( 69,319)( 70,320)( 71,326)( 72,327)
( 73,328)( 74,329)( 75,330)( 76,351)( 77,352)( 78,353)( 79,354)( 80,355)
( 81,346)( 82,347)( 83,348)( 84,349)( 85,350)( 86,356)( 87,357)( 88,358)
( 89,359)( 90,360)( 91,246)( 92,247)( 93,248)( 94,249)( 95,250)( 96,241)
( 97,242)( 98,243)( 99,244)(100,245)(101,251)(102,252)(103,253)(104,254)
(105,255)(106,231)(107,232)(108,233)(109,234)(110,235)(111,226)(112,227)
(113,228)(114,229)(115,230)(116,236)(117,237)(118,238)(119,239)(120,240)
(121,261)(122,262)(123,263)(124,264)(125,265)(126,256)(127,257)(128,258)
(129,259)(130,260)(131,266)(132,267)(133,268)(134,269)(135,270)(136,201)
(137,202)(138,203)(139,204)(140,205)(141,196)(142,197)(143,198)(144,199)
(145,200)(146,206)(147,207)(148,208)(149,209)(150,210)(151,186)(152,187)
(153,188)(154,189)(155,190)(156,181)(157,182)(158,183)(159,184)(160,185)
(161,191)(162,192)(163,193)(164,194)(165,195)(166,216)(167,217)(168,218)
(169,219)(170,220)(171,211)(172,212)(173,213)(174,214)(175,215)(176,221)
(177,222)(178,223)(179,224)(180,225);;
s2 := (  1,181)(  2,185)(  3,184)(  4,183)(  5,182)(  6,191)(  7,195)(  8,194)
(  9,193)( 10,192)( 11,186)( 12,190)( 13,189)( 14,188)( 15,187)( 16,196)
( 17,200)( 18,199)( 19,198)( 20,197)( 21,206)( 22,210)( 23,209)( 24,208)
( 25,207)( 26,201)( 27,205)( 28,204)( 29,203)( 30,202)( 31,211)( 32,215)
( 33,214)( 34,213)( 35,212)( 36,221)( 37,225)( 38,224)( 39,223)( 40,222)
( 41,216)( 42,220)( 43,219)( 44,218)( 45,217)( 46,226)( 47,230)( 48,229)
( 49,228)( 50,227)( 51,236)( 52,240)( 53,239)( 54,238)( 55,237)( 56,231)
( 57,235)( 58,234)( 59,233)( 60,232)( 61,241)( 62,245)( 63,244)( 64,243)
( 65,242)( 66,251)( 67,255)( 68,254)( 69,253)( 70,252)( 71,246)( 72,250)
( 73,249)( 74,248)( 75,247)( 76,256)( 77,260)( 78,259)( 79,258)( 80,257)
( 81,266)( 82,270)( 83,269)( 84,268)( 85,267)( 86,261)( 87,265)( 88,264)
( 89,263)( 90,262)( 91,316)( 92,320)( 93,319)( 94,318)( 95,317)( 96,326)
( 97,330)( 98,329)( 99,328)(100,327)(101,321)(102,325)(103,324)(104,323)
(105,322)(106,331)(107,335)(108,334)(109,333)(110,332)(111,341)(112,345)
(113,344)(114,343)(115,342)(116,336)(117,340)(118,339)(119,338)(120,337)
(121,346)(122,350)(123,349)(124,348)(125,347)(126,356)(127,360)(128,359)
(129,358)(130,357)(131,351)(132,355)(133,354)(134,353)(135,352)(136,271)
(137,275)(138,274)(139,273)(140,272)(141,281)(142,285)(143,284)(144,283)
(145,282)(146,276)(147,280)(148,279)(149,278)(150,277)(151,286)(152,290)
(153,289)(154,288)(155,287)(156,296)(157,300)(158,299)(159,298)(160,297)
(161,291)(162,295)(163,294)(164,293)(165,292)(166,301)(167,305)(168,304)
(169,303)(170,302)(171,311)(172,315)(173,314)(174,313)(175,312)(176,306)
(177,310)(178,309)(179,308)(180,307);;
s3 := (  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)( 18, 20)
( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 32)( 33, 35)( 36, 37)( 38, 40)
( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)( 58, 60)
( 61, 62)( 63, 65)( 66, 67)( 68, 70)( 71, 72)( 73, 75)( 76, 77)( 78, 80)
( 81, 82)( 83, 85)( 86, 87)( 88, 90)( 91,137)( 92,136)( 93,140)( 94,139)
( 95,138)( 96,142)( 97,141)( 98,145)( 99,144)(100,143)(101,147)(102,146)
(103,150)(104,149)(105,148)(106,152)(107,151)(108,155)(109,154)(110,153)
(111,157)(112,156)(113,160)(114,159)(115,158)(116,162)(117,161)(118,165)
(119,164)(120,163)(121,167)(122,166)(123,170)(124,169)(125,168)(126,172)
(127,171)(128,175)(129,174)(130,173)(131,177)(132,176)(133,180)(134,179)
(135,178)(181,227)(182,226)(183,230)(184,229)(185,228)(186,232)(187,231)
(188,235)(189,234)(190,233)(191,237)(192,236)(193,240)(194,239)(195,238)
(196,242)(197,241)(198,245)(199,244)(200,243)(201,247)(202,246)(203,250)
(204,249)(205,248)(206,252)(207,251)(208,255)(209,254)(210,253)(211,257)
(212,256)(213,260)(214,259)(215,258)(216,262)(217,261)(218,265)(219,264)
(220,263)(221,267)(222,266)(223,270)(224,269)(225,268)(271,272)(273,275)
(276,277)(278,280)(281,282)(283,285)(286,287)(288,290)(291,292)(293,295)
(296,297)(298,300)(301,302)(303,305)(306,307)(308,310)(311,312)(313,315)
(316,317)(318,320)(321,322)(323,325)(326,327)(328,330)(331,332)(333,335)
(336,337)(338,340)(341,342)(343,345)(346,347)(348,350)(351,352)(353,355)
(356,357)(358,360);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(360)!(  1,271)(  2,272)(  3,273)(  4,274)(  5,275)(  6,281)(  7,282)
(  8,283)(  9,284)( 10,285)( 11,276)( 12,277)( 13,278)( 14,279)( 15,280)
( 16,301)( 17,302)( 18,303)( 19,304)( 20,305)( 21,311)( 22,312)( 23,313)
( 24,314)( 25,315)( 26,306)( 27,307)( 28,308)( 29,309)( 30,310)( 31,286)
( 32,287)( 33,288)( 34,289)( 35,290)( 36,296)( 37,297)( 38,298)( 39,299)
( 40,300)( 41,291)( 42,292)( 43,293)( 44,294)( 45,295)( 46,316)( 47,317)
( 48,318)( 49,319)( 50,320)( 51,326)( 52,327)( 53,328)( 54,329)( 55,330)
( 56,321)( 57,322)( 58,323)( 59,324)( 60,325)( 61,346)( 62,347)( 63,348)
( 64,349)( 65,350)( 66,356)( 67,357)( 68,358)( 69,359)( 70,360)( 71,351)
( 72,352)( 73,353)( 74,354)( 75,355)( 76,331)( 77,332)( 78,333)( 79,334)
( 80,335)( 81,341)( 82,342)( 83,343)( 84,344)( 85,345)( 86,336)( 87,337)
( 88,338)( 89,339)( 90,340)( 91,226)( 92,227)( 93,228)( 94,229)( 95,230)
( 96,236)( 97,237)( 98,238)( 99,239)(100,240)(101,231)(102,232)(103,233)
(104,234)(105,235)(106,256)(107,257)(108,258)(109,259)(110,260)(111,266)
(112,267)(113,268)(114,269)(115,270)(116,261)(117,262)(118,263)(119,264)
(120,265)(121,241)(122,242)(123,243)(124,244)(125,245)(126,251)(127,252)
(128,253)(129,254)(130,255)(131,246)(132,247)(133,248)(134,249)(135,250)
(136,181)(137,182)(138,183)(139,184)(140,185)(141,191)(142,192)(143,193)
(144,194)(145,195)(146,186)(147,187)(148,188)(149,189)(150,190)(151,211)
(152,212)(153,213)(154,214)(155,215)(156,221)(157,222)(158,223)(159,224)
(160,225)(161,216)(162,217)(163,218)(164,219)(165,220)(166,196)(167,197)
(168,198)(169,199)(170,200)(171,206)(172,207)(173,208)(174,209)(175,210)
(176,201)(177,202)(178,203)(179,204)(180,205);
s1 := Sym(360)!(  1,291)(  2,292)(  3,293)(  4,294)(  5,295)(  6,286)(  7,287)
(  8,288)(  9,289)( 10,290)( 11,296)( 12,297)( 13,298)( 14,299)( 15,300)
( 16,276)( 17,277)( 18,278)( 19,279)( 20,280)( 21,271)( 22,272)( 23,273)
( 24,274)( 25,275)( 26,281)( 27,282)( 28,283)( 29,284)( 30,285)( 31,306)
( 32,307)( 33,308)( 34,309)( 35,310)( 36,301)( 37,302)( 38,303)( 39,304)
( 40,305)( 41,311)( 42,312)( 43,313)( 44,314)( 45,315)( 46,336)( 47,337)
( 48,338)( 49,339)( 50,340)( 51,331)( 52,332)( 53,333)( 54,334)( 55,335)
( 56,341)( 57,342)( 58,343)( 59,344)( 60,345)( 61,321)( 62,322)( 63,323)
( 64,324)( 65,325)( 66,316)( 67,317)( 68,318)( 69,319)( 70,320)( 71,326)
( 72,327)( 73,328)( 74,329)( 75,330)( 76,351)( 77,352)( 78,353)( 79,354)
( 80,355)( 81,346)( 82,347)( 83,348)( 84,349)( 85,350)( 86,356)( 87,357)
( 88,358)( 89,359)( 90,360)( 91,246)( 92,247)( 93,248)( 94,249)( 95,250)
( 96,241)( 97,242)( 98,243)( 99,244)(100,245)(101,251)(102,252)(103,253)
(104,254)(105,255)(106,231)(107,232)(108,233)(109,234)(110,235)(111,226)
(112,227)(113,228)(114,229)(115,230)(116,236)(117,237)(118,238)(119,239)
(120,240)(121,261)(122,262)(123,263)(124,264)(125,265)(126,256)(127,257)
(128,258)(129,259)(130,260)(131,266)(132,267)(133,268)(134,269)(135,270)
(136,201)(137,202)(138,203)(139,204)(140,205)(141,196)(142,197)(143,198)
(144,199)(145,200)(146,206)(147,207)(148,208)(149,209)(150,210)(151,186)
(152,187)(153,188)(154,189)(155,190)(156,181)(157,182)(158,183)(159,184)
(160,185)(161,191)(162,192)(163,193)(164,194)(165,195)(166,216)(167,217)
(168,218)(169,219)(170,220)(171,211)(172,212)(173,213)(174,214)(175,215)
(176,221)(177,222)(178,223)(179,224)(180,225);
s2 := Sym(360)!(  1,181)(  2,185)(  3,184)(  4,183)(  5,182)(  6,191)(  7,195)
(  8,194)(  9,193)( 10,192)( 11,186)( 12,190)( 13,189)( 14,188)( 15,187)
( 16,196)( 17,200)( 18,199)( 19,198)( 20,197)( 21,206)( 22,210)( 23,209)
( 24,208)( 25,207)( 26,201)( 27,205)( 28,204)( 29,203)( 30,202)( 31,211)
( 32,215)( 33,214)( 34,213)( 35,212)( 36,221)( 37,225)( 38,224)( 39,223)
( 40,222)( 41,216)( 42,220)( 43,219)( 44,218)( 45,217)( 46,226)( 47,230)
( 48,229)( 49,228)( 50,227)( 51,236)( 52,240)( 53,239)( 54,238)( 55,237)
( 56,231)( 57,235)( 58,234)( 59,233)( 60,232)( 61,241)( 62,245)( 63,244)
( 64,243)( 65,242)( 66,251)( 67,255)( 68,254)( 69,253)( 70,252)( 71,246)
( 72,250)( 73,249)( 74,248)( 75,247)( 76,256)( 77,260)( 78,259)( 79,258)
( 80,257)( 81,266)( 82,270)( 83,269)( 84,268)( 85,267)( 86,261)( 87,265)
( 88,264)( 89,263)( 90,262)( 91,316)( 92,320)( 93,319)( 94,318)( 95,317)
( 96,326)( 97,330)( 98,329)( 99,328)(100,327)(101,321)(102,325)(103,324)
(104,323)(105,322)(106,331)(107,335)(108,334)(109,333)(110,332)(111,341)
(112,345)(113,344)(114,343)(115,342)(116,336)(117,340)(118,339)(119,338)
(120,337)(121,346)(122,350)(123,349)(124,348)(125,347)(126,356)(127,360)
(128,359)(129,358)(130,357)(131,351)(132,355)(133,354)(134,353)(135,352)
(136,271)(137,275)(138,274)(139,273)(140,272)(141,281)(142,285)(143,284)
(144,283)(145,282)(146,276)(147,280)(148,279)(149,278)(150,277)(151,286)
(152,290)(153,289)(154,288)(155,287)(156,296)(157,300)(158,299)(159,298)
(160,297)(161,291)(162,295)(163,294)(164,293)(165,292)(166,301)(167,305)
(168,304)(169,303)(170,302)(171,311)(172,315)(173,314)(174,313)(175,312)
(176,306)(177,310)(178,309)(179,308)(180,307);
s3 := Sym(360)!(  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 17)
( 18, 20)( 21, 22)( 23, 25)( 26, 27)( 28, 30)( 31, 32)( 33, 35)( 36, 37)
( 38, 40)( 41, 42)( 43, 45)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)
( 58, 60)( 61, 62)( 63, 65)( 66, 67)( 68, 70)( 71, 72)( 73, 75)( 76, 77)
( 78, 80)( 81, 82)( 83, 85)( 86, 87)( 88, 90)( 91,137)( 92,136)( 93,140)
( 94,139)( 95,138)( 96,142)( 97,141)( 98,145)( 99,144)(100,143)(101,147)
(102,146)(103,150)(104,149)(105,148)(106,152)(107,151)(108,155)(109,154)
(110,153)(111,157)(112,156)(113,160)(114,159)(115,158)(116,162)(117,161)
(118,165)(119,164)(120,163)(121,167)(122,166)(123,170)(124,169)(125,168)
(126,172)(127,171)(128,175)(129,174)(130,173)(131,177)(132,176)(133,180)
(134,179)(135,178)(181,227)(182,226)(183,230)(184,229)(185,228)(186,232)
(187,231)(188,235)(189,234)(190,233)(191,237)(192,236)(193,240)(194,239)
(195,238)(196,242)(197,241)(198,245)(199,244)(200,243)(201,247)(202,246)
(203,250)(204,249)(205,248)(206,252)(207,251)(208,255)(209,254)(210,253)
(211,257)(212,256)(213,260)(214,259)(215,258)(216,262)(217,261)(218,265)
(219,264)(220,263)(221,267)(222,266)(223,270)(224,269)(225,268)(271,272)
(273,275)(276,277)(278,280)(281,282)(283,285)(286,287)(288,290)(291,292)
(293,295)(296,297)(298,300)(301,302)(303,305)(306,307)(308,310)(311,312)
(313,315)(316,317)(318,320)(321,322)(323,325)(326,327)(328,330)(331,332)
(333,335)(336,337)(338,340)(341,342)(343,345)(346,347)(348,350)(351,352)
(353,355)(356,357)(358,360);
poly := sub<Sym(360)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
to this polytope