Polytope of Type {36,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,12}*1728c
if this polytope has a name.
Group : SmallGroup(1728,30173)
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
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {36,6}*864
3-fold quotients : {12,12}*576f
4-fold quotients : {18,12}*432a, {36,6}*432c
6-fold quotients : {12,6}*288a
8-fold quotients : {18,6}*216a
9-fold quotients : {4,12}*192b
12-fold quotients : {18,4}*144a, {6,12}*144a, {12,6}*144d
18-fold quotients : {4,12}*96b, {4,12}*96c, {4,6}*96
24-fold quotients : {18,2}*72, {6,6}*72a
36-fold quotients : {2,12}*48, {6,4}*48a, {4,3}*48, {4,6}*48b, {4,6}*48c
48-fold quotients : {9,2}*36
72-fold quotients : {4,3}*24, {2,6}*24, {6,2}*24
108-fold quotients : {2,4}*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 := ( 1, 3)( 2, 4)( 5, 11)( 6, 12)( 7, 9)( 8, 10)( 13, 35)( 14, 36)
( 15, 33)( 16, 34)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 27)( 22, 28)
( 23, 25)( 24, 26)( 37, 39)( 38, 40)( 41, 47)( 42, 48)( 43, 45)( 44, 46)
( 49, 71)( 50, 72)( 51, 69)( 52, 70)( 53, 67)( 54, 68)( 55, 65)( 56, 66)
( 57, 63)( 58, 64)( 59, 61)( 60, 62)( 73, 75)( 74, 76)( 77, 83)( 78, 84)
( 79, 81)( 80, 82)( 85,107)( 86,108)( 87,105)( 88,106)( 89,103)( 90,104)
( 91,101)( 92,102)( 93, 99)( 94,100)( 95, 97)( 96, 98)(109,111)(110,112)
(113,119)(114,120)(115,117)(116,118)(121,143)(122,144)(123,141)(124,142)
(125,139)(126,140)(127,137)(128,138)(129,135)(130,136)(131,133)(132,134)
(145,147)(146,148)(149,155)(150,156)(151,153)(152,154)(157,179)(158,180)
(159,177)(160,178)(161,175)(162,176)(163,173)(164,174)(165,171)(166,172)
(167,169)(168,170)(181,183)(182,184)(185,191)(186,192)(187,189)(188,190)
(193,215)(194,216)(195,213)(196,214)(197,211)(198,212)(199,209)(200,210)
(201,207)(202,208)(203,205)(204,206)(217,219)(218,220)(221,227)(222,228)
(223,225)(224,226)(229,251)(230,252)(231,249)(232,250)(233,247)(234,248)
(235,245)(236,246)(237,243)(238,244)(239,241)(240,242)(253,255)(254,256)
(257,263)(258,264)(259,261)(260,262)(265,287)(266,288)(267,285)(268,286)
(269,283)(270,284)(271,281)(272,282)(273,279)(274,280)(275,277)(276,278)
(289,291)(290,292)(293,299)(294,300)(295,297)(296,298)(301,323)(302,324)
(303,321)(304,322)(305,319)(306,320)(307,317)(308,318)(309,315)(310,316)
(311,313)(312,314)(325,327)(326,328)(329,335)(330,336)(331,333)(332,334)
(337,359)(338,360)(339,357)(340,358)(341,355)(342,356)(343,353)(344,354)
(345,351)(346,352)(347,349)(348,350)(361,363)(362,364)(365,371)(366,372)
(367,369)(368,370)(373,395)(374,396)(375,393)(376,394)(377,391)(378,392)
(379,389)(380,390)(381,387)(382,388)(383,385)(384,386)(397,399)(398,400)
(401,407)(402,408)(403,405)(404,406)(409,431)(410,432)(411,429)(412,430)
(413,427)(414,428)(415,425)(416,426)(417,423)(418,424)(419,421)(420,422);;
s1 := ( 1, 13)( 2, 14)( 3, 16)( 4, 15)( 5, 21)( 6, 22)( 7, 24)( 8, 23)
( 9, 17)( 10, 18)( 11, 20)( 12, 19)( 25, 33)( 26, 34)( 27, 36)( 28, 35)
( 31, 32)( 37, 85)( 38, 86)( 39, 88)( 40, 87)( 41, 93)( 42, 94)( 43, 96)
( 44, 95)( 45, 89)( 46, 90)( 47, 92)( 48, 91)( 49, 73)( 50, 74)( 51, 76)
( 52, 75)( 53, 81)( 54, 82)( 55, 84)( 56, 83)( 57, 77)( 58, 78)( 59, 80)
( 60, 79)( 61,105)( 62,106)( 63,108)( 64,107)( 65,101)( 66,102)( 67,104)
( 68,103)( 69, 97)( 70, 98)( 71,100)( 72, 99)(109,121)(110,122)(111,124)
(112,123)(113,129)(114,130)(115,132)(116,131)(117,125)(118,126)(119,128)
(120,127)(133,141)(134,142)(135,144)(136,143)(139,140)(145,193)(146,194)
(147,196)(148,195)(149,201)(150,202)(151,204)(152,203)(153,197)(154,198)
(155,200)(156,199)(157,181)(158,182)(159,184)(160,183)(161,189)(162,190)
(163,192)(164,191)(165,185)(166,186)(167,188)(168,187)(169,213)(170,214)
(171,216)(172,215)(173,209)(174,210)(175,212)(176,211)(177,205)(178,206)
(179,208)(180,207)(217,337)(218,338)(219,340)(220,339)(221,345)(222,346)
(223,348)(224,347)(225,341)(226,342)(227,344)(228,343)(229,325)(230,326)
(231,328)(232,327)(233,333)(234,334)(235,336)(236,335)(237,329)(238,330)
(239,332)(240,331)(241,357)(242,358)(243,360)(244,359)(245,353)(246,354)
(247,356)(248,355)(249,349)(250,350)(251,352)(252,351)(253,409)(254,410)
(255,412)(256,411)(257,417)(258,418)(259,420)(260,419)(261,413)(262,414)
(263,416)(264,415)(265,397)(266,398)(267,400)(268,399)(269,405)(270,406)
(271,408)(272,407)(273,401)(274,402)(275,404)(276,403)(277,429)(278,430)
(279,432)(280,431)(281,425)(282,426)(283,428)(284,427)(285,421)(286,422)
(287,424)(288,423)(289,373)(290,374)(291,376)(292,375)(293,381)(294,382)
(295,384)(296,383)(297,377)(298,378)(299,380)(300,379)(301,361)(302,362)
(303,364)(304,363)(305,369)(306,370)(307,372)(308,371)(309,365)(310,366)
(311,368)(312,367)(313,393)(314,394)(315,396)(316,395)(317,389)(318,390)
(319,392)(320,391)(321,385)(322,386)(323,388)(324,387);;
s2 := ( 1,253)( 2,256)( 3,255)( 4,254)( 5,257)( 6,260)( 7,259)( 8,258)
( 9,261)( 10,264)( 11,263)( 12,262)( 13,265)( 14,268)( 15,267)( 16,266)
( 17,269)( 18,272)( 19,271)( 20,270)( 21,273)( 22,276)( 23,275)( 24,274)
( 25,277)( 26,280)( 27,279)( 28,278)( 29,281)( 30,284)( 31,283)( 32,282)
( 33,285)( 34,288)( 35,287)( 36,286)( 37,217)( 38,220)( 39,219)( 40,218)
( 41,221)( 42,224)( 43,223)( 44,222)( 45,225)( 46,228)( 47,227)( 48,226)
( 49,229)( 50,232)( 51,231)( 52,230)( 53,233)( 54,236)( 55,235)( 56,234)
( 57,237)( 58,240)( 59,239)( 60,238)( 61,241)( 62,244)( 63,243)( 64,242)
( 65,245)( 66,248)( 67,247)( 68,246)( 69,249)( 70,252)( 71,251)( 72,250)
( 73,289)( 74,292)( 75,291)( 76,290)( 77,293)( 78,296)( 79,295)( 80,294)
( 81,297)( 82,300)( 83,299)( 84,298)( 85,301)( 86,304)( 87,303)( 88,302)
( 89,305)( 90,308)( 91,307)( 92,306)( 93,309)( 94,312)( 95,311)( 96,310)
( 97,313)( 98,316)( 99,315)(100,314)(101,317)(102,320)(103,319)(104,318)
(105,321)(106,324)(107,323)(108,322)(109,361)(110,364)(111,363)(112,362)
(113,365)(114,368)(115,367)(116,366)(117,369)(118,372)(119,371)(120,370)
(121,373)(122,376)(123,375)(124,374)(125,377)(126,380)(127,379)(128,378)
(129,381)(130,384)(131,383)(132,382)(133,385)(134,388)(135,387)(136,386)
(137,389)(138,392)(139,391)(140,390)(141,393)(142,396)(143,395)(144,394)
(145,325)(146,328)(147,327)(148,326)(149,329)(150,332)(151,331)(152,330)
(153,333)(154,336)(155,335)(156,334)(157,337)(158,340)(159,339)(160,338)
(161,341)(162,344)(163,343)(164,342)(165,345)(166,348)(167,347)(168,346)
(169,349)(170,352)(171,351)(172,350)(173,353)(174,356)(175,355)(176,354)
(177,357)(178,360)(179,359)(180,358)(181,397)(182,400)(183,399)(184,398)
(185,401)(186,404)(187,403)(188,402)(189,405)(190,408)(191,407)(192,406)
(193,409)(194,412)(195,411)(196,410)(197,413)(198,416)(199,415)(200,414)
(201,417)(202,420)(203,419)(204,418)(205,421)(206,424)(207,423)(208,422)
(209,425)(210,428)(211,427)(212,426)(213,429)(214,432)(215,431)(216,430);;
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*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*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,
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*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 1, 3)( 2, 4)( 5, 11)( 6, 12)( 7, 9)( 8, 10)( 13, 35)
( 14, 36)( 15, 33)( 16, 34)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 27)
( 22, 28)( 23, 25)( 24, 26)( 37, 39)( 38, 40)( 41, 47)( 42, 48)( 43, 45)
( 44, 46)( 49, 71)( 50, 72)( 51, 69)( 52, 70)( 53, 67)( 54, 68)( 55, 65)
( 56, 66)( 57, 63)( 58, 64)( 59, 61)( 60, 62)( 73, 75)( 74, 76)( 77, 83)
( 78, 84)( 79, 81)( 80, 82)( 85,107)( 86,108)( 87,105)( 88,106)( 89,103)
( 90,104)( 91,101)( 92,102)( 93, 99)( 94,100)( 95, 97)( 96, 98)(109,111)
(110,112)(113,119)(114,120)(115,117)(116,118)(121,143)(122,144)(123,141)
(124,142)(125,139)(126,140)(127,137)(128,138)(129,135)(130,136)(131,133)
(132,134)(145,147)(146,148)(149,155)(150,156)(151,153)(152,154)(157,179)
(158,180)(159,177)(160,178)(161,175)(162,176)(163,173)(164,174)(165,171)
(166,172)(167,169)(168,170)(181,183)(182,184)(185,191)(186,192)(187,189)
(188,190)(193,215)(194,216)(195,213)(196,214)(197,211)(198,212)(199,209)
(200,210)(201,207)(202,208)(203,205)(204,206)(217,219)(218,220)(221,227)
(222,228)(223,225)(224,226)(229,251)(230,252)(231,249)(232,250)(233,247)
(234,248)(235,245)(236,246)(237,243)(238,244)(239,241)(240,242)(253,255)
(254,256)(257,263)(258,264)(259,261)(260,262)(265,287)(266,288)(267,285)
(268,286)(269,283)(270,284)(271,281)(272,282)(273,279)(274,280)(275,277)
(276,278)(289,291)(290,292)(293,299)(294,300)(295,297)(296,298)(301,323)
(302,324)(303,321)(304,322)(305,319)(306,320)(307,317)(308,318)(309,315)
(310,316)(311,313)(312,314)(325,327)(326,328)(329,335)(330,336)(331,333)
(332,334)(337,359)(338,360)(339,357)(340,358)(341,355)(342,356)(343,353)
(344,354)(345,351)(346,352)(347,349)(348,350)(361,363)(362,364)(365,371)
(366,372)(367,369)(368,370)(373,395)(374,396)(375,393)(376,394)(377,391)
(378,392)(379,389)(380,390)(381,387)(382,388)(383,385)(384,386)(397,399)
(398,400)(401,407)(402,408)(403,405)(404,406)(409,431)(410,432)(411,429)
(412,430)(413,427)(414,428)(415,425)(416,426)(417,423)(418,424)(419,421)
(420,422);
s1 := Sym(432)!( 1, 13)( 2, 14)( 3, 16)( 4, 15)( 5, 21)( 6, 22)( 7, 24)
( 8, 23)( 9, 17)( 10, 18)( 11, 20)( 12, 19)( 25, 33)( 26, 34)( 27, 36)
( 28, 35)( 31, 32)( 37, 85)( 38, 86)( 39, 88)( 40, 87)( 41, 93)( 42, 94)
( 43, 96)( 44, 95)( 45, 89)( 46, 90)( 47, 92)( 48, 91)( 49, 73)( 50, 74)
( 51, 76)( 52, 75)( 53, 81)( 54, 82)( 55, 84)( 56, 83)( 57, 77)( 58, 78)
( 59, 80)( 60, 79)( 61,105)( 62,106)( 63,108)( 64,107)( 65,101)( 66,102)
( 67,104)( 68,103)( 69, 97)( 70, 98)( 71,100)( 72, 99)(109,121)(110,122)
(111,124)(112,123)(113,129)(114,130)(115,132)(116,131)(117,125)(118,126)
(119,128)(120,127)(133,141)(134,142)(135,144)(136,143)(139,140)(145,193)
(146,194)(147,196)(148,195)(149,201)(150,202)(151,204)(152,203)(153,197)
(154,198)(155,200)(156,199)(157,181)(158,182)(159,184)(160,183)(161,189)
(162,190)(163,192)(164,191)(165,185)(166,186)(167,188)(168,187)(169,213)
(170,214)(171,216)(172,215)(173,209)(174,210)(175,212)(176,211)(177,205)
(178,206)(179,208)(180,207)(217,337)(218,338)(219,340)(220,339)(221,345)
(222,346)(223,348)(224,347)(225,341)(226,342)(227,344)(228,343)(229,325)
(230,326)(231,328)(232,327)(233,333)(234,334)(235,336)(236,335)(237,329)
(238,330)(239,332)(240,331)(241,357)(242,358)(243,360)(244,359)(245,353)
(246,354)(247,356)(248,355)(249,349)(250,350)(251,352)(252,351)(253,409)
(254,410)(255,412)(256,411)(257,417)(258,418)(259,420)(260,419)(261,413)
(262,414)(263,416)(264,415)(265,397)(266,398)(267,400)(268,399)(269,405)
(270,406)(271,408)(272,407)(273,401)(274,402)(275,404)(276,403)(277,429)
(278,430)(279,432)(280,431)(281,425)(282,426)(283,428)(284,427)(285,421)
(286,422)(287,424)(288,423)(289,373)(290,374)(291,376)(292,375)(293,381)
(294,382)(295,384)(296,383)(297,377)(298,378)(299,380)(300,379)(301,361)
(302,362)(303,364)(304,363)(305,369)(306,370)(307,372)(308,371)(309,365)
(310,366)(311,368)(312,367)(313,393)(314,394)(315,396)(316,395)(317,389)
(318,390)(319,392)(320,391)(321,385)(322,386)(323,388)(324,387);
s2 := Sym(432)!( 1,253)( 2,256)( 3,255)( 4,254)( 5,257)( 6,260)( 7,259)
( 8,258)( 9,261)( 10,264)( 11,263)( 12,262)( 13,265)( 14,268)( 15,267)
( 16,266)( 17,269)( 18,272)( 19,271)( 20,270)( 21,273)( 22,276)( 23,275)
( 24,274)( 25,277)( 26,280)( 27,279)( 28,278)( 29,281)( 30,284)( 31,283)
( 32,282)( 33,285)( 34,288)( 35,287)( 36,286)( 37,217)( 38,220)( 39,219)
( 40,218)( 41,221)( 42,224)( 43,223)( 44,222)( 45,225)( 46,228)( 47,227)
( 48,226)( 49,229)( 50,232)( 51,231)( 52,230)( 53,233)( 54,236)( 55,235)
( 56,234)( 57,237)( 58,240)( 59,239)( 60,238)( 61,241)( 62,244)( 63,243)
( 64,242)( 65,245)( 66,248)( 67,247)( 68,246)( 69,249)( 70,252)( 71,251)
( 72,250)( 73,289)( 74,292)( 75,291)( 76,290)( 77,293)( 78,296)( 79,295)
( 80,294)( 81,297)( 82,300)( 83,299)( 84,298)( 85,301)( 86,304)( 87,303)
( 88,302)( 89,305)( 90,308)( 91,307)( 92,306)( 93,309)( 94,312)( 95,311)
( 96,310)( 97,313)( 98,316)( 99,315)(100,314)(101,317)(102,320)(103,319)
(104,318)(105,321)(106,324)(107,323)(108,322)(109,361)(110,364)(111,363)
(112,362)(113,365)(114,368)(115,367)(116,366)(117,369)(118,372)(119,371)
(120,370)(121,373)(122,376)(123,375)(124,374)(125,377)(126,380)(127,379)
(128,378)(129,381)(130,384)(131,383)(132,382)(133,385)(134,388)(135,387)
(136,386)(137,389)(138,392)(139,391)(140,390)(141,393)(142,396)(143,395)
(144,394)(145,325)(146,328)(147,327)(148,326)(149,329)(150,332)(151,331)
(152,330)(153,333)(154,336)(155,335)(156,334)(157,337)(158,340)(159,339)
(160,338)(161,341)(162,344)(163,343)(164,342)(165,345)(166,348)(167,347)
(168,346)(169,349)(170,352)(171,351)(172,350)(173,353)(174,356)(175,355)
(176,354)(177,357)(178,360)(179,359)(180,358)(181,397)(182,400)(183,399)
(184,398)(185,401)(186,404)(187,403)(188,402)(189,405)(190,408)(191,407)
(192,406)(193,409)(194,412)(195,411)(196,410)(197,413)(198,416)(199,415)
(200,414)(201,417)(202,420)(203,419)(204,418)(205,421)(206,424)(207,423)
(208,422)(209,425)(210,428)(211,427)(212,426)(213,429)(214,432)(215,431)
(216,430);
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*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*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,
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*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1 >;
References : None.
to this polytope