Polytope of Type {36,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,6,3}*1728
Also Known As : {{36,6|2},{6,3}4}. if this polytope has another name.
Group : SmallGroup(1728,30174)
Rank : 4
Schlafli Type : {36,6,3}
Number of vertices, edges, etc : 36, 144, 12, 4
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {18,6,3}*864
3-fold quotients : {12,6,3}*576
6-fold quotients : {6,6,3}*288
9-fold quotients : {4,6,3}*192
18-fold quotients : {2,6,3}*96
36-fold quotients : {2,3,3}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 5, 9)( 6, 10)( 7, 11)( 8, 12)( 13, 33)( 14, 34)( 15, 35)( 16, 36)
( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 25)( 22, 26)( 23, 27)( 24, 28)
( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 69)( 50, 70)( 51, 71)( 52, 72)
( 53, 65)( 54, 66)( 55, 67)( 56, 68)( 57, 61)( 58, 62)( 59, 63)( 60, 64)
( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 85,105)( 86,106)( 87,107)( 88,108)
( 89,101)( 90,102)( 91,103)( 92,104)( 93, 97)( 94, 98)( 95, 99)( 96,100)
(113,117)(114,118)(115,119)(116,120)(121,141)(122,142)(123,143)(124,144)
(125,137)(126,138)(127,139)(128,140)(129,133)(130,134)(131,135)(132,136)
(149,153)(150,154)(151,155)(152,156)(157,177)(158,178)(159,179)(160,180)
(161,173)(162,174)(163,175)(164,176)(165,169)(166,170)(167,171)(168,172)
(185,189)(186,190)(187,191)(188,192)(193,213)(194,214)(195,215)(196,216)
(197,209)(198,210)(199,211)(200,212)(201,205)(202,206)(203,207)(204,208)
(217,325)(218,326)(219,327)(220,328)(221,333)(222,334)(223,335)(224,336)
(225,329)(226,330)(227,331)(228,332)(229,357)(230,358)(231,359)(232,360)
(233,353)(234,354)(235,355)(236,356)(237,349)(238,350)(239,351)(240,352)
(241,345)(242,346)(243,347)(244,348)(245,341)(246,342)(247,343)(248,344)
(249,337)(250,338)(251,339)(252,340)(253,361)(254,362)(255,363)(256,364)
(257,369)(258,370)(259,371)(260,372)(261,365)(262,366)(263,367)(264,368)
(265,393)(266,394)(267,395)(268,396)(269,389)(270,390)(271,391)(272,392)
(273,385)(274,386)(275,387)(276,388)(277,381)(278,382)(279,383)(280,384)
(281,377)(282,378)(283,379)(284,380)(285,373)(286,374)(287,375)(288,376)
(289,397)(290,398)(291,399)(292,400)(293,405)(294,406)(295,407)(296,408)
(297,401)(298,402)(299,403)(300,404)(301,429)(302,430)(303,431)(304,432)
(305,425)(306,426)(307,427)(308,428)(309,421)(310,422)(311,423)(312,424)
(313,417)(314,418)(315,419)(316,420)(317,413)(318,414)(319,415)(320,416)
(321,409)(322,410)(323,411)(324,412);;
s1 := ( 1,229)( 2,230)( 3,232)( 4,231)( 5,237)( 6,238)( 7,240)( 8,239)
( 9,233)( 10,234)( 11,236)( 12,235)( 13,217)( 14,218)( 15,220)( 16,219)
( 17,225)( 18,226)( 19,228)( 20,227)( 21,221)( 22,222)( 23,224)( 24,223)
( 25,249)( 26,250)( 27,252)( 28,251)( 29,245)( 30,246)( 31,248)( 32,247)
( 33,241)( 34,242)( 35,244)( 36,243)( 37,301)( 38,302)( 39,304)( 40,303)
( 41,309)( 42,310)( 43,312)( 44,311)( 45,305)( 46,306)( 47,308)( 48,307)
( 49,289)( 50,290)( 51,292)( 52,291)( 53,297)( 54,298)( 55,300)( 56,299)
( 57,293)( 58,294)( 59,296)( 60,295)( 61,321)( 62,322)( 63,324)( 64,323)
( 65,317)( 66,318)( 67,320)( 68,319)( 69,313)( 70,314)( 71,316)( 72,315)
( 73,265)( 74,266)( 75,268)( 76,267)( 77,273)( 78,274)( 79,276)( 80,275)
( 81,269)( 82,270)( 83,272)( 84,271)( 85,253)( 86,254)( 87,256)( 88,255)
( 89,261)( 90,262)( 91,264)( 92,263)( 93,257)( 94,258)( 95,260)( 96,259)
( 97,285)( 98,286)( 99,288)(100,287)(101,281)(102,282)(103,284)(104,283)
(105,277)(106,278)(107,280)(108,279)(109,337)(110,338)(111,340)(112,339)
(113,345)(114,346)(115,348)(116,347)(117,341)(118,342)(119,344)(120,343)
(121,325)(122,326)(123,328)(124,327)(125,333)(126,334)(127,336)(128,335)
(129,329)(130,330)(131,332)(132,331)(133,357)(134,358)(135,360)(136,359)
(137,353)(138,354)(139,356)(140,355)(141,349)(142,350)(143,352)(144,351)
(145,409)(146,410)(147,412)(148,411)(149,417)(150,418)(151,420)(152,419)
(153,413)(154,414)(155,416)(156,415)(157,397)(158,398)(159,400)(160,399)
(161,405)(162,406)(163,408)(164,407)(165,401)(166,402)(167,404)(168,403)
(169,429)(170,430)(171,432)(172,431)(173,425)(174,426)(175,428)(176,427)
(177,421)(178,422)(179,424)(180,423)(181,373)(182,374)(183,376)(184,375)
(185,381)(186,382)(187,384)(188,383)(189,377)(190,378)(191,380)(192,379)
(193,361)(194,362)(195,364)(196,363)(197,369)(198,370)(199,372)(200,371)
(201,365)(202,366)(203,368)(204,367)(205,393)(206,394)(207,396)(208,395)
(209,389)(210,390)(211,392)(212,391)(213,385)(214,386)(215,388)(216,387);;
s2 := ( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 41)( 6, 44)( 7, 43)( 8, 42)
( 9, 45)( 10, 48)( 11, 47)( 12, 46)( 13, 49)( 14, 52)( 15, 51)( 16, 50)
( 17, 53)( 18, 56)( 19, 55)( 20, 54)( 21, 57)( 22, 60)( 23, 59)( 24, 58)
( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 29, 65)( 30, 68)( 31, 67)( 32, 66)
( 33, 69)( 34, 72)( 35, 71)( 36, 70)( 74, 76)( 78, 80)( 82, 84)( 86, 88)
( 90, 92)( 94, 96)( 98,100)(102,104)(106,108)(109,145)(110,148)(111,147)
(112,146)(113,149)(114,152)(115,151)(116,150)(117,153)(118,156)(119,155)
(120,154)(121,157)(122,160)(123,159)(124,158)(125,161)(126,164)(127,163)
(128,162)(129,165)(130,168)(131,167)(132,166)(133,169)(134,172)(135,171)
(136,170)(137,173)(138,176)(139,175)(140,174)(141,177)(142,180)(143,179)
(144,178)(182,184)(186,188)(190,192)(194,196)(198,200)(202,204)(206,208)
(210,212)(214,216)(217,253)(218,256)(219,255)(220,254)(221,257)(222,260)
(223,259)(224,258)(225,261)(226,264)(227,263)(228,262)(229,265)(230,268)
(231,267)(232,266)(233,269)(234,272)(235,271)(236,270)(237,273)(238,276)
(239,275)(240,274)(241,277)(242,280)(243,279)(244,278)(245,281)(246,284)
(247,283)(248,282)(249,285)(250,288)(251,287)(252,286)(290,292)(294,296)
(298,300)(302,304)(306,308)(310,312)(314,316)(318,320)(322,324)(325,361)
(326,364)(327,363)(328,362)(329,365)(330,368)(331,367)(332,366)(333,369)
(334,372)(335,371)(336,370)(337,373)(338,376)(339,375)(340,374)(341,377)
(342,380)(343,379)(344,378)(345,381)(346,384)(347,383)(348,382)(349,385)
(350,388)(351,387)(352,386)(353,389)(354,392)(355,391)(356,390)(357,393)
(358,396)(359,395)(360,394)(398,400)(402,404)(406,408)(410,412)(414,416)
(418,420)(422,424)(426,428)(430,432);;
s3 := ( 1, 2)( 5, 6)( 9, 10)( 13, 14)( 17, 18)( 21, 22)( 25, 26)( 29, 30)
( 33, 34)( 37, 74)( 38, 73)( 39, 75)( 40, 76)( 41, 78)( 42, 77)( 43, 79)
( 44, 80)( 45, 82)( 46, 81)( 47, 83)( 48, 84)( 49, 86)( 50, 85)( 51, 87)
( 52, 88)( 53, 90)( 54, 89)( 55, 91)( 56, 92)( 57, 94)( 58, 93)( 59, 95)
( 60, 96)( 61, 98)( 62, 97)( 63, 99)( 64,100)( 65,102)( 66,101)( 67,103)
( 68,104)( 69,106)( 70,105)( 71,107)( 72,108)(109,110)(113,114)(117,118)
(121,122)(125,126)(129,130)(133,134)(137,138)(141,142)(145,182)(146,181)
(147,183)(148,184)(149,186)(150,185)(151,187)(152,188)(153,190)(154,189)
(155,191)(156,192)(157,194)(158,193)(159,195)(160,196)(161,198)(162,197)
(163,199)(164,200)(165,202)(166,201)(167,203)(168,204)(169,206)(170,205)
(171,207)(172,208)(173,210)(174,209)(175,211)(176,212)(177,214)(178,213)
(179,215)(180,216)(217,218)(221,222)(225,226)(229,230)(233,234)(237,238)
(241,242)(245,246)(249,250)(253,290)(254,289)(255,291)(256,292)(257,294)
(258,293)(259,295)(260,296)(261,298)(262,297)(263,299)(264,300)(265,302)
(266,301)(267,303)(268,304)(269,306)(270,305)(271,307)(272,308)(273,310)
(274,309)(275,311)(276,312)(277,314)(278,313)(279,315)(280,316)(281,318)
(282,317)(283,319)(284,320)(285,322)(286,321)(287,323)(288,324)(325,326)
(329,330)(333,334)(337,338)(341,342)(345,346)(349,350)(353,354)(357,358)
(361,398)(362,397)(363,399)(364,400)(365,402)(366,401)(367,403)(368,404)
(369,406)(370,405)(371,407)(372,408)(373,410)(374,409)(375,411)(376,412)
(377,414)(378,413)(379,415)(380,416)(381,418)(382,417)(383,419)(384,420)
(385,422)(386,421)(387,423)(388,424)(389,426)(390,425)(391,427)(392,428)
(393,430)(394,429)(395,431)(396,432);;
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, s2*s3*s2*s3*s2*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*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*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(432)!( 5, 9)( 6, 10)( 7, 11)( 8, 12)( 13, 33)( 14, 34)( 15, 35)
( 16, 36)( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 25)( 22, 26)( 23, 27)
( 24, 28)( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 69)( 50, 70)( 51, 71)
( 52, 72)( 53, 65)( 54, 66)( 55, 67)( 56, 68)( 57, 61)( 58, 62)( 59, 63)
( 60, 64)( 77, 81)( 78, 82)( 79, 83)( 80, 84)( 85,105)( 86,106)( 87,107)
( 88,108)( 89,101)( 90,102)( 91,103)( 92,104)( 93, 97)( 94, 98)( 95, 99)
( 96,100)(113,117)(114,118)(115,119)(116,120)(121,141)(122,142)(123,143)
(124,144)(125,137)(126,138)(127,139)(128,140)(129,133)(130,134)(131,135)
(132,136)(149,153)(150,154)(151,155)(152,156)(157,177)(158,178)(159,179)
(160,180)(161,173)(162,174)(163,175)(164,176)(165,169)(166,170)(167,171)
(168,172)(185,189)(186,190)(187,191)(188,192)(193,213)(194,214)(195,215)
(196,216)(197,209)(198,210)(199,211)(200,212)(201,205)(202,206)(203,207)
(204,208)(217,325)(218,326)(219,327)(220,328)(221,333)(222,334)(223,335)
(224,336)(225,329)(226,330)(227,331)(228,332)(229,357)(230,358)(231,359)
(232,360)(233,353)(234,354)(235,355)(236,356)(237,349)(238,350)(239,351)
(240,352)(241,345)(242,346)(243,347)(244,348)(245,341)(246,342)(247,343)
(248,344)(249,337)(250,338)(251,339)(252,340)(253,361)(254,362)(255,363)
(256,364)(257,369)(258,370)(259,371)(260,372)(261,365)(262,366)(263,367)
(264,368)(265,393)(266,394)(267,395)(268,396)(269,389)(270,390)(271,391)
(272,392)(273,385)(274,386)(275,387)(276,388)(277,381)(278,382)(279,383)
(280,384)(281,377)(282,378)(283,379)(284,380)(285,373)(286,374)(287,375)
(288,376)(289,397)(290,398)(291,399)(292,400)(293,405)(294,406)(295,407)
(296,408)(297,401)(298,402)(299,403)(300,404)(301,429)(302,430)(303,431)
(304,432)(305,425)(306,426)(307,427)(308,428)(309,421)(310,422)(311,423)
(312,424)(313,417)(314,418)(315,419)(316,420)(317,413)(318,414)(319,415)
(320,416)(321,409)(322,410)(323,411)(324,412);
s1 := Sym(432)!( 1,229)( 2,230)( 3,232)( 4,231)( 5,237)( 6,238)( 7,240)
( 8,239)( 9,233)( 10,234)( 11,236)( 12,235)( 13,217)( 14,218)( 15,220)
( 16,219)( 17,225)( 18,226)( 19,228)( 20,227)( 21,221)( 22,222)( 23,224)
( 24,223)( 25,249)( 26,250)( 27,252)( 28,251)( 29,245)( 30,246)( 31,248)
( 32,247)( 33,241)( 34,242)( 35,244)( 36,243)( 37,301)( 38,302)( 39,304)
( 40,303)( 41,309)( 42,310)( 43,312)( 44,311)( 45,305)( 46,306)( 47,308)
( 48,307)( 49,289)( 50,290)( 51,292)( 52,291)( 53,297)( 54,298)( 55,300)
( 56,299)( 57,293)( 58,294)( 59,296)( 60,295)( 61,321)( 62,322)( 63,324)
( 64,323)( 65,317)( 66,318)( 67,320)( 68,319)( 69,313)( 70,314)( 71,316)
( 72,315)( 73,265)( 74,266)( 75,268)( 76,267)( 77,273)( 78,274)( 79,276)
( 80,275)( 81,269)( 82,270)( 83,272)( 84,271)( 85,253)( 86,254)( 87,256)
( 88,255)( 89,261)( 90,262)( 91,264)( 92,263)( 93,257)( 94,258)( 95,260)
( 96,259)( 97,285)( 98,286)( 99,288)(100,287)(101,281)(102,282)(103,284)
(104,283)(105,277)(106,278)(107,280)(108,279)(109,337)(110,338)(111,340)
(112,339)(113,345)(114,346)(115,348)(116,347)(117,341)(118,342)(119,344)
(120,343)(121,325)(122,326)(123,328)(124,327)(125,333)(126,334)(127,336)
(128,335)(129,329)(130,330)(131,332)(132,331)(133,357)(134,358)(135,360)
(136,359)(137,353)(138,354)(139,356)(140,355)(141,349)(142,350)(143,352)
(144,351)(145,409)(146,410)(147,412)(148,411)(149,417)(150,418)(151,420)
(152,419)(153,413)(154,414)(155,416)(156,415)(157,397)(158,398)(159,400)
(160,399)(161,405)(162,406)(163,408)(164,407)(165,401)(166,402)(167,404)
(168,403)(169,429)(170,430)(171,432)(172,431)(173,425)(174,426)(175,428)
(176,427)(177,421)(178,422)(179,424)(180,423)(181,373)(182,374)(183,376)
(184,375)(185,381)(186,382)(187,384)(188,383)(189,377)(190,378)(191,380)
(192,379)(193,361)(194,362)(195,364)(196,363)(197,369)(198,370)(199,372)
(200,371)(201,365)(202,366)(203,368)(204,367)(205,393)(206,394)(207,396)
(208,395)(209,389)(210,390)(211,392)(212,391)(213,385)(214,386)(215,388)
(216,387);
s2 := Sym(432)!( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 41)( 6, 44)( 7, 43)
( 8, 42)( 9, 45)( 10, 48)( 11, 47)( 12, 46)( 13, 49)( 14, 52)( 15, 51)
( 16, 50)( 17, 53)( 18, 56)( 19, 55)( 20, 54)( 21, 57)( 22, 60)( 23, 59)
( 24, 58)( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 29, 65)( 30, 68)( 31, 67)
( 32, 66)( 33, 69)( 34, 72)( 35, 71)( 36, 70)( 74, 76)( 78, 80)( 82, 84)
( 86, 88)( 90, 92)( 94, 96)( 98,100)(102,104)(106,108)(109,145)(110,148)
(111,147)(112,146)(113,149)(114,152)(115,151)(116,150)(117,153)(118,156)
(119,155)(120,154)(121,157)(122,160)(123,159)(124,158)(125,161)(126,164)
(127,163)(128,162)(129,165)(130,168)(131,167)(132,166)(133,169)(134,172)
(135,171)(136,170)(137,173)(138,176)(139,175)(140,174)(141,177)(142,180)
(143,179)(144,178)(182,184)(186,188)(190,192)(194,196)(198,200)(202,204)
(206,208)(210,212)(214,216)(217,253)(218,256)(219,255)(220,254)(221,257)
(222,260)(223,259)(224,258)(225,261)(226,264)(227,263)(228,262)(229,265)
(230,268)(231,267)(232,266)(233,269)(234,272)(235,271)(236,270)(237,273)
(238,276)(239,275)(240,274)(241,277)(242,280)(243,279)(244,278)(245,281)
(246,284)(247,283)(248,282)(249,285)(250,288)(251,287)(252,286)(290,292)
(294,296)(298,300)(302,304)(306,308)(310,312)(314,316)(318,320)(322,324)
(325,361)(326,364)(327,363)(328,362)(329,365)(330,368)(331,367)(332,366)
(333,369)(334,372)(335,371)(336,370)(337,373)(338,376)(339,375)(340,374)
(341,377)(342,380)(343,379)(344,378)(345,381)(346,384)(347,383)(348,382)
(349,385)(350,388)(351,387)(352,386)(353,389)(354,392)(355,391)(356,390)
(357,393)(358,396)(359,395)(360,394)(398,400)(402,404)(406,408)(410,412)
(414,416)(418,420)(422,424)(426,428)(430,432);
s3 := Sym(432)!( 1, 2)( 5, 6)( 9, 10)( 13, 14)( 17, 18)( 21, 22)( 25, 26)
( 29, 30)( 33, 34)( 37, 74)( 38, 73)( 39, 75)( 40, 76)( 41, 78)( 42, 77)
( 43, 79)( 44, 80)( 45, 82)( 46, 81)( 47, 83)( 48, 84)( 49, 86)( 50, 85)
( 51, 87)( 52, 88)( 53, 90)( 54, 89)( 55, 91)( 56, 92)( 57, 94)( 58, 93)
( 59, 95)( 60, 96)( 61, 98)( 62, 97)( 63, 99)( 64,100)( 65,102)( 66,101)
( 67,103)( 68,104)( 69,106)( 70,105)( 71,107)( 72,108)(109,110)(113,114)
(117,118)(121,122)(125,126)(129,130)(133,134)(137,138)(141,142)(145,182)
(146,181)(147,183)(148,184)(149,186)(150,185)(151,187)(152,188)(153,190)
(154,189)(155,191)(156,192)(157,194)(158,193)(159,195)(160,196)(161,198)
(162,197)(163,199)(164,200)(165,202)(166,201)(167,203)(168,204)(169,206)
(170,205)(171,207)(172,208)(173,210)(174,209)(175,211)(176,212)(177,214)
(178,213)(179,215)(180,216)(217,218)(221,222)(225,226)(229,230)(233,234)
(237,238)(241,242)(245,246)(249,250)(253,290)(254,289)(255,291)(256,292)
(257,294)(258,293)(259,295)(260,296)(261,298)(262,297)(263,299)(264,300)
(265,302)(266,301)(267,303)(268,304)(269,306)(270,305)(271,307)(272,308)
(273,310)(274,309)(275,311)(276,312)(277,314)(278,313)(279,315)(280,316)
(281,318)(282,317)(283,319)(284,320)(285,322)(286,321)(287,323)(288,324)
(325,326)(329,330)(333,334)(337,338)(341,342)(345,346)(349,350)(353,354)
(357,358)(361,398)(362,397)(363,399)(364,400)(365,402)(366,401)(367,403)
(368,404)(369,406)(370,405)(371,407)(372,408)(373,410)(374,409)(375,411)
(376,412)(377,414)(378,413)(379,415)(380,416)(381,418)(382,417)(383,419)
(384,420)(385,422)(386,421)(387,423)(388,424)(389,426)(390,425)(391,427)
(392,428)(393,430)(394,429)(395,431)(396,432);
poly := sub<Sym(432)|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,
s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*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*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