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