Polytope of Type {12,36}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,36}*1728b
if this polytope has a name.
Group : SmallGroup(1728,3530)
Rank : 3
Schlafli Type : {12,36}
Number of vertices, edges, etc : 24, 432, 72
Order of s0s1s2 : 36
Order of s0s1s2s1 : 12
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 : {12,36}*864b
3-fold quotients : {4,36}*576a, {12,12}*576b
4-fold quotients : {6,36}*432b, {12,18}*432b
6-fold quotients : {4,36}*288a, {12,12}*288b
8-fold quotients : {6,18}*216b
9-fold quotients : {4,12}*192a
12-fold quotients : {2,36}*144, {4,18}*144a, {6,12}*144b, {12,6}*144c
16-fold quotients : {6,9}*108
18-fold quotients : {4,12}*96a
24-fold quotients : {2,18}*72, {6,6}*72b
27-fold quotients : {4,4}*64
36-fold quotients : {2,12}*48, {4,6}*48a
48-fold quotients : {2,9}*36, {6,3}*36
54-fold quotients : {4,4}*32
72-fold quotients : {2,6}*24
108-fold quotients : {2,4}*16, {4,2}*16
144-fold quotients : {2,3}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)
( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)( 68, 98)
( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)( 76,106)
( 77,107)( 78,108)( 79,103)( 80,104)( 81,105)(112,115)(113,116)(114,117)
(121,124)(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)
(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(163,190)
(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)(171,195)
(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)(179,203)
(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)(187,211)
(188,212)(189,213)(217,325)(218,326)(219,327)(220,331)(221,332)(222,333)
(223,328)(224,329)(225,330)(226,334)(227,335)(228,336)(229,340)(230,341)
(231,342)(232,337)(233,338)(234,339)(235,343)(236,344)(237,345)(238,349)
(239,350)(240,351)(241,346)(242,347)(243,348)(244,352)(245,353)(246,354)
(247,358)(248,359)(249,360)(250,355)(251,356)(252,357)(253,361)(254,362)
(255,363)(256,367)(257,368)(258,369)(259,364)(260,365)(261,366)(262,370)
(263,371)(264,372)(265,376)(266,377)(267,378)(268,373)(269,374)(270,375)
(271,406)(272,407)(273,408)(274,412)(275,413)(276,414)(277,409)(278,410)
(279,411)(280,415)(281,416)(282,417)(283,421)(284,422)(285,423)(286,418)
(287,419)(288,420)(289,424)(290,425)(291,426)(292,430)(293,431)(294,432)
(295,427)(296,428)(297,429)(298,379)(299,380)(300,381)(301,385)(302,386)
(303,387)(304,382)(305,383)(306,384)(307,388)(308,389)(309,390)(310,394)
(311,395)(312,396)(313,391)(314,392)(315,393)(316,397)(317,398)(318,399)
(319,403)(320,404)(321,405)(322,400)(323,401)(324,402);;
s1 := ( 1,220)( 2,222)( 3,221)( 4,217)( 5,219)( 6,218)( 7,223)( 8,225)
( 9,224)( 10,240)( 11,239)( 12,238)( 13,237)( 14,236)( 15,235)( 16,243)
( 17,242)( 18,241)( 19,231)( 20,230)( 21,229)( 22,228)( 23,227)( 24,226)
( 25,234)( 26,233)( 27,232)( 28,247)( 29,249)( 30,248)( 31,244)( 32,246)
( 33,245)( 34,250)( 35,252)( 36,251)( 37,267)( 38,266)( 39,265)( 40,264)
( 41,263)( 42,262)( 43,270)( 44,269)( 45,268)( 46,258)( 47,257)( 48,256)
( 49,255)( 50,254)( 51,253)( 52,261)( 53,260)( 54,259)( 55,274)( 56,276)
( 57,275)( 58,271)( 59,273)( 60,272)( 61,277)( 62,279)( 63,278)( 64,294)
( 65,293)( 66,292)( 67,291)( 68,290)( 69,289)( 70,297)( 71,296)( 72,295)
( 73,285)( 74,284)( 75,283)( 76,282)( 77,281)( 78,280)( 79,288)( 80,287)
( 81,286)( 82,301)( 83,303)( 84,302)( 85,298)( 86,300)( 87,299)( 88,304)
( 89,306)( 90,305)( 91,321)( 92,320)( 93,319)( 94,318)( 95,317)( 96,316)
( 97,324)( 98,323)( 99,322)(100,312)(101,311)(102,310)(103,309)(104,308)
(105,307)(106,315)(107,314)(108,313)(109,328)(110,330)(111,329)(112,325)
(113,327)(114,326)(115,331)(116,333)(117,332)(118,348)(119,347)(120,346)
(121,345)(122,344)(123,343)(124,351)(125,350)(126,349)(127,339)(128,338)
(129,337)(130,336)(131,335)(132,334)(133,342)(134,341)(135,340)(136,355)
(137,357)(138,356)(139,352)(140,354)(141,353)(142,358)(143,360)(144,359)
(145,375)(146,374)(147,373)(148,372)(149,371)(150,370)(151,378)(152,377)
(153,376)(154,366)(155,365)(156,364)(157,363)(158,362)(159,361)(160,369)
(161,368)(162,367)(163,382)(164,384)(165,383)(166,379)(167,381)(168,380)
(169,385)(170,387)(171,386)(172,402)(173,401)(174,400)(175,399)(176,398)
(177,397)(178,405)(179,404)(180,403)(181,393)(182,392)(183,391)(184,390)
(185,389)(186,388)(187,396)(188,395)(189,394)(190,409)(191,411)(192,410)
(193,406)(194,408)(195,407)(196,412)(197,414)(198,413)(199,429)(200,428)
(201,427)(202,426)(203,425)(204,424)(205,432)(206,431)(207,430)(208,420)
(209,419)(210,418)(211,417)(212,416)(213,415)(214,423)(215,422)(216,421);;
s2 := ( 1, 10)( 2, 12)( 3, 11)( 4, 16)( 5, 18)( 6, 17)( 7, 13)( 8, 15)
( 9, 14)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 37)( 29, 39)( 30, 38)
( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 48)( 49, 54)
( 50, 53)( 51, 52)( 55, 91)( 56, 93)( 57, 92)( 58, 97)( 59, 99)( 60, 98)
( 61, 94)( 62, 96)( 63, 95)( 64, 82)( 65, 84)( 66, 83)( 67, 88)( 68, 90)
( 69, 89)( 70, 85)( 71, 87)( 72, 86)( 73,102)( 74,101)( 75,100)( 76,108)
( 77,107)( 78,106)( 79,105)( 80,104)( 81,103)(109,145)(110,147)(111,146)
(112,151)(113,153)(114,152)(115,148)(116,150)(117,149)(118,136)(119,138)
(120,137)(121,142)(122,144)(123,143)(124,139)(125,141)(126,140)(127,156)
(128,155)(129,154)(130,162)(131,161)(132,160)(133,159)(134,158)(135,157)
(163,172)(164,174)(165,173)(166,178)(167,180)(168,179)(169,175)(170,177)
(171,176)(181,183)(184,189)(185,188)(186,187)(190,199)(191,201)(192,200)
(193,205)(194,207)(195,206)(196,202)(197,204)(198,203)(208,210)(211,216)
(212,215)(213,214)(217,415)(218,417)(219,416)(220,421)(221,423)(222,422)
(223,418)(224,420)(225,419)(226,406)(227,408)(228,407)(229,412)(230,414)
(231,413)(232,409)(233,411)(234,410)(235,426)(236,425)(237,424)(238,432)
(239,431)(240,430)(241,429)(242,428)(243,427)(244,388)(245,390)(246,389)
(247,394)(248,396)(249,395)(250,391)(251,393)(252,392)(253,379)(254,381)
(255,380)(256,385)(257,387)(258,386)(259,382)(260,384)(261,383)(262,399)
(263,398)(264,397)(265,405)(266,404)(267,403)(268,402)(269,401)(270,400)
(271,334)(272,336)(273,335)(274,340)(275,342)(276,341)(277,337)(278,339)
(279,338)(280,325)(281,327)(282,326)(283,331)(284,333)(285,332)(286,328)
(287,330)(288,329)(289,345)(290,344)(291,343)(292,351)(293,350)(294,349)
(295,348)(296,347)(297,346)(298,361)(299,363)(300,362)(301,367)(302,369)
(303,368)(304,364)(305,366)(306,365)(307,352)(308,354)(309,353)(310,358)
(311,360)(312,359)(313,355)(314,357)(315,356)(316,372)(317,371)(318,370)
(319,378)(320,377)(321,376)(322,375)(323,374)(324,373);;
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,
s2*s0*s1*s0*s1*s0*s1*s0*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,
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*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)!( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)
( 60, 90)( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)
( 68, 98)( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)
( 76,106)( 77,107)( 78,108)( 79,103)( 80,104)( 81,105)(112,115)(113,116)
(114,117)(121,124)(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)
(140,143)(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)
(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)
(171,195)(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)
(179,203)(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)
(187,211)(188,212)(189,213)(217,325)(218,326)(219,327)(220,331)(221,332)
(222,333)(223,328)(224,329)(225,330)(226,334)(227,335)(228,336)(229,340)
(230,341)(231,342)(232,337)(233,338)(234,339)(235,343)(236,344)(237,345)
(238,349)(239,350)(240,351)(241,346)(242,347)(243,348)(244,352)(245,353)
(246,354)(247,358)(248,359)(249,360)(250,355)(251,356)(252,357)(253,361)
(254,362)(255,363)(256,367)(257,368)(258,369)(259,364)(260,365)(261,366)
(262,370)(263,371)(264,372)(265,376)(266,377)(267,378)(268,373)(269,374)
(270,375)(271,406)(272,407)(273,408)(274,412)(275,413)(276,414)(277,409)
(278,410)(279,411)(280,415)(281,416)(282,417)(283,421)(284,422)(285,423)
(286,418)(287,419)(288,420)(289,424)(290,425)(291,426)(292,430)(293,431)
(294,432)(295,427)(296,428)(297,429)(298,379)(299,380)(300,381)(301,385)
(302,386)(303,387)(304,382)(305,383)(306,384)(307,388)(308,389)(309,390)
(310,394)(311,395)(312,396)(313,391)(314,392)(315,393)(316,397)(317,398)
(318,399)(319,403)(320,404)(321,405)(322,400)(323,401)(324,402);
s1 := Sym(432)!( 1,220)( 2,222)( 3,221)( 4,217)( 5,219)( 6,218)( 7,223)
( 8,225)( 9,224)( 10,240)( 11,239)( 12,238)( 13,237)( 14,236)( 15,235)
( 16,243)( 17,242)( 18,241)( 19,231)( 20,230)( 21,229)( 22,228)( 23,227)
( 24,226)( 25,234)( 26,233)( 27,232)( 28,247)( 29,249)( 30,248)( 31,244)
( 32,246)( 33,245)( 34,250)( 35,252)( 36,251)( 37,267)( 38,266)( 39,265)
( 40,264)( 41,263)( 42,262)( 43,270)( 44,269)( 45,268)( 46,258)( 47,257)
( 48,256)( 49,255)( 50,254)( 51,253)( 52,261)( 53,260)( 54,259)( 55,274)
( 56,276)( 57,275)( 58,271)( 59,273)( 60,272)( 61,277)( 62,279)( 63,278)
( 64,294)( 65,293)( 66,292)( 67,291)( 68,290)( 69,289)( 70,297)( 71,296)
( 72,295)( 73,285)( 74,284)( 75,283)( 76,282)( 77,281)( 78,280)( 79,288)
( 80,287)( 81,286)( 82,301)( 83,303)( 84,302)( 85,298)( 86,300)( 87,299)
( 88,304)( 89,306)( 90,305)( 91,321)( 92,320)( 93,319)( 94,318)( 95,317)
( 96,316)( 97,324)( 98,323)( 99,322)(100,312)(101,311)(102,310)(103,309)
(104,308)(105,307)(106,315)(107,314)(108,313)(109,328)(110,330)(111,329)
(112,325)(113,327)(114,326)(115,331)(116,333)(117,332)(118,348)(119,347)
(120,346)(121,345)(122,344)(123,343)(124,351)(125,350)(126,349)(127,339)
(128,338)(129,337)(130,336)(131,335)(132,334)(133,342)(134,341)(135,340)
(136,355)(137,357)(138,356)(139,352)(140,354)(141,353)(142,358)(143,360)
(144,359)(145,375)(146,374)(147,373)(148,372)(149,371)(150,370)(151,378)
(152,377)(153,376)(154,366)(155,365)(156,364)(157,363)(158,362)(159,361)
(160,369)(161,368)(162,367)(163,382)(164,384)(165,383)(166,379)(167,381)
(168,380)(169,385)(170,387)(171,386)(172,402)(173,401)(174,400)(175,399)
(176,398)(177,397)(178,405)(179,404)(180,403)(181,393)(182,392)(183,391)
(184,390)(185,389)(186,388)(187,396)(188,395)(189,394)(190,409)(191,411)
(192,410)(193,406)(194,408)(195,407)(196,412)(197,414)(198,413)(199,429)
(200,428)(201,427)(202,426)(203,425)(204,424)(205,432)(206,431)(207,430)
(208,420)(209,419)(210,418)(211,417)(212,416)(213,415)(214,423)(215,422)
(216,421);
s2 := Sym(432)!( 1, 10)( 2, 12)( 3, 11)( 4, 16)( 5, 18)( 6, 17)( 7, 13)
( 8, 15)( 9, 14)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 37)( 29, 39)
( 30, 38)( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 48)
( 49, 54)( 50, 53)( 51, 52)( 55, 91)( 56, 93)( 57, 92)( 58, 97)( 59, 99)
( 60, 98)( 61, 94)( 62, 96)( 63, 95)( 64, 82)( 65, 84)( 66, 83)( 67, 88)
( 68, 90)( 69, 89)( 70, 85)( 71, 87)( 72, 86)( 73,102)( 74,101)( 75,100)
( 76,108)( 77,107)( 78,106)( 79,105)( 80,104)( 81,103)(109,145)(110,147)
(111,146)(112,151)(113,153)(114,152)(115,148)(116,150)(117,149)(118,136)
(119,138)(120,137)(121,142)(122,144)(123,143)(124,139)(125,141)(126,140)
(127,156)(128,155)(129,154)(130,162)(131,161)(132,160)(133,159)(134,158)
(135,157)(163,172)(164,174)(165,173)(166,178)(167,180)(168,179)(169,175)
(170,177)(171,176)(181,183)(184,189)(185,188)(186,187)(190,199)(191,201)
(192,200)(193,205)(194,207)(195,206)(196,202)(197,204)(198,203)(208,210)
(211,216)(212,215)(213,214)(217,415)(218,417)(219,416)(220,421)(221,423)
(222,422)(223,418)(224,420)(225,419)(226,406)(227,408)(228,407)(229,412)
(230,414)(231,413)(232,409)(233,411)(234,410)(235,426)(236,425)(237,424)
(238,432)(239,431)(240,430)(241,429)(242,428)(243,427)(244,388)(245,390)
(246,389)(247,394)(248,396)(249,395)(250,391)(251,393)(252,392)(253,379)
(254,381)(255,380)(256,385)(257,387)(258,386)(259,382)(260,384)(261,383)
(262,399)(263,398)(264,397)(265,405)(266,404)(267,403)(268,402)(269,401)
(270,400)(271,334)(272,336)(273,335)(274,340)(275,342)(276,341)(277,337)
(278,339)(279,338)(280,325)(281,327)(282,326)(283,331)(284,333)(285,332)
(286,328)(287,330)(288,329)(289,345)(290,344)(291,343)(292,351)(293,350)
(294,349)(295,348)(296,347)(297,346)(298,361)(299,363)(300,362)(301,367)
(302,369)(303,368)(304,364)(305,366)(306,365)(307,352)(308,354)(309,353)
(310,358)(311,360)(312,359)(313,355)(314,357)(315,356)(316,372)(317,371)
(318,370)(319,378)(320,377)(321,376)(322,375)(323,374)(324,373);
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,
s2*s0*s1*s0*s1*s0*s1*s0*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,
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*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