Polytope of Type {36,24}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,24}*1728c
Also Known As : {36,24|2}. if this polytope has another name.
Group : SmallGroup(1728,4743)
Rank : 3
Schlafli Type : {36,24}
Number of vertices, edges, etc : 36, 432, 24
Order of s0s1s2 : 72
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
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 : {18,24}*864a, {36,12}*864a
3-fold quotients : {36,8}*576a, {12,24}*576c
4-fold quotients : {36,6}*432a, {18,12}*432a
6-fold quotients : {36,4}*288a, {18,8}*288, {6,24}*288a, {12,12}*288a
8-fold quotients : {18,6}*216a
9-fold quotients : {4,24}*192a, {12,8}*192a
12-fold quotients : {36,2}*144, {18,4}*144a, {6,12}*144a, {12,6}*144a
18-fold quotients : {4,12}*96a, {12,4}*96a, {2,24}*96, {6,8}*96
24-fold quotients : {18,2}*72, {6,6}*72a
27-fold quotients : {4,8}*64a
36-fold quotients : {2,12}*48, {12,2}*48, {4,6}*48a, {6,4}*48a
48-fold quotients : {9,2}*36
54-fold quotients : {4,4}*32, {2,8}*32
72-fold quotients : {2,6}*24, {6,2}*24
108-fold quotients : {2,4}*16, {4,2}*16
144-fold quotients : {2,3}*12, {3,2}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 10, 21)( 11, 20)( 12, 19)( 13, 24)( 14, 23)
( 15, 22)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 32, 33)( 35, 36)( 37, 48)
( 38, 47)( 39, 46)( 40, 51)( 41, 50)( 42, 49)( 43, 54)( 44, 53)( 45, 52)
( 56, 57)( 59, 60)( 62, 63)( 64, 75)( 65, 74)( 66, 73)( 67, 78)( 68, 77)
( 69, 76)( 70, 81)( 71, 80)( 72, 79)( 83, 84)( 86, 87)( 89, 90)( 91,102)
( 92,101)( 93,100)( 94,105)( 95,104)( 96,103)( 97,108)( 98,107)( 99,106)
(110,111)(113,114)(116,117)(118,129)(119,128)(120,127)(121,132)(122,131)
(123,130)(124,135)(125,134)(126,133)(137,138)(140,141)(143,144)(145,156)
(146,155)(147,154)(148,159)(149,158)(150,157)(151,162)(152,161)(153,160)
(164,165)(167,168)(170,171)(172,183)(173,182)(174,181)(175,186)(176,185)
(177,184)(178,189)(179,188)(180,187)(191,192)(194,195)(197,198)(199,210)
(200,209)(201,208)(202,213)(203,212)(204,211)(205,216)(206,215)(207,214)
(217,271)(218,273)(219,272)(220,274)(221,276)(222,275)(223,277)(224,279)
(225,278)(226,291)(227,290)(228,289)(229,294)(230,293)(231,292)(232,297)
(233,296)(234,295)(235,282)(236,281)(237,280)(238,285)(239,284)(240,283)
(241,288)(242,287)(243,286)(244,298)(245,300)(246,299)(247,301)(248,303)
(249,302)(250,304)(251,306)(252,305)(253,318)(254,317)(255,316)(256,321)
(257,320)(258,319)(259,324)(260,323)(261,322)(262,309)(263,308)(264,307)
(265,312)(266,311)(267,310)(268,315)(269,314)(270,313)(325,379)(326,381)
(327,380)(328,382)(329,384)(330,383)(331,385)(332,387)(333,386)(334,399)
(335,398)(336,397)(337,402)(338,401)(339,400)(340,405)(341,404)(342,403)
(343,390)(344,389)(345,388)(346,393)(347,392)(348,391)(349,396)(350,395)
(351,394)(352,406)(353,408)(354,407)(355,409)(356,411)(357,410)(358,412)
(359,414)(360,413)(361,426)(362,425)(363,424)(364,429)(365,428)(366,427)
(367,432)(368,431)(369,430)(370,417)(371,416)(372,415)(373,420)(374,419)
(375,418)(376,423)(377,422)(378,421);;
s1 := ( 1,226)( 2,228)( 3,227)( 4,232)( 5,234)( 6,233)( 7,229)( 8,231)
( 9,230)( 10,217)( 11,219)( 12,218)( 13,223)( 14,225)( 15,224)( 16,220)
( 17,222)( 18,221)( 19,237)( 20,236)( 21,235)( 22,243)( 23,242)( 24,241)
( 25,240)( 26,239)( 27,238)( 28,253)( 29,255)( 30,254)( 31,259)( 32,261)
( 33,260)( 34,256)( 35,258)( 36,257)( 37,244)( 38,246)( 39,245)( 40,250)
( 41,252)( 42,251)( 43,247)( 44,249)( 45,248)( 46,264)( 47,263)( 48,262)
( 49,270)( 50,269)( 51,268)( 52,267)( 53,266)( 54,265)( 55,280)( 56,282)
( 57,281)( 58,286)( 59,288)( 60,287)( 61,283)( 62,285)( 63,284)( 64,271)
( 65,273)( 66,272)( 67,277)( 68,279)( 69,278)( 70,274)( 71,276)( 72,275)
( 73,291)( 74,290)( 75,289)( 76,297)( 77,296)( 78,295)( 79,294)( 80,293)
( 81,292)( 82,307)( 83,309)( 84,308)( 85,313)( 86,315)( 87,314)( 88,310)
( 89,312)( 90,311)( 91,298)( 92,300)( 93,299)( 94,304)( 95,306)( 96,305)
( 97,301)( 98,303)( 99,302)(100,318)(101,317)(102,316)(103,324)(104,323)
(105,322)(106,321)(107,320)(108,319)(109,361)(110,363)(111,362)(112,367)
(113,369)(114,368)(115,364)(116,366)(117,365)(118,352)(119,354)(120,353)
(121,358)(122,360)(123,359)(124,355)(125,357)(126,356)(127,372)(128,371)
(129,370)(130,378)(131,377)(132,376)(133,375)(134,374)(135,373)(136,334)
(137,336)(138,335)(139,340)(140,342)(141,341)(142,337)(143,339)(144,338)
(145,325)(146,327)(147,326)(148,331)(149,333)(150,332)(151,328)(152,330)
(153,329)(154,345)(155,344)(156,343)(157,351)(158,350)(159,349)(160,348)
(161,347)(162,346)(163,415)(164,417)(165,416)(166,421)(167,423)(168,422)
(169,418)(170,420)(171,419)(172,406)(173,408)(174,407)(175,412)(176,414)
(177,413)(178,409)(179,411)(180,410)(181,426)(182,425)(183,424)(184,432)
(185,431)(186,430)(187,429)(188,428)(189,427)(190,388)(191,390)(192,389)
(193,394)(194,396)(195,395)(196,391)(197,393)(198,392)(199,379)(200,381)
(201,380)(202,385)(203,387)(204,386)(205,382)(206,384)(207,383)(208,399)
(209,398)(210,397)(211,405)(212,404)(213,403)(214,402)(215,401)(216,400);;
s2 := ( 1, 4)( 2, 5)( 3, 6)( 10, 13)( 11, 14)( 12, 15)( 19, 22)( 20, 23)
( 21, 24)( 28, 31)( 29, 32)( 30, 33)( 37, 40)( 38, 41)( 39, 42)( 46, 49)
( 47, 50)( 48, 51)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)( 66, 69)
( 73, 76)( 74, 77)( 75, 78)( 82, 85)( 83, 86)( 84, 87)( 91, 94)( 92, 95)
( 93, 96)(100,103)(101,104)(102,105)(109,139)(110,140)(111,141)(112,136)
(113,137)(114,138)(115,142)(116,143)(117,144)(118,148)(119,149)(120,150)
(121,145)(122,146)(123,147)(124,151)(125,152)(126,153)(127,157)(128,158)
(129,159)(130,154)(131,155)(132,156)(133,160)(134,161)(135,162)(163,193)
(164,194)(165,195)(166,190)(167,191)(168,192)(169,196)(170,197)(171,198)
(172,202)(173,203)(174,204)(175,199)(176,200)(177,201)(178,205)(179,206)
(180,207)(181,211)(182,212)(183,213)(184,208)(185,209)(186,210)(187,214)
(188,215)(189,216)(217,328)(218,329)(219,330)(220,325)(221,326)(222,327)
(223,331)(224,332)(225,333)(226,337)(227,338)(228,339)(229,334)(230,335)
(231,336)(232,340)(233,341)(234,342)(235,346)(236,347)(237,348)(238,343)
(239,344)(240,345)(241,349)(242,350)(243,351)(244,355)(245,356)(246,357)
(247,352)(248,353)(249,354)(250,358)(251,359)(252,360)(253,364)(254,365)
(255,366)(256,361)(257,362)(258,363)(259,367)(260,368)(261,369)(262,373)
(263,374)(264,375)(265,370)(266,371)(267,372)(268,376)(269,377)(270,378)
(271,382)(272,383)(273,384)(274,379)(275,380)(276,381)(277,385)(278,386)
(279,387)(280,391)(281,392)(282,393)(283,388)(284,389)(285,390)(286,394)
(287,395)(288,396)(289,400)(290,401)(291,402)(292,397)(293,398)(294,399)
(295,403)(296,404)(297,405)(298,409)(299,410)(300,411)(301,406)(302,407)
(303,408)(304,412)(305,413)(306,414)(307,418)(308,419)(309,420)(310,415)
(311,416)(312,417)(313,421)(314,422)(315,423)(316,427)(317,428)(318,429)
(319,424)(320,425)(321,426)(322,430)(323,431)(324,432);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*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)!( 2, 3)( 5, 6)( 8, 9)( 10, 21)( 11, 20)( 12, 19)( 13, 24)
( 14, 23)( 15, 22)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 32, 33)( 35, 36)
( 37, 48)( 38, 47)( 39, 46)( 40, 51)( 41, 50)( 42, 49)( 43, 54)( 44, 53)
( 45, 52)( 56, 57)( 59, 60)( 62, 63)( 64, 75)( 65, 74)( 66, 73)( 67, 78)
( 68, 77)( 69, 76)( 70, 81)( 71, 80)( 72, 79)( 83, 84)( 86, 87)( 89, 90)
( 91,102)( 92,101)( 93,100)( 94,105)( 95,104)( 96,103)( 97,108)( 98,107)
( 99,106)(110,111)(113,114)(116,117)(118,129)(119,128)(120,127)(121,132)
(122,131)(123,130)(124,135)(125,134)(126,133)(137,138)(140,141)(143,144)
(145,156)(146,155)(147,154)(148,159)(149,158)(150,157)(151,162)(152,161)
(153,160)(164,165)(167,168)(170,171)(172,183)(173,182)(174,181)(175,186)
(176,185)(177,184)(178,189)(179,188)(180,187)(191,192)(194,195)(197,198)
(199,210)(200,209)(201,208)(202,213)(203,212)(204,211)(205,216)(206,215)
(207,214)(217,271)(218,273)(219,272)(220,274)(221,276)(222,275)(223,277)
(224,279)(225,278)(226,291)(227,290)(228,289)(229,294)(230,293)(231,292)
(232,297)(233,296)(234,295)(235,282)(236,281)(237,280)(238,285)(239,284)
(240,283)(241,288)(242,287)(243,286)(244,298)(245,300)(246,299)(247,301)
(248,303)(249,302)(250,304)(251,306)(252,305)(253,318)(254,317)(255,316)
(256,321)(257,320)(258,319)(259,324)(260,323)(261,322)(262,309)(263,308)
(264,307)(265,312)(266,311)(267,310)(268,315)(269,314)(270,313)(325,379)
(326,381)(327,380)(328,382)(329,384)(330,383)(331,385)(332,387)(333,386)
(334,399)(335,398)(336,397)(337,402)(338,401)(339,400)(340,405)(341,404)
(342,403)(343,390)(344,389)(345,388)(346,393)(347,392)(348,391)(349,396)
(350,395)(351,394)(352,406)(353,408)(354,407)(355,409)(356,411)(357,410)
(358,412)(359,414)(360,413)(361,426)(362,425)(363,424)(364,429)(365,428)
(366,427)(367,432)(368,431)(369,430)(370,417)(371,416)(372,415)(373,420)
(374,419)(375,418)(376,423)(377,422)(378,421);
s1 := Sym(432)!( 1,226)( 2,228)( 3,227)( 4,232)( 5,234)( 6,233)( 7,229)
( 8,231)( 9,230)( 10,217)( 11,219)( 12,218)( 13,223)( 14,225)( 15,224)
( 16,220)( 17,222)( 18,221)( 19,237)( 20,236)( 21,235)( 22,243)( 23,242)
( 24,241)( 25,240)( 26,239)( 27,238)( 28,253)( 29,255)( 30,254)( 31,259)
( 32,261)( 33,260)( 34,256)( 35,258)( 36,257)( 37,244)( 38,246)( 39,245)
( 40,250)( 41,252)( 42,251)( 43,247)( 44,249)( 45,248)( 46,264)( 47,263)
( 48,262)( 49,270)( 50,269)( 51,268)( 52,267)( 53,266)( 54,265)( 55,280)
( 56,282)( 57,281)( 58,286)( 59,288)( 60,287)( 61,283)( 62,285)( 63,284)
( 64,271)( 65,273)( 66,272)( 67,277)( 68,279)( 69,278)( 70,274)( 71,276)
( 72,275)( 73,291)( 74,290)( 75,289)( 76,297)( 77,296)( 78,295)( 79,294)
( 80,293)( 81,292)( 82,307)( 83,309)( 84,308)( 85,313)( 86,315)( 87,314)
( 88,310)( 89,312)( 90,311)( 91,298)( 92,300)( 93,299)( 94,304)( 95,306)
( 96,305)( 97,301)( 98,303)( 99,302)(100,318)(101,317)(102,316)(103,324)
(104,323)(105,322)(106,321)(107,320)(108,319)(109,361)(110,363)(111,362)
(112,367)(113,369)(114,368)(115,364)(116,366)(117,365)(118,352)(119,354)
(120,353)(121,358)(122,360)(123,359)(124,355)(125,357)(126,356)(127,372)
(128,371)(129,370)(130,378)(131,377)(132,376)(133,375)(134,374)(135,373)
(136,334)(137,336)(138,335)(139,340)(140,342)(141,341)(142,337)(143,339)
(144,338)(145,325)(146,327)(147,326)(148,331)(149,333)(150,332)(151,328)
(152,330)(153,329)(154,345)(155,344)(156,343)(157,351)(158,350)(159,349)
(160,348)(161,347)(162,346)(163,415)(164,417)(165,416)(166,421)(167,423)
(168,422)(169,418)(170,420)(171,419)(172,406)(173,408)(174,407)(175,412)
(176,414)(177,413)(178,409)(179,411)(180,410)(181,426)(182,425)(183,424)
(184,432)(185,431)(186,430)(187,429)(188,428)(189,427)(190,388)(191,390)
(192,389)(193,394)(194,396)(195,395)(196,391)(197,393)(198,392)(199,379)
(200,381)(201,380)(202,385)(203,387)(204,386)(205,382)(206,384)(207,383)
(208,399)(209,398)(210,397)(211,405)(212,404)(213,403)(214,402)(215,401)
(216,400);
s2 := Sym(432)!( 1, 4)( 2, 5)( 3, 6)( 10, 13)( 11, 14)( 12, 15)( 19, 22)
( 20, 23)( 21, 24)( 28, 31)( 29, 32)( 30, 33)( 37, 40)( 38, 41)( 39, 42)
( 46, 49)( 47, 50)( 48, 51)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)
( 66, 69)( 73, 76)( 74, 77)( 75, 78)( 82, 85)( 83, 86)( 84, 87)( 91, 94)
( 92, 95)( 93, 96)(100,103)(101,104)(102,105)(109,139)(110,140)(111,141)
(112,136)(113,137)(114,138)(115,142)(116,143)(117,144)(118,148)(119,149)
(120,150)(121,145)(122,146)(123,147)(124,151)(125,152)(126,153)(127,157)
(128,158)(129,159)(130,154)(131,155)(132,156)(133,160)(134,161)(135,162)
(163,193)(164,194)(165,195)(166,190)(167,191)(168,192)(169,196)(170,197)
(171,198)(172,202)(173,203)(174,204)(175,199)(176,200)(177,201)(178,205)
(179,206)(180,207)(181,211)(182,212)(183,213)(184,208)(185,209)(186,210)
(187,214)(188,215)(189,216)(217,328)(218,329)(219,330)(220,325)(221,326)
(222,327)(223,331)(224,332)(225,333)(226,337)(227,338)(228,339)(229,334)
(230,335)(231,336)(232,340)(233,341)(234,342)(235,346)(236,347)(237,348)
(238,343)(239,344)(240,345)(241,349)(242,350)(243,351)(244,355)(245,356)
(246,357)(247,352)(248,353)(249,354)(250,358)(251,359)(252,360)(253,364)
(254,365)(255,366)(256,361)(257,362)(258,363)(259,367)(260,368)(261,369)
(262,373)(263,374)(264,375)(265,370)(266,371)(267,372)(268,376)(269,377)
(270,378)(271,382)(272,383)(273,384)(274,379)(275,380)(276,381)(277,385)
(278,386)(279,387)(280,391)(281,392)(282,393)(283,388)(284,389)(285,390)
(286,394)(287,395)(288,396)(289,400)(290,401)(291,402)(292,397)(293,398)
(294,399)(295,403)(296,404)(297,405)(298,409)(299,410)(300,411)(301,406)
(302,407)(303,408)(304,412)(305,413)(306,414)(307,418)(308,419)(309,420)
(310,415)(311,416)(312,417)(313,421)(314,422)(315,423)(316,427)(317,428)
(318,429)(319,424)(320,425)(321,426)(322,430)(323,431)(324,432);
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, s0*s1*s2*s1*s0*s1*s2*s1,
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*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