Polytope of Type {36,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,12}*1728a
if this polytope has a name.
Group : SmallGroup(1728,3530)
Rank : 3
Schlafli Type : {36,12}
Number of vertices, edges, etc : 72, 432, 24
Order of s0s1s2 : 36
Order of s0s1s2s1 : 4
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 : {36,12}*864a
3-fold quotients : {36,4}*576a, {12,12}*576a
4-fold quotients : {36,6}*432a, {18,12}*432a
6-fold quotients : {36,4}*288a, {12,12}*288a
8-fold quotients : {18,6}*216a
9-fold quotients : {4,12}*192a, {12,4}*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
24-fold quotients : {18,2}*72, {6,6}*72a
27-fold quotients : {4,4}*64
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
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)
(109,136)(110,138)(111,137)(112,139)(113,141)(114,140)(115,142)(116,144)
(117,143)(118,156)(119,155)(120,154)(121,159)(122,158)(123,157)(124,162)
(125,161)(126,160)(127,147)(128,146)(129,145)(130,150)(131,149)(132,148)
(133,153)(134,152)(135,151)(163,190)(164,192)(165,191)(166,193)(167,195)
(168,194)(169,196)(170,198)(171,197)(172,210)(173,209)(174,208)(175,213)
(176,212)(177,211)(178,216)(179,215)(180,214)(181,201)(182,200)(183,199)
(184,204)(185,203)(186,202)(187,207)(188,206)(189,205)(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)(325,406)(326,408)(327,407)(328,409)
(329,411)(330,410)(331,412)(332,414)(333,413)(334,426)(335,425)(336,424)
(337,429)(338,428)(339,427)(340,432)(341,431)(342,430)(343,417)(344,416)
(345,415)(346,420)(347,419)(348,418)(349,423)(350,422)(351,421)(352,379)
(353,381)(354,380)(355,382)(356,384)(357,383)(358,385)(359,387)(360,386)
(361,399)(362,398)(363,397)(364,402)(365,401)(366,400)(367,405)(368,404)
(369,403)(370,390)(371,389)(372,388)(373,393)(374,392)(375,391)(376,396)
(377,395)(378,394);;
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)(109,334)(110,336)(111,335)(112,340)
(113,342)(114,341)(115,337)(116,339)(117,338)(118,325)(119,327)(120,326)
(121,331)(122,333)(123,332)(124,328)(125,330)(126,329)(127,345)(128,344)
(129,343)(130,351)(131,350)(132,349)(133,348)(134,347)(135,346)(136,361)
(137,363)(138,362)(139,367)(140,369)(141,368)(142,364)(143,366)(144,365)
(145,352)(146,354)(147,353)(148,358)(149,360)(150,359)(151,355)(152,357)
(153,356)(154,372)(155,371)(156,370)(157,378)(158,377)(159,376)(160,375)
(161,374)(162,373)(163,388)(164,390)(165,389)(166,394)(167,396)(168,395)
(169,391)(170,393)(171,392)(172,379)(173,381)(174,380)(175,385)(176,387)
(177,386)(178,382)(179,384)(180,383)(181,399)(182,398)(183,397)(184,405)
(185,404)(186,403)(187,402)(188,401)(189,400)(190,415)(191,417)(192,416)
(193,421)(194,423)(195,422)(196,418)(197,420)(198,419)(199,406)(200,408)
(201,407)(202,412)(203,414)(204,413)(205,409)(206,411)(207,410)(208,426)
(209,425)(210,424)(211,432)(212,431)(213,430)(214,429)(215,428)(216,427);;
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, 85)( 56, 86)( 57, 87)( 58, 82)( 59, 83)( 60, 84)
( 61, 88)( 62, 89)( 63, 90)( 64, 94)( 65, 95)( 66, 96)( 67, 91)( 68, 92)
( 69, 93)( 70, 97)( 71, 98)( 72, 99)( 73,103)( 74,104)( 75,105)( 76,100)
( 77,101)( 78,102)( 79,106)( 80,107)( 81,108)(109,112)(110,113)(111,114)
(118,121)(119,122)(120,123)(127,130)(128,131)(129,132)(136,139)(137,140)
(138,141)(145,148)(146,149)(147,150)(154,157)(155,158)(156,159)(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)
(271,409)(272,410)(273,411)(274,406)(275,407)(276,408)(277,412)(278,413)
(279,414)(280,418)(281,419)(282,420)(283,415)(284,416)(285,417)(286,421)
(287,422)(288,423)(289,427)(290,428)(291,429)(292,424)(293,425)(294,426)
(295,430)(296,431)(297,432)(298,382)(299,383)(300,384)(301,379)(302,380)
(303,381)(304,385)(305,386)(306,387)(307,391)(308,392)(309,393)(310,388)
(311,389)(312,390)(313,394)(314,395)(315,396)(316,400)(317,401)(318,402)
(319,397)(320,398)(321,399)(322,403)(323,404)(324,405);;
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*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*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,
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)(109,136)(110,138)(111,137)(112,139)(113,141)(114,140)(115,142)
(116,144)(117,143)(118,156)(119,155)(120,154)(121,159)(122,158)(123,157)
(124,162)(125,161)(126,160)(127,147)(128,146)(129,145)(130,150)(131,149)
(132,148)(133,153)(134,152)(135,151)(163,190)(164,192)(165,191)(166,193)
(167,195)(168,194)(169,196)(170,198)(171,197)(172,210)(173,209)(174,208)
(175,213)(176,212)(177,211)(178,216)(179,215)(180,214)(181,201)(182,200)
(183,199)(184,204)(185,203)(186,202)(187,207)(188,206)(189,205)(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)(325,406)(326,408)(327,407)
(328,409)(329,411)(330,410)(331,412)(332,414)(333,413)(334,426)(335,425)
(336,424)(337,429)(338,428)(339,427)(340,432)(341,431)(342,430)(343,417)
(344,416)(345,415)(346,420)(347,419)(348,418)(349,423)(350,422)(351,421)
(352,379)(353,381)(354,380)(355,382)(356,384)(357,383)(358,385)(359,387)
(360,386)(361,399)(362,398)(363,397)(364,402)(365,401)(366,400)(367,405)
(368,404)(369,403)(370,390)(371,389)(372,388)(373,393)(374,392)(375,391)
(376,396)(377,395)(378,394);
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)(109,334)(110,336)(111,335)
(112,340)(113,342)(114,341)(115,337)(116,339)(117,338)(118,325)(119,327)
(120,326)(121,331)(122,333)(123,332)(124,328)(125,330)(126,329)(127,345)
(128,344)(129,343)(130,351)(131,350)(132,349)(133,348)(134,347)(135,346)
(136,361)(137,363)(138,362)(139,367)(140,369)(141,368)(142,364)(143,366)
(144,365)(145,352)(146,354)(147,353)(148,358)(149,360)(150,359)(151,355)
(152,357)(153,356)(154,372)(155,371)(156,370)(157,378)(158,377)(159,376)
(160,375)(161,374)(162,373)(163,388)(164,390)(165,389)(166,394)(167,396)
(168,395)(169,391)(170,393)(171,392)(172,379)(173,381)(174,380)(175,385)
(176,387)(177,386)(178,382)(179,384)(180,383)(181,399)(182,398)(183,397)
(184,405)(185,404)(186,403)(187,402)(188,401)(189,400)(190,415)(191,417)
(192,416)(193,421)(194,423)(195,422)(196,418)(197,420)(198,419)(199,406)
(200,408)(201,407)(202,412)(203,414)(204,413)(205,409)(206,411)(207,410)
(208,426)(209,425)(210,424)(211,432)(212,431)(213,430)(214,429)(215,428)
(216,427);
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, 85)( 56, 86)( 57, 87)( 58, 82)( 59, 83)
( 60, 84)( 61, 88)( 62, 89)( 63, 90)( 64, 94)( 65, 95)( 66, 96)( 67, 91)
( 68, 92)( 69, 93)( 70, 97)( 71, 98)( 72, 99)( 73,103)( 74,104)( 75,105)
( 76,100)( 77,101)( 78,102)( 79,106)( 80,107)( 81,108)(109,112)(110,113)
(111,114)(118,121)(119,122)(120,123)(127,130)(128,131)(129,132)(136,139)
(137,140)(138,141)(145,148)(146,149)(147,150)(154,157)(155,158)(156,159)
(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)(271,409)(272,410)(273,411)(274,406)(275,407)(276,408)(277,412)
(278,413)(279,414)(280,418)(281,419)(282,420)(283,415)(284,416)(285,417)
(286,421)(287,422)(288,423)(289,427)(290,428)(291,429)(292,424)(293,425)
(294,426)(295,430)(296,431)(297,432)(298,382)(299,383)(300,384)(301,379)
(302,380)(303,381)(304,385)(305,386)(306,387)(307,391)(308,392)(309,393)
(310,388)(311,389)(312,390)(313,394)(314,395)(315,396)(316,400)(317,401)
(318,402)(319,397)(320,398)(321,399)(322,403)(323,404)(324,405);
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*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*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,
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