Polytope of Type {36,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,12}*1728i
if this polytope has a name.
Group : SmallGroup(1728,46098)
Rank : 3
Schlafli Type : {36,12}
Number of vertices, edges, etc : 72, 432, 24
Order of s0s1s2 : 9
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-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) :
3-fold quotients : {12,12}*576l
4-fold quotients : {36,6}*432c, {18,12}*432c
12-fold quotients : {18,4}*144c, {6,12}*144d, {12,6}*144d
24-fold quotients : {9,4}*72
36-fold quotients : {4,6}*48b, {6,4}*48b
72-fold quotients : {3,4}*24, {4,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 9)( 2, 10)( 3, 12)( 4, 11)( 5, 13)( 6, 14)( 7, 16)( 8, 15)
( 17, 41)( 18, 42)( 19, 44)( 20, 43)( 21, 45)( 22, 46)( 23, 48)( 24, 47)
( 25, 33)( 26, 34)( 27, 36)( 28, 35)( 29, 37)( 30, 38)( 31, 40)( 32, 39)
( 49, 57)( 50, 58)( 51, 60)( 52, 59)( 53, 61)( 54, 62)( 55, 64)( 56, 63)
( 65, 89)( 66, 90)( 67, 92)( 68, 91)( 69, 93)( 70, 94)( 71, 96)( 72, 95)
( 73, 81)( 74, 82)( 75, 84)( 76, 83)( 77, 85)( 78, 86)( 79, 88)( 80, 87)
( 97,105)( 98,106)( 99,108)(100,107)(101,109)(102,110)(103,112)(104,111)
(113,137)(114,138)(115,140)(116,139)(117,141)(118,142)(119,144)(120,143)
(121,129)(122,130)(123,132)(124,131)(125,133)(126,134)(127,136)(128,135)
(145,329)(146,330)(147,332)(148,331)(149,333)(150,334)(151,336)(152,335)
(153,321)(154,322)(155,324)(156,323)(157,325)(158,326)(159,328)(160,327)
(161,313)(162,314)(163,316)(164,315)(165,317)(166,318)(167,320)(168,319)
(169,305)(170,306)(171,308)(172,307)(173,309)(174,310)(175,312)(176,311)
(177,297)(178,298)(179,300)(180,299)(181,301)(182,302)(183,304)(184,303)
(185,289)(186,290)(187,292)(188,291)(189,293)(190,294)(191,296)(192,295)
(193,377)(194,378)(195,380)(196,379)(197,381)(198,382)(199,384)(200,383)
(201,369)(202,370)(203,372)(204,371)(205,373)(206,374)(207,376)(208,375)
(209,361)(210,362)(211,364)(212,363)(213,365)(214,366)(215,368)(216,367)
(217,353)(218,354)(219,356)(220,355)(221,357)(222,358)(223,360)(224,359)
(225,345)(226,346)(227,348)(228,347)(229,349)(230,350)(231,352)(232,351)
(233,337)(234,338)(235,340)(236,339)(237,341)(238,342)(239,344)(240,343)
(241,425)(242,426)(243,428)(244,427)(245,429)(246,430)(247,432)(248,431)
(249,417)(250,418)(251,420)(252,419)(253,421)(254,422)(255,424)(256,423)
(257,409)(258,410)(259,412)(260,411)(261,413)(262,414)(263,416)(264,415)
(265,401)(266,402)(267,404)(268,403)(269,405)(270,406)(271,408)(272,407)
(273,393)(274,394)(275,396)(276,395)(277,397)(278,398)(279,400)(280,399)
(281,385)(282,386)(283,388)(284,387)(285,389)(286,390)(287,392)(288,391);;
s1 := ( 1,145)( 2,148)( 3,147)( 4,146)( 5,149)( 6,152)( 7,151)( 8,150)
( 9,157)( 10,160)( 11,159)( 12,158)( 13,153)( 14,156)( 15,155)( 16,154)
( 17,177)( 18,180)( 19,179)( 20,178)( 21,181)( 22,184)( 23,183)( 24,182)
( 25,189)( 26,192)( 27,191)( 28,190)( 29,185)( 30,188)( 31,187)( 32,186)
( 33,161)( 34,164)( 35,163)( 36,162)( 37,165)( 38,168)( 39,167)( 40,166)
( 41,173)( 42,176)( 43,175)( 44,174)( 45,169)( 46,172)( 47,171)( 48,170)
( 49,241)( 50,244)( 51,243)( 52,242)( 53,245)( 54,248)( 55,247)( 56,246)
( 57,253)( 58,256)( 59,255)( 60,254)( 61,249)( 62,252)( 63,251)( 64,250)
( 65,273)( 66,276)( 67,275)( 68,274)( 69,277)( 70,280)( 71,279)( 72,278)
( 73,285)( 74,288)( 75,287)( 76,286)( 77,281)( 78,284)( 79,283)( 80,282)
( 81,257)( 82,260)( 83,259)( 84,258)( 85,261)( 86,264)( 87,263)( 88,262)
( 89,269)( 90,272)( 91,271)( 92,270)( 93,265)( 94,268)( 95,267)( 96,266)
( 97,193)( 98,196)( 99,195)(100,194)(101,197)(102,200)(103,199)(104,198)
(105,205)(106,208)(107,207)(108,206)(109,201)(110,204)(111,203)(112,202)
(113,225)(114,228)(115,227)(116,226)(117,229)(118,232)(119,231)(120,230)
(121,237)(122,240)(123,239)(124,238)(125,233)(126,236)(127,235)(128,234)
(129,209)(130,212)(131,211)(132,210)(133,213)(134,216)(135,215)(136,214)
(137,221)(138,224)(139,223)(140,222)(141,217)(142,220)(143,219)(144,218)
(289,321)(290,324)(291,323)(292,322)(293,325)(294,328)(295,327)(296,326)
(297,333)(298,336)(299,335)(300,334)(301,329)(302,332)(303,331)(304,330)
(306,308)(310,312)(313,317)(314,320)(315,319)(316,318)(337,417)(338,420)
(339,419)(340,418)(341,421)(342,424)(343,423)(344,422)(345,429)(346,432)
(347,431)(348,430)(349,425)(350,428)(351,427)(352,426)(353,401)(354,404)
(355,403)(356,402)(357,405)(358,408)(359,407)(360,406)(361,413)(362,416)
(363,415)(364,414)(365,409)(366,412)(367,411)(368,410)(369,385)(370,388)
(371,387)(372,386)(373,389)(374,392)(375,391)(376,390)(377,397)(378,400)
(379,399)(380,398)(381,393)(382,396)(383,395)(384,394);;
s2 := ( 1, 50)( 2, 49)( 3, 52)( 4, 51)( 5, 62)( 6, 61)( 7, 64)( 8, 63)
( 9, 58)( 10, 57)( 11, 60)( 12, 59)( 13, 54)( 14, 53)( 15, 56)( 16, 55)
( 17, 66)( 18, 65)( 19, 68)( 20, 67)( 21, 78)( 22, 77)( 23, 80)( 24, 79)
( 25, 74)( 26, 73)( 27, 76)( 28, 75)( 29, 70)( 30, 69)( 31, 72)( 32, 71)
( 33, 82)( 34, 81)( 35, 84)( 36, 83)( 37, 94)( 38, 93)( 39, 96)( 40, 95)
( 41, 90)( 42, 89)( 43, 92)( 44, 91)( 45, 86)( 46, 85)( 47, 88)( 48, 87)
( 97, 98)( 99,100)(101,110)(102,109)(103,112)(104,111)(105,106)(107,108)
(113,114)(115,116)(117,126)(118,125)(119,128)(120,127)(121,122)(123,124)
(129,130)(131,132)(133,142)(134,141)(135,144)(136,143)(137,138)(139,140)
(145,194)(146,193)(147,196)(148,195)(149,206)(150,205)(151,208)(152,207)
(153,202)(154,201)(155,204)(156,203)(157,198)(158,197)(159,200)(160,199)
(161,210)(162,209)(163,212)(164,211)(165,222)(166,221)(167,224)(168,223)
(169,218)(170,217)(171,220)(172,219)(173,214)(174,213)(175,216)(176,215)
(177,226)(178,225)(179,228)(180,227)(181,238)(182,237)(183,240)(184,239)
(185,234)(186,233)(187,236)(188,235)(189,230)(190,229)(191,232)(192,231)
(241,242)(243,244)(245,254)(246,253)(247,256)(248,255)(249,250)(251,252)
(257,258)(259,260)(261,270)(262,269)(263,272)(264,271)(265,266)(267,268)
(273,274)(275,276)(277,286)(278,285)(279,288)(280,287)(281,282)(283,284)
(289,338)(290,337)(291,340)(292,339)(293,350)(294,349)(295,352)(296,351)
(297,346)(298,345)(299,348)(300,347)(301,342)(302,341)(303,344)(304,343)
(305,354)(306,353)(307,356)(308,355)(309,366)(310,365)(311,368)(312,367)
(313,362)(314,361)(315,364)(316,363)(317,358)(318,357)(319,360)(320,359)
(321,370)(322,369)(323,372)(324,371)(325,382)(326,381)(327,384)(328,383)
(329,378)(330,377)(331,380)(332,379)(333,374)(334,373)(335,376)(336,375)
(385,386)(387,388)(389,398)(390,397)(391,400)(392,399)(393,394)(395,396)
(401,402)(403,404)(405,414)(406,413)(407,416)(408,415)(409,410)(411,412)
(417,418)(419,420)(421,430)(422,429)(423,432)(424,431)(425,426)(427,428);;
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, s2*s0*s1*s2*s1*s2*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*s1,
s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 1, 9)( 2, 10)( 3, 12)( 4, 11)( 5, 13)( 6, 14)( 7, 16)
( 8, 15)( 17, 41)( 18, 42)( 19, 44)( 20, 43)( 21, 45)( 22, 46)( 23, 48)
( 24, 47)( 25, 33)( 26, 34)( 27, 36)( 28, 35)( 29, 37)( 30, 38)( 31, 40)
( 32, 39)( 49, 57)( 50, 58)( 51, 60)( 52, 59)( 53, 61)( 54, 62)( 55, 64)
( 56, 63)( 65, 89)( 66, 90)( 67, 92)( 68, 91)( 69, 93)( 70, 94)( 71, 96)
( 72, 95)( 73, 81)( 74, 82)( 75, 84)( 76, 83)( 77, 85)( 78, 86)( 79, 88)
( 80, 87)( 97,105)( 98,106)( 99,108)(100,107)(101,109)(102,110)(103,112)
(104,111)(113,137)(114,138)(115,140)(116,139)(117,141)(118,142)(119,144)
(120,143)(121,129)(122,130)(123,132)(124,131)(125,133)(126,134)(127,136)
(128,135)(145,329)(146,330)(147,332)(148,331)(149,333)(150,334)(151,336)
(152,335)(153,321)(154,322)(155,324)(156,323)(157,325)(158,326)(159,328)
(160,327)(161,313)(162,314)(163,316)(164,315)(165,317)(166,318)(167,320)
(168,319)(169,305)(170,306)(171,308)(172,307)(173,309)(174,310)(175,312)
(176,311)(177,297)(178,298)(179,300)(180,299)(181,301)(182,302)(183,304)
(184,303)(185,289)(186,290)(187,292)(188,291)(189,293)(190,294)(191,296)
(192,295)(193,377)(194,378)(195,380)(196,379)(197,381)(198,382)(199,384)
(200,383)(201,369)(202,370)(203,372)(204,371)(205,373)(206,374)(207,376)
(208,375)(209,361)(210,362)(211,364)(212,363)(213,365)(214,366)(215,368)
(216,367)(217,353)(218,354)(219,356)(220,355)(221,357)(222,358)(223,360)
(224,359)(225,345)(226,346)(227,348)(228,347)(229,349)(230,350)(231,352)
(232,351)(233,337)(234,338)(235,340)(236,339)(237,341)(238,342)(239,344)
(240,343)(241,425)(242,426)(243,428)(244,427)(245,429)(246,430)(247,432)
(248,431)(249,417)(250,418)(251,420)(252,419)(253,421)(254,422)(255,424)
(256,423)(257,409)(258,410)(259,412)(260,411)(261,413)(262,414)(263,416)
(264,415)(265,401)(266,402)(267,404)(268,403)(269,405)(270,406)(271,408)
(272,407)(273,393)(274,394)(275,396)(276,395)(277,397)(278,398)(279,400)
(280,399)(281,385)(282,386)(283,388)(284,387)(285,389)(286,390)(287,392)
(288,391);
s1 := Sym(432)!( 1,145)( 2,148)( 3,147)( 4,146)( 5,149)( 6,152)( 7,151)
( 8,150)( 9,157)( 10,160)( 11,159)( 12,158)( 13,153)( 14,156)( 15,155)
( 16,154)( 17,177)( 18,180)( 19,179)( 20,178)( 21,181)( 22,184)( 23,183)
( 24,182)( 25,189)( 26,192)( 27,191)( 28,190)( 29,185)( 30,188)( 31,187)
( 32,186)( 33,161)( 34,164)( 35,163)( 36,162)( 37,165)( 38,168)( 39,167)
( 40,166)( 41,173)( 42,176)( 43,175)( 44,174)( 45,169)( 46,172)( 47,171)
( 48,170)( 49,241)( 50,244)( 51,243)( 52,242)( 53,245)( 54,248)( 55,247)
( 56,246)( 57,253)( 58,256)( 59,255)( 60,254)( 61,249)( 62,252)( 63,251)
( 64,250)( 65,273)( 66,276)( 67,275)( 68,274)( 69,277)( 70,280)( 71,279)
( 72,278)( 73,285)( 74,288)( 75,287)( 76,286)( 77,281)( 78,284)( 79,283)
( 80,282)( 81,257)( 82,260)( 83,259)( 84,258)( 85,261)( 86,264)( 87,263)
( 88,262)( 89,269)( 90,272)( 91,271)( 92,270)( 93,265)( 94,268)( 95,267)
( 96,266)( 97,193)( 98,196)( 99,195)(100,194)(101,197)(102,200)(103,199)
(104,198)(105,205)(106,208)(107,207)(108,206)(109,201)(110,204)(111,203)
(112,202)(113,225)(114,228)(115,227)(116,226)(117,229)(118,232)(119,231)
(120,230)(121,237)(122,240)(123,239)(124,238)(125,233)(126,236)(127,235)
(128,234)(129,209)(130,212)(131,211)(132,210)(133,213)(134,216)(135,215)
(136,214)(137,221)(138,224)(139,223)(140,222)(141,217)(142,220)(143,219)
(144,218)(289,321)(290,324)(291,323)(292,322)(293,325)(294,328)(295,327)
(296,326)(297,333)(298,336)(299,335)(300,334)(301,329)(302,332)(303,331)
(304,330)(306,308)(310,312)(313,317)(314,320)(315,319)(316,318)(337,417)
(338,420)(339,419)(340,418)(341,421)(342,424)(343,423)(344,422)(345,429)
(346,432)(347,431)(348,430)(349,425)(350,428)(351,427)(352,426)(353,401)
(354,404)(355,403)(356,402)(357,405)(358,408)(359,407)(360,406)(361,413)
(362,416)(363,415)(364,414)(365,409)(366,412)(367,411)(368,410)(369,385)
(370,388)(371,387)(372,386)(373,389)(374,392)(375,391)(376,390)(377,397)
(378,400)(379,399)(380,398)(381,393)(382,396)(383,395)(384,394);
s2 := Sym(432)!( 1, 50)( 2, 49)( 3, 52)( 4, 51)( 5, 62)( 6, 61)( 7, 64)
( 8, 63)( 9, 58)( 10, 57)( 11, 60)( 12, 59)( 13, 54)( 14, 53)( 15, 56)
( 16, 55)( 17, 66)( 18, 65)( 19, 68)( 20, 67)( 21, 78)( 22, 77)( 23, 80)
( 24, 79)( 25, 74)( 26, 73)( 27, 76)( 28, 75)( 29, 70)( 30, 69)( 31, 72)
( 32, 71)( 33, 82)( 34, 81)( 35, 84)( 36, 83)( 37, 94)( 38, 93)( 39, 96)
( 40, 95)( 41, 90)( 42, 89)( 43, 92)( 44, 91)( 45, 86)( 46, 85)( 47, 88)
( 48, 87)( 97, 98)( 99,100)(101,110)(102,109)(103,112)(104,111)(105,106)
(107,108)(113,114)(115,116)(117,126)(118,125)(119,128)(120,127)(121,122)
(123,124)(129,130)(131,132)(133,142)(134,141)(135,144)(136,143)(137,138)
(139,140)(145,194)(146,193)(147,196)(148,195)(149,206)(150,205)(151,208)
(152,207)(153,202)(154,201)(155,204)(156,203)(157,198)(158,197)(159,200)
(160,199)(161,210)(162,209)(163,212)(164,211)(165,222)(166,221)(167,224)
(168,223)(169,218)(170,217)(171,220)(172,219)(173,214)(174,213)(175,216)
(176,215)(177,226)(178,225)(179,228)(180,227)(181,238)(182,237)(183,240)
(184,239)(185,234)(186,233)(187,236)(188,235)(189,230)(190,229)(191,232)
(192,231)(241,242)(243,244)(245,254)(246,253)(247,256)(248,255)(249,250)
(251,252)(257,258)(259,260)(261,270)(262,269)(263,272)(264,271)(265,266)
(267,268)(273,274)(275,276)(277,286)(278,285)(279,288)(280,287)(281,282)
(283,284)(289,338)(290,337)(291,340)(292,339)(293,350)(294,349)(295,352)
(296,351)(297,346)(298,345)(299,348)(300,347)(301,342)(302,341)(303,344)
(304,343)(305,354)(306,353)(307,356)(308,355)(309,366)(310,365)(311,368)
(312,367)(313,362)(314,361)(315,364)(316,363)(317,358)(318,357)(319,360)
(320,359)(321,370)(322,369)(323,372)(324,371)(325,382)(326,381)(327,384)
(328,383)(329,378)(330,377)(331,380)(332,379)(333,374)(334,373)(335,376)
(336,375)(385,386)(387,388)(389,398)(390,397)(391,400)(392,399)(393,394)
(395,396)(401,402)(403,404)(405,414)(406,413)(407,416)(408,415)(409,410)
(411,412)(417,418)(419,420)(421,430)(422,429)(423,432)(424,431)(425,426)
(427,428);
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, s2*s0*s1*s2*s1*s2*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*s1,
s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope