Polytope of Type {36,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,6}*1728c
if this polytope has a name.
Group : SmallGroup(1728,30201)
Rank : 3
Schlafli Type : {36,6}
Number of vertices, edges, etc : 144, 432, 24
Order of s0s1s2 : 72
Order of s0s1s2s1 : 12
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 : {18,6}*864
3-fold quotients : {12,6}*576d
4-fold quotients : {9,6}*432
6-fold quotients : {6,6}*288b
8-fold quotients : {18,6}*216b
9-fold quotients : {12,6}*192b
12-fold quotients : {3,6}*144
16-fold quotients : {9,6}*108
18-fold quotients : {12,3}*96, {6,6}*96
24-fold quotients : {18,2}*72, {6,6}*72c
36-fold quotients : {3,6}*48, {6,3}*48
48-fold quotients : {9,2}*36, {3,6}*36
72-fold quotients : {3,3}*24, {6,2}*24
144-fold quotients : {3,2}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)( 13, 23)
( 14, 24)( 15, 21)( 16, 22)( 25, 65)( 26, 66)( 27, 68)( 28, 67)( 29, 71)
( 30, 72)( 31, 69)( 32, 70)( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 37, 63)
( 38, 64)( 39, 61)( 40, 62)( 41, 49)( 42, 50)( 43, 52)( 44, 51)( 45, 55)
( 46, 56)( 47, 53)( 48, 54)( 73,145)( 74,146)( 75,148)( 76,147)( 77,151)
( 78,152)( 79,149)( 80,150)( 81,161)( 82,162)( 83,164)( 84,163)( 85,167)
( 86,168)( 87,165)( 88,166)( 89,153)( 90,154)( 91,156)( 92,155)( 93,159)
( 94,160)( 95,157)( 96,158)( 97,209)( 98,210)( 99,212)(100,211)(101,215)
(102,216)(103,213)(104,214)(105,201)(106,202)(107,204)(108,203)(109,207)
(110,208)(111,205)(112,206)(113,193)(114,194)(115,196)(116,195)(117,199)
(118,200)(119,197)(120,198)(121,185)(122,186)(123,188)(124,187)(125,191)
(126,192)(127,189)(128,190)(129,177)(130,178)(131,180)(132,179)(133,183)
(134,184)(135,181)(136,182)(137,169)(138,170)(139,172)(140,171)(141,175)
(142,176)(143,173)(144,174)(217,218)(221,224)(222,223)(225,234)(226,233)
(227,235)(228,236)(229,240)(230,239)(231,238)(232,237)(241,282)(242,281)
(243,283)(244,284)(245,288)(246,287)(247,286)(248,285)(249,274)(250,273)
(251,275)(252,276)(253,280)(254,279)(255,278)(256,277)(257,266)(258,265)
(259,267)(260,268)(261,272)(262,271)(263,270)(264,269)(289,362)(290,361)
(291,363)(292,364)(293,368)(294,367)(295,366)(296,365)(297,378)(298,377)
(299,379)(300,380)(301,384)(302,383)(303,382)(304,381)(305,370)(306,369)
(307,371)(308,372)(309,376)(310,375)(311,374)(312,373)(313,426)(314,425)
(315,427)(316,428)(317,432)(318,431)(319,430)(320,429)(321,418)(322,417)
(323,419)(324,420)(325,424)(326,423)(327,422)(328,421)(329,410)(330,409)
(331,411)(332,412)(333,416)(334,415)(335,414)(336,413)(337,402)(338,401)
(339,403)(340,404)(341,408)(342,407)(343,406)(344,405)(345,394)(346,393)
(347,395)(348,396)(349,400)(350,399)(351,398)(352,397)(353,386)(354,385)
(355,387)(356,388)(357,392)(358,391)(359,390)(360,389);;
s1 := ( 1,313)( 2,314)( 3,319)( 4,320)( 5,318)( 6,317)( 7,315)( 8,316)
( 9,329)( 10,330)( 11,335)( 12,336)( 13,334)( 14,333)( 15,331)( 16,332)
( 17,321)( 18,322)( 19,327)( 20,328)( 21,326)( 22,325)( 23,323)( 24,324)
( 25,289)( 26,290)( 27,295)( 28,296)( 29,294)( 30,293)( 31,291)( 32,292)
( 33,305)( 34,306)( 35,311)( 36,312)( 37,310)( 38,309)( 39,307)( 40,308)
( 41,297)( 42,298)( 43,303)( 44,304)( 45,302)( 46,301)( 47,299)( 48,300)
( 49,353)( 50,354)( 51,359)( 52,360)( 53,358)( 54,357)( 55,355)( 56,356)
( 57,345)( 58,346)( 59,351)( 60,352)( 61,350)( 62,349)( 63,347)( 64,348)
( 65,337)( 66,338)( 67,343)( 68,344)( 69,342)( 70,341)( 71,339)( 72,340)
( 73,241)( 74,242)( 75,247)( 76,248)( 77,246)( 78,245)( 79,243)( 80,244)
( 81,257)( 82,258)( 83,263)( 84,264)( 85,262)( 86,261)( 87,259)( 88,260)
( 89,249)( 90,250)( 91,255)( 92,256)( 93,254)( 94,253)( 95,251)( 96,252)
( 97,217)( 98,218)( 99,223)(100,224)(101,222)(102,221)(103,219)(104,220)
(105,233)(106,234)(107,239)(108,240)(109,238)(110,237)(111,235)(112,236)
(113,225)(114,226)(115,231)(116,232)(117,230)(118,229)(119,227)(120,228)
(121,281)(122,282)(123,287)(124,288)(125,286)(126,285)(127,283)(128,284)
(129,273)(130,274)(131,279)(132,280)(133,278)(134,277)(135,275)(136,276)
(137,265)(138,266)(139,271)(140,272)(141,270)(142,269)(143,267)(144,268)
(145,385)(146,386)(147,391)(148,392)(149,390)(150,389)(151,387)(152,388)
(153,401)(154,402)(155,407)(156,408)(157,406)(158,405)(159,403)(160,404)
(161,393)(162,394)(163,399)(164,400)(165,398)(166,397)(167,395)(168,396)
(169,361)(170,362)(171,367)(172,368)(173,366)(174,365)(175,363)(176,364)
(177,377)(178,378)(179,383)(180,384)(181,382)(182,381)(183,379)(184,380)
(185,369)(186,370)(187,375)(188,376)(189,374)(190,373)(191,371)(192,372)
(193,425)(194,426)(195,431)(196,432)(197,430)(198,429)(199,427)(200,428)
(201,417)(202,418)(203,423)(204,424)(205,422)(206,421)(207,419)(208,420)
(209,409)(210,410)(211,415)(212,416)(213,414)(214,413)(215,411)(216,412);;
s2 := ( 1,219)( 2,220)( 3,217)( 4,218)( 5,222)( 6,221)( 7,223)( 8,224)
( 9,227)( 10,228)( 11,225)( 12,226)( 13,230)( 14,229)( 15,231)( 16,232)
( 17,235)( 18,236)( 19,233)( 20,234)( 21,238)( 22,237)( 23,239)( 24,240)
( 25,243)( 26,244)( 27,241)( 28,242)( 29,246)( 30,245)( 31,247)( 32,248)
( 33,251)( 34,252)( 35,249)( 36,250)( 37,254)( 38,253)( 39,255)( 40,256)
( 41,259)( 42,260)( 43,257)( 44,258)( 45,262)( 46,261)( 47,263)( 48,264)
( 49,267)( 50,268)( 51,265)( 52,266)( 53,270)( 54,269)( 55,271)( 56,272)
( 57,275)( 58,276)( 59,273)( 60,274)( 61,278)( 62,277)( 63,279)( 64,280)
( 65,283)( 66,284)( 67,281)( 68,282)( 69,286)( 70,285)( 71,287)( 72,288)
( 73,363)( 74,364)( 75,361)( 76,362)( 77,366)( 78,365)( 79,367)( 80,368)
( 81,371)( 82,372)( 83,369)( 84,370)( 85,374)( 86,373)( 87,375)( 88,376)
( 89,379)( 90,380)( 91,377)( 92,378)( 93,382)( 94,381)( 95,383)( 96,384)
( 97,387)( 98,388)( 99,385)(100,386)(101,390)(102,389)(103,391)(104,392)
(105,395)(106,396)(107,393)(108,394)(109,398)(110,397)(111,399)(112,400)
(113,403)(114,404)(115,401)(116,402)(117,406)(118,405)(119,407)(120,408)
(121,411)(122,412)(123,409)(124,410)(125,414)(126,413)(127,415)(128,416)
(129,419)(130,420)(131,417)(132,418)(133,422)(134,421)(135,423)(136,424)
(137,427)(138,428)(139,425)(140,426)(141,430)(142,429)(143,431)(144,432)
(145,291)(146,292)(147,289)(148,290)(149,294)(150,293)(151,295)(152,296)
(153,299)(154,300)(155,297)(156,298)(157,302)(158,301)(159,303)(160,304)
(161,307)(162,308)(163,305)(164,306)(165,310)(166,309)(167,311)(168,312)
(169,315)(170,316)(171,313)(172,314)(173,318)(174,317)(175,319)(176,320)
(177,323)(178,324)(179,321)(180,322)(181,326)(182,325)(183,327)(184,328)
(185,331)(186,332)(187,329)(188,330)(189,334)(190,333)(191,335)(192,336)
(193,339)(194,340)(195,337)(196,338)(197,342)(198,341)(199,343)(200,344)
(201,347)(202,348)(203,345)(204,346)(205,350)(206,349)(207,351)(208,352)
(209,355)(210,356)(211,353)(212,354)(213,358)(214,357)(215,359)(216,360);;
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*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*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(432)!( 3, 4)( 5, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)
( 13, 23)( 14, 24)( 15, 21)( 16, 22)( 25, 65)( 26, 66)( 27, 68)( 28, 67)
( 29, 71)( 30, 72)( 31, 69)( 32, 70)( 33, 57)( 34, 58)( 35, 60)( 36, 59)
( 37, 63)( 38, 64)( 39, 61)( 40, 62)( 41, 49)( 42, 50)( 43, 52)( 44, 51)
( 45, 55)( 46, 56)( 47, 53)( 48, 54)( 73,145)( 74,146)( 75,148)( 76,147)
( 77,151)( 78,152)( 79,149)( 80,150)( 81,161)( 82,162)( 83,164)( 84,163)
( 85,167)( 86,168)( 87,165)( 88,166)( 89,153)( 90,154)( 91,156)( 92,155)
( 93,159)( 94,160)( 95,157)( 96,158)( 97,209)( 98,210)( 99,212)(100,211)
(101,215)(102,216)(103,213)(104,214)(105,201)(106,202)(107,204)(108,203)
(109,207)(110,208)(111,205)(112,206)(113,193)(114,194)(115,196)(116,195)
(117,199)(118,200)(119,197)(120,198)(121,185)(122,186)(123,188)(124,187)
(125,191)(126,192)(127,189)(128,190)(129,177)(130,178)(131,180)(132,179)
(133,183)(134,184)(135,181)(136,182)(137,169)(138,170)(139,172)(140,171)
(141,175)(142,176)(143,173)(144,174)(217,218)(221,224)(222,223)(225,234)
(226,233)(227,235)(228,236)(229,240)(230,239)(231,238)(232,237)(241,282)
(242,281)(243,283)(244,284)(245,288)(246,287)(247,286)(248,285)(249,274)
(250,273)(251,275)(252,276)(253,280)(254,279)(255,278)(256,277)(257,266)
(258,265)(259,267)(260,268)(261,272)(262,271)(263,270)(264,269)(289,362)
(290,361)(291,363)(292,364)(293,368)(294,367)(295,366)(296,365)(297,378)
(298,377)(299,379)(300,380)(301,384)(302,383)(303,382)(304,381)(305,370)
(306,369)(307,371)(308,372)(309,376)(310,375)(311,374)(312,373)(313,426)
(314,425)(315,427)(316,428)(317,432)(318,431)(319,430)(320,429)(321,418)
(322,417)(323,419)(324,420)(325,424)(326,423)(327,422)(328,421)(329,410)
(330,409)(331,411)(332,412)(333,416)(334,415)(335,414)(336,413)(337,402)
(338,401)(339,403)(340,404)(341,408)(342,407)(343,406)(344,405)(345,394)
(346,393)(347,395)(348,396)(349,400)(350,399)(351,398)(352,397)(353,386)
(354,385)(355,387)(356,388)(357,392)(358,391)(359,390)(360,389);
s1 := Sym(432)!( 1,313)( 2,314)( 3,319)( 4,320)( 5,318)( 6,317)( 7,315)
( 8,316)( 9,329)( 10,330)( 11,335)( 12,336)( 13,334)( 14,333)( 15,331)
( 16,332)( 17,321)( 18,322)( 19,327)( 20,328)( 21,326)( 22,325)( 23,323)
( 24,324)( 25,289)( 26,290)( 27,295)( 28,296)( 29,294)( 30,293)( 31,291)
( 32,292)( 33,305)( 34,306)( 35,311)( 36,312)( 37,310)( 38,309)( 39,307)
( 40,308)( 41,297)( 42,298)( 43,303)( 44,304)( 45,302)( 46,301)( 47,299)
( 48,300)( 49,353)( 50,354)( 51,359)( 52,360)( 53,358)( 54,357)( 55,355)
( 56,356)( 57,345)( 58,346)( 59,351)( 60,352)( 61,350)( 62,349)( 63,347)
( 64,348)( 65,337)( 66,338)( 67,343)( 68,344)( 69,342)( 70,341)( 71,339)
( 72,340)( 73,241)( 74,242)( 75,247)( 76,248)( 77,246)( 78,245)( 79,243)
( 80,244)( 81,257)( 82,258)( 83,263)( 84,264)( 85,262)( 86,261)( 87,259)
( 88,260)( 89,249)( 90,250)( 91,255)( 92,256)( 93,254)( 94,253)( 95,251)
( 96,252)( 97,217)( 98,218)( 99,223)(100,224)(101,222)(102,221)(103,219)
(104,220)(105,233)(106,234)(107,239)(108,240)(109,238)(110,237)(111,235)
(112,236)(113,225)(114,226)(115,231)(116,232)(117,230)(118,229)(119,227)
(120,228)(121,281)(122,282)(123,287)(124,288)(125,286)(126,285)(127,283)
(128,284)(129,273)(130,274)(131,279)(132,280)(133,278)(134,277)(135,275)
(136,276)(137,265)(138,266)(139,271)(140,272)(141,270)(142,269)(143,267)
(144,268)(145,385)(146,386)(147,391)(148,392)(149,390)(150,389)(151,387)
(152,388)(153,401)(154,402)(155,407)(156,408)(157,406)(158,405)(159,403)
(160,404)(161,393)(162,394)(163,399)(164,400)(165,398)(166,397)(167,395)
(168,396)(169,361)(170,362)(171,367)(172,368)(173,366)(174,365)(175,363)
(176,364)(177,377)(178,378)(179,383)(180,384)(181,382)(182,381)(183,379)
(184,380)(185,369)(186,370)(187,375)(188,376)(189,374)(190,373)(191,371)
(192,372)(193,425)(194,426)(195,431)(196,432)(197,430)(198,429)(199,427)
(200,428)(201,417)(202,418)(203,423)(204,424)(205,422)(206,421)(207,419)
(208,420)(209,409)(210,410)(211,415)(212,416)(213,414)(214,413)(215,411)
(216,412);
s2 := Sym(432)!( 1,219)( 2,220)( 3,217)( 4,218)( 5,222)( 6,221)( 7,223)
( 8,224)( 9,227)( 10,228)( 11,225)( 12,226)( 13,230)( 14,229)( 15,231)
( 16,232)( 17,235)( 18,236)( 19,233)( 20,234)( 21,238)( 22,237)( 23,239)
( 24,240)( 25,243)( 26,244)( 27,241)( 28,242)( 29,246)( 30,245)( 31,247)
( 32,248)( 33,251)( 34,252)( 35,249)( 36,250)( 37,254)( 38,253)( 39,255)
( 40,256)( 41,259)( 42,260)( 43,257)( 44,258)( 45,262)( 46,261)( 47,263)
( 48,264)( 49,267)( 50,268)( 51,265)( 52,266)( 53,270)( 54,269)( 55,271)
( 56,272)( 57,275)( 58,276)( 59,273)( 60,274)( 61,278)( 62,277)( 63,279)
( 64,280)( 65,283)( 66,284)( 67,281)( 68,282)( 69,286)( 70,285)( 71,287)
( 72,288)( 73,363)( 74,364)( 75,361)( 76,362)( 77,366)( 78,365)( 79,367)
( 80,368)( 81,371)( 82,372)( 83,369)( 84,370)( 85,374)( 86,373)( 87,375)
( 88,376)( 89,379)( 90,380)( 91,377)( 92,378)( 93,382)( 94,381)( 95,383)
( 96,384)( 97,387)( 98,388)( 99,385)(100,386)(101,390)(102,389)(103,391)
(104,392)(105,395)(106,396)(107,393)(108,394)(109,398)(110,397)(111,399)
(112,400)(113,403)(114,404)(115,401)(116,402)(117,406)(118,405)(119,407)
(120,408)(121,411)(122,412)(123,409)(124,410)(125,414)(126,413)(127,415)
(128,416)(129,419)(130,420)(131,417)(132,418)(133,422)(134,421)(135,423)
(136,424)(137,427)(138,428)(139,425)(140,426)(141,430)(142,429)(143,431)
(144,432)(145,291)(146,292)(147,289)(148,290)(149,294)(150,293)(151,295)
(152,296)(153,299)(154,300)(155,297)(156,298)(157,302)(158,301)(159,303)
(160,304)(161,307)(162,308)(163,305)(164,306)(165,310)(166,309)(167,311)
(168,312)(169,315)(170,316)(171,313)(172,314)(173,318)(174,317)(175,319)
(176,320)(177,323)(178,324)(179,321)(180,322)(181,326)(182,325)(183,327)
(184,328)(185,331)(186,332)(187,329)(188,330)(189,334)(190,333)(191,335)
(192,336)(193,339)(194,340)(195,337)(196,338)(197,342)(198,341)(199,343)
(200,344)(201,347)(202,348)(203,345)(204,346)(205,350)(206,349)(207,351)
(208,352)(209,355)(210,356)(211,353)(212,354)(213,358)(214,357)(215,359)
(216,360);
poly := sub<Sym(432)|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*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*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