Polytope of Type {12,36}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,36}*1728g
if this polytope has a name.
Group : SmallGroup(1728,30229)
Rank : 3
Schlafli Type : {12,36}
Number of vertices, edges, etc : 24, 432, 72
Order of s0s1s2 : 18
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 : {12,18}*864b
3-fold quotients : {4,36}*576c, {12,12}*576i
4-fold quotients : {12,9}*432
6-fold quotients : {4,18}*288, {12,6}*288b
8-fold quotients : {6,18}*216b
9-fold quotients : {4,12}*192c
12-fold quotients : {4,9}*144, {4,18}*144b, {4,18}*144c, {12,3}*144
16-fold quotients : {6,9}*108
18-fold quotients : {4,6}*96
24-fold quotients : {4,9}*72, {2,18}*72, {6,6}*72b
36-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
48-fold quotients : {2,9}*36, {6,3}*36
72-fold quotients : {4,3}*24, {2,6}*24
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 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 27)( 14, 28)
( 15, 25)( 16, 26)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 35)( 22, 36)
( 23, 33)( 24, 34)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 63)( 50, 64)( 51, 61)( 52, 62)( 53, 67)( 54, 68)( 55, 65)( 56, 66)
( 57, 71)( 58, 72)( 59, 69)( 60, 70)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,103)( 90,104)
( 91,101)( 92,102)( 93,107)( 94,108)( 95,105)( 96,106)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,135)(122,136)(123,133)(124,134)
(125,139)(126,140)(127,137)(128,138)(129,143)(130,144)(131,141)(132,142)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,171)(158,172)
(159,169)(160,170)(161,175)(162,176)(163,173)(164,174)(165,179)(166,180)
(167,177)(168,178)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,207)(194,208)(195,205)(196,206)(197,211)(198,212)(199,209)(200,210)
(201,215)(202,216)(203,213)(204,214)(217,327)(218,328)(219,325)(220,326)
(221,331)(222,332)(223,329)(224,330)(225,335)(226,336)(227,333)(228,334)
(229,351)(230,352)(231,349)(232,350)(233,355)(234,356)(235,353)(236,354)
(237,359)(238,360)(239,357)(240,358)(241,339)(242,340)(243,337)(244,338)
(245,343)(246,344)(247,341)(248,342)(249,347)(250,348)(251,345)(252,346)
(253,363)(254,364)(255,361)(256,362)(257,367)(258,368)(259,365)(260,366)
(261,371)(262,372)(263,369)(264,370)(265,387)(266,388)(267,385)(268,386)
(269,391)(270,392)(271,389)(272,390)(273,395)(274,396)(275,393)(276,394)
(277,375)(278,376)(279,373)(280,374)(281,379)(282,380)(283,377)(284,378)
(285,383)(286,384)(287,381)(288,382)(289,399)(290,400)(291,397)(292,398)
(293,403)(294,404)(295,401)(296,402)(297,407)(298,408)(299,405)(300,406)
(301,423)(302,424)(303,421)(304,422)(305,427)(306,428)(307,425)(308,426)
(309,431)(310,432)(311,429)(312,430)(313,411)(314,412)(315,409)(316,410)
(317,415)(318,416)(319,413)(320,414)(321,419)(322,420)(323,417)(324,418);;
s1 := ( 1,229)( 2,230)( 3,232)( 4,231)( 5,237)( 6,238)( 7,240)( 8,239)
( 9,233)( 10,234)( 11,236)( 12,235)( 13,217)( 14,218)( 15,220)( 16,219)
( 17,225)( 18,226)( 19,228)( 20,227)( 21,221)( 22,222)( 23,224)( 24,223)
( 25,241)( 26,242)( 27,244)( 28,243)( 29,249)( 30,250)( 31,252)( 32,251)
( 33,245)( 34,246)( 35,248)( 36,247)( 37,309)( 38,310)( 39,312)( 40,311)
( 41,305)( 42,306)( 43,308)( 44,307)( 45,301)( 46,302)( 47,304)( 48,303)
( 49,297)( 50,298)( 51,300)( 52,299)( 53,293)( 54,294)( 55,296)( 56,295)
( 57,289)( 58,290)( 59,292)( 60,291)( 61,321)( 62,322)( 63,324)( 64,323)
( 65,317)( 66,318)( 67,320)( 68,319)( 69,313)( 70,314)( 71,316)( 72,315)
( 73,273)( 74,274)( 75,276)( 76,275)( 77,269)( 78,270)( 79,272)( 80,271)
( 81,265)( 82,266)( 83,268)( 84,267)( 85,261)( 86,262)( 87,264)( 88,263)
( 89,257)( 90,258)( 91,260)( 92,259)( 93,253)( 94,254)( 95,256)( 96,255)
( 97,285)( 98,286)( 99,288)(100,287)(101,281)(102,282)(103,284)(104,283)
(105,277)(106,278)(107,280)(108,279)(109,337)(110,338)(111,340)(112,339)
(113,345)(114,346)(115,348)(116,347)(117,341)(118,342)(119,344)(120,343)
(121,325)(122,326)(123,328)(124,327)(125,333)(126,334)(127,336)(128,335)
(129,329)(130,330)(131,332)(132,331)(133,349)(134,350)(135,352)(136,351)
(137,357)(138,358)(139,360)(140,359)(141,353)(142,354)(143,356)(144,355)
(145,417)(146,418)(147,420)(148,419)(149,413)(150,414)(151,416)(152,415)
(153,409)(154,410)(155,412)(156,411)(157,405)(158,406)(159,408)(160,407)
(161,401)(162,402)(163,404)(164,403)(165,397)(166,398)(167,400)(168,399)
(169,429)(170,430)(171,432)(172,431)(173,425)(174,426)(175,428)(176,427)
(177,421)(178,422)(179,424)(180,423)(181,381)(182,382)(183,384)(184,383)
(185,377)(186,378)(187,380)(188,379)(189,373)(190,374)(191,376)(192,375)
(193,369)(194,370)(195,372)(196,371)(197,365)(198,366)(199,368)(200,367)
(201,361)(202,362)(203,364)(204,363)(205,393)(206,394)(207,396)(208,395)
(209,389)(210,390)(211,392)(212,391)(213,385)(214,386)(215,388)(216,387);;
s2 := ( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 45)( 6, 48)( 7, 47)( 8, 46)
( 9, 41)( 10, 44)( 11, 43)( 12, 42)( 13, 61)( 14, 64)( 15, 63)( 16, 62)
( 17, 69)( 18, 72)( 19, 71)( 20, 70)( 21, 65)( 22, 68)( 23, 67)( 24, 66)
( 25, 49)( 26, 52)( 27, 51)( 28, 50)( 29, 57)( 30, 60)( 31, 59)( 32, 58)
( 33, 53)( 34, 56)( 35, 55)( 36, 54)( 73, 81)( 74, 84)( 75, 83)( 76, 82)
( 78, 80)( 85,105)( 86,108)( 87,107)( 88,106)( 89,101)( 90,104)( 91,103)
( 92,102)( 93, 97)( 94,100)( 95, 99)( 96, 98)(109,145)(110,148)(111,147)
(112,146)(113,153)(114,156)(115,155)(116,154)(117,149)(118,152)(119,151)
(120,150)(121,169)(122,172)(123,171)(124,170)(125,177)(126,180)(127,179)
(128,178)(129,173)(130,176)(131,175)(132,174)(133,157)(134,160)(135,159)
(136,158)(137,165)(138,168)(139,167)(140,166)(141,161)(142,164)(143,163)
(144,162)(181,189)(182,192)(183,191)(184,190)(186,188)(193,213)(194,216)
(195,215)(196,214)(197,209)(198,212)(199,211)(200,210)(201,205)(202,208)
(203,207)(204,206)(217,361)(218,364)(219,363)(220,362)(221,369)(222,372)
(223,371)(224,370)(225,365)(226,368)(227,367)(228,366)(229,385)(230,388)
(231,387)(232,386)(233,393)(234,396)(235,395)(236,394)(237,389)(238,392)
(239,391)(240,390)(241,373)(242,376)(243,375)(244,374)(245,381)(246,384)
(247,383)(248,382)(249,377)(250,380)(251,379)(252,378)(253,325)(254,328)
(255,327)(256,326)(257,333)(258,336)(259,335)(260,334)(261,329)(262,332)
(263,331)(264,330)(265,349)(266,352)(267,351)(268,350)(269,357)(270,360)
(271,359)(272,358)(273,353)(274,356)(275,355)(276,354)(277,337)(278,340)
(279,339)(280,338)(281,345)(282,348)(283,347)(284,346)(285,341)(286,344)
(287,343)(288,342)(289,405)(290,408)(291,407)(292,406)(293,401)(294,404)
(295,403)(296,402)(297,397)(298,400)(299,399)(300,398)(301,429)(302,432)
(303,431)(304,430)(305,425)(306,428)(307,427)(308,426)(309,421)(310,424)
(311,423)(312,422)(313,417)(314,420)(315,419)(316,418)(317,413)(318,416)
(319,415)(320,414)(321,409)(322,412)(323,411)(324,410);;
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*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*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,
s2*s0*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*s2*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 27)
( 14, 28)( 15, 25)( 16, 26)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 35)
( 22, 36)( 23, 33)( 24, 34)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 63)( 50, 64)( 51, 61)( 52, 62)( 53, 67)( 54, 68)( 55, 65)
( 56, 66)( 57, 71)( 58, 72)( 59, 69)( 60, 70)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,103)
( 90,104)( 91,101)( 92,102)( 93,107)( 94,108)( 95,105)( 96,106)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,135)(122,136)(123,133)
(124,134)(125,139)(126,140)(127,137)(128,138)(129,143)(130,144)(131,141)
(132,142)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,171)
(158,172)(159,169)(160,170)(161,175)(162,176)(163,173)(164,174)(165,179)
(166,180)(167,177)(168,178)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,207)(194,208)(195,205)(196,206)(197,211)(198,212)(199,209)
(200,210)(201,215)(202,216)(203,213)(204,214)(217,327)(218,328)(219,325)
(220,326)(221,331)(222,332)(223,329)(224,330)(225,335)(226,336)(227,333)
(228,334)(229,351)(230,352)(231,349)(232,350)(233,355)(234,356)(235,353)
(236,354)(237,359)(238,360)(239,357)(240,358)(241,339)(242,340)(243,337)
(244,338)(245,343)(246,344)(247,341)(248,342)(249,347)(250,348)(251,345)
(252,346)(253,363)(254,364)(255,361)(256,362)(257,367)(258,368)(259,365)
(260,366)(261,371)(262,372)(263,369)(264,370)(265,387)(266,388)(267,385)
(268,386)(269,391)(270,392)(271,389)(272,390)(273,395)(274,396)(275,393)
(276,394)(277,375)(278,376)(279,373)(280,374)(281,379)(282,380)(283,377)
(284,378)(285,383)(286,384)(287,381)(288,382)(289,399)(290,400)(291,397)
(292,398)(293,403)(294,404)(295,401)(296,402)(297,407)(298,408)(299,405)
(300,406)(301,423)(302,424)(303,421)(304,422)(305,427)(306,428)(307,425)
(308,426)(309,431)(310,432)(311,429)(312,430)(313,411)(314,412)(315,409)
(316,410)(317,415)(318,416)(319,413)(320,414)(321,419)(322,420)(323,417)
(324,418);
s1 := Sym(432)!( 1,229)( 2,230)( 3,232)( 4,231)( 5,237)( 6,238)( 7,240)
( 8,239)( 9,233)( 10,234)( 11,236)( 12,235)( 13,217)( 14,218)( 15,220)
( 16,219)( 17,225)( 18,226)( 19,228)( 20,227)( 21,221)( 22,222)( 23,224)
( 24,223)( 25,241)( 26,242)( 27,244)( 28,243)( 29,249)( 30,250)( 31,252)
( 32,251)( 33,245)( 34,246)( 35,248)( 36,247)( 37,309)( 38,310)( 39,312)
( 40,311)( 41,305)( 42,306)( 43,308)( 44,307)( 45,301)( 46,302)( 47,304)
( 48,303)( 49,297)( 50,298)( 51,300)( 52,299)( 53,293)( 54,294)( 55,296)
( 56,295)( 57,289)( 58,290)( 59,292)( 60,291)( 61,321)( 62,322)( 63,324)
( 64,323)( 65,317)( 66,318)( 67,320)( 68,319)( 69,313)( 70,314)( 71,316)
( 72,315)( 73,273)( 74,274)( 75,276)( 76,275)( 77,269)( 78,270)( 79,272)
( 80,271)( 81,265)( 82,266)( 83,268)( 84,267)( 85,261)( 86,262)( 87,264)
( 88,263)( 89,257)( 90,258)( 91,260)( 92,259)( 93,253)( 94,254)( 95,256)
( 96,255)( 97,285)( 98,286)( 99,288)(100,287)(101,281)(102,282)(103,284)
(104,283)(105,277)(106,278)(107,280)(108,279)(109,337)(110,338)(111,340)
(112,339)(113,345)(114,346)(115,348)(116,347)(117,341)(118,342)(119,344)
(120,343)(121,325)(122,326)(123,328)(124,327)(125,333)(126,334)(127,336)
(128,335)(129,329)(130,330)(131,332)(132,331)(133,349)(134,350)(135,352)
(136,351)(137,357)(138,358)(139,360)(140,359)(141,353)(142,354)(143,356)
(144,355)(145,417)(146,418)(147,420)(148,419)(149,413)(150,414)(151,416)
(152,415)(153,409)(154,410)(155,412)(156,411)(157,405)(158,406)(159,408)
(160,407)(161,401)(162,402)(163,404)(164,403)(165,397)(166,398)(167,400)
(168,399)(169,429)(170,430)(171,432)(172,431)(173,425)(174,426)(175,428)
(176,427)(177,421)(178,422)(179,424)(180,423)(181,381)(182,382)(183,384)
(184,383)(185,377)(186,378)(187,380)(188,379)(189,373)(190,374)(191,376)
(192,375)(193,369)(194,370)(195,372)(196,371)(197,365)(198,366)(199,368)
(200,367)(201,361)(202,362)(203,364)(204,363)(205,393)(206,394)(207,396)
(208,395)(209,389)(210,390)(211,392)(212,391)(213,385)(214,386)(215,388)
(216,387);
s2 := Sym(432)!( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 45)( 6, 48)( 7, 47)
( 8, 46)( 9, 41)( 10, 44)( 11, 43)( 12, 42)( 13, 61)( 14, 64)( 15, 63)
( 16, 62)( 17, 69)( 18, 72)( 19, 71)( 20, 70)( 21, 65)( 22, 68)( 23, 67)
( 24, 66)( 25, 49)( 26, 52)( 27, 51)( 28, 50)( 29, 57)( 30, 60)( 31, 59)
( 32, 58)( 33, 53)( 34, 56)( 35, 55)( 36, 54)( 73, 81)( 74, 84)( 75, 83)
( 76, 82)( 78, 80)( 85,105)( 86,108)( 87,107)( 88,106)( 89,101)( 90,104)
( 91,103)( 92,102)( 93, 97)( 94,100)( 95, 99)( 96, 98)(109,145)(110,148)
(111,147)(112,146)(113,153)(114,156)(115,155)(116,154)(117,149)(118,152)
(119,151)(120,150)(121,169)(122,172)(123,171)(124,170)(125,177)(126,180)
(127,179)(128,178)(129,173)(130,176)(131,175)(132,174)(133,157)(134,160)
(135,159)(136,158)(137,165)(138,168)(139,167)(140,166)(141,161)(142,164)
(143,163)(144,162)(181,189)(182,192)(183,191)(184,190)(186,188)(193,213)
(194,216)(195,215)(196,214)(197,209)(198,212)(199,211)(200,210)(201,205)
(202,208)(203,207)(204,206)(217,361)(218,364)(219,363)(220,362)(221,369)
(222,372)(223,371)(224,370)(225,365)(226,368)(227,367)(228,366)(229,385)
(230,388)(231,387)(232,386)(233,393)(234,396)(235,395)(236,394)(237,389)
(238,392)(239,391)(240,390)(241,373)(242,376)(243,375)(244,374)(245,381)
(246,384)(247,383)(248,382)(249,377)(250,380)(251,379)(252,378)(253,325)
(254,328)(255,327)(256,326)(257,333)(258,336)(259,335)(260,334)(261,329)
(262,332)(263,331)(264,330)(265,349)(266,352)(267,351)(268,350)(269,357)
(270,360)(271,359)(272,358)(273,353)(274,356)(275,355)(276,354)(277,337)
(278,340)(279,339)(280,338)(281,345)(282,348)(283,347)(284,346)(285,341)
(286,344)(287,343)(288,342)(289,405)(290,408)(291,407)(292,406)(293,401)
(294,404)(295,403)(296,402)(297,397)(298,400)(299,399)(300,398)(301,429)
(302,432)(303,431)(304,430)(305,425)(306,428)(307,427)(308,426)(309,421)
(310,424)(311,423)(312,422)(313,417)(314,420)(315,419)(316,418)(317,413)
(318,416)(319,415)(320,414)(321,409)(322,412)(323,411)(324,410);
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*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*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,
s2*s0*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*s2*s1*s2*s1 >;
References : None.
to this polytope