Questions?
See the FAQ
or other info.

# Polytope of Type {12,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,36}*1728i
if this polytope has a name.
Group : SmallGroup(1728,46098)
Rank : 3
Schlafli Type : {12,36}
Number of vertices, edges, etc : 24, 432, 72
Order of s0s1s2 : 9
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-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) :
3-fold quotients : {12,12}*576l
4-fold quotients : {6,36}*432c, {12,18}*432c
12-fold quotients : {4,18}*144c, {6,12}*144d, {12,6}*144d
24-fold quotients : {4,9}*72
36-fold quotients : {4,6}*48b, {6,4}*48b
72-fold quotients : {3,4}*24, {4,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 15)( 10, 16)( 11, 13)( 12, 14)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 31)( 26, 32)( 27, 29)( 28, 30)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 47)( 42, 48)( 43, 45)( 44, 46)
( 49, 99)( 50,100)( 51, 97)( 52, 98)( 53,103)( 54,104)( 55,101)( 56,102)
( 57,111)( 58,112)( 59,109)( 60,110)( 61,107)( 62,108)( 63,105)( 64,106)
( 65,115)( 66,116)( 67,113)( 68,114)( 69,119)( 70,120)( 71,117)( 72,118)
( 73,127)( 74,128)( 75,125)( 76,126)( 77,123)( 78,124)( 79,121)( 80,122)
( 81,131)( 82,132)( 83,129)( 84,130)( 85,135)( 86,136)( 87,133)( 88,134)
( 89,143)( 90,144)( 91,141)( 92,142)( 93,139)( 94,140)( 95,137)( 96,138)
(145,147)(146,148)(149,151)(150,152)(153,159)(154,160)(155,157)(156,158)
(161,163)(162,164)(165,167)(166,168)(169,175)(170,176)(171,173)(172,174)
(177,179)(178,180)(181,183)(182,184)(185,191)(186,192)(187,189)(188,190)
(193,243)(194,244)(195,241)(196,242)(197,247)(198,248)(199,245)(200,246)
(201,255)(202,256)(203,253)(204,254)(205,251)(206,252)(207,249)(208,250)
(209,259)(210,260)(211,257)(212,258)(213,263)(214,264)(215,261)(216,262)
(217,271)(218,272)(219,269)(220,270)(221,267)(222,268)(223,265)(224,266)
(225,275)(226,276)(227,273)(228,274)(229,279)(230,280)(231,277)(232,278)
(233,287)(234,288)(235,285)(236,286)(237,283)(238,284)(239,281)(240,282)
(289,291)(290,292)(293,295)(294,296)(297,303)(298,304)(299,301)(300,302)
(305,307)(306,308)(309,311)(310,312)(313,319)(314,320)(315,317)(316,318)
(321,323)(322,324)(325,327)(326,328)(329,335)(330,336)(331,333)(332,334)
(337,387)(338,388)(339,385)(340,386)(341,391)(342,392)(343,389)(344,390)
(345,399)(346,400)(347,397)(348,398)(349,395)(350,396)(351,393)(352,394)
(353,403)(354,404)(355,401)(356,402)(357,407)(358,408)(359,405)(360,406)
(361,415)(362,416)(363,413)(364,414)(365,411)(366,412)(367,409)(368,410)
(369,419)(370,420)(371,417)(372,418)(373,423)(374,424)(375,421)(376,422)
(377,431)(378,432)(379,429)(380,430)(381,427)(382,428)(383,425)(384,426);;
s1 := (  1, 49)(  2, 50)(  3, 52)(  4, 51)(  5, 61)(  6, 62)(  7, 64)(  8, 63)
(  9, 57)( 10, 58)( 11, 60)( 12, 59)( 13, 53)( 14, 54)( 15, 56)( 16, 55)
( 17, 81)( 18, 82)( 19, 84)( 20, 83)( 21, 93)( 22, 94)( 23, 96)( 24, 95)
( 25, 89)( 26, 90)( 27, 92)( 28, 91)( 29, 85)( 30, 86)( 31, 88)( 32, 87)
( 33, 65)( 34, 66)( 35, 68)( 36, 67)( 37, 77)( 38, 78)( 39, 80)( 40, 79)
( 41, 73)( 42, 74)( 43, 76)( 44, 75)( 45, 69)( 46, 70)( 47, 72)( 48, 71)
( 99,100)(101,109)(102,110)(103,112)(104,111)(107,108)(113,129)(114,130)
(115,132)(116,131)(117,141)(118,142)(119,144)(120,143)(121,137)(122,138)
(123,140)(124,139)(125,133)(126,134)(127,136)(128,135)(145,369)(146,370)
(147,372)(148,371)(149,381)(150,382)(151,384)(152,383)(153,377)(154,378)
(155,380)(156,379)(157,373)(158,374)(159,376)(160,375)(161,353)(162,354)
(163,356)(164,355)(165,365)(166,366)(167,368)(168,367)(169,361)(170,362)
(171,364)(172,363)(173,357)(174,358)(175,360)(176,359)(177,337)(178,338)
(179,340)(180,339)(181,349)(182,350)(183,352)(184,351)(185,345)(186,346)
(187,348)(188,347)(189,341)(190,342)(191,344)(192,343)(193,321)(194,322)
(195,324)(196,323)(197,333)(198,334)(199,336)(200,335)(201,329)(202,330)
(203,332)(204,331)(205,325)(206,326)(207,328)(208,327)(209,305)(210,306)
(211,308)(212,307)(213,317)(214,318)(215,320)(216,319)(217,313)(218,314)
(219,316)(220,315)(221,309)(222,310)(223,312)(224,311)(225,289)(226,290)
(227,292)(228,291)(229,301)(230,302)(231,304)(232,303)(233,297)(234,298)
(235,300)(236,299)(237,293)(238,294)(239,296)(240,295)(241,417)(242,418)
(243,420)(244,419)(245,429)(246,430)(247,432)(248,431)(249,425)(250,426)
(251,428)(252,427)(253,421)(254,422)(255,424)(256,423)(257,401)(258,402)
(259,404)(260,403)(261,413)(262,414)(263,416)(264,415)(265,409)(266,410)
(267,412)(268,411)(269,405)(270,406)(271,408)(272,407)(273,385)(274,386)
(275,388)(276,387)(277,397)(278,398)(279,400)(280,399)(281,393)(282,394)
(283,396)(284,395)(285,389)(286,390)(287,392)(288,391);;
s2 := (  1,149)(  2,152)(  3,151)(  4,150)(  5,145)(  6,148)(  7,147)(  8,146)
(  9,157)( 10,160)( 11,159)( 12,158)( 13,153)( 14,156)( 15,155)( 16,154)
( 17,181)( 18,184)( 19,183)( 20,182)( 21,177)( 22,180)( 23,179)( 24,178)
( 25,189)( 26,192)( 27,191)( 28,190)( 29,185)( 30,188)( 31,187)( 32,186)
( 33,165)( 34,168)( 35,167)( 36,166)( 37,161)( 38,164)( 39,163)( 40,162)
( 41,173)( 42,176)( 43,175)( 44,174)( 45,169)( 46,172)( 47,171)( 48,170)
( 49,197)( 50,200)( 51,199)( 52,198)( 53,193)( 54,196)( 55,195)( 56,194)
( 57,205)( 58,208)( 59,207)( 60,206)( 61,201)( 62,204)( 63,203)( 64,202)
( 65,229)( 66,232)( 67,231)( 68,230)( 69,225)( 70,228)( 71,227)( 72,226)
( 73,237)( 74,240)( 75,239)( 76,238)( 77,233)( 78,236)( 79,235)( 80,234)
( 81,213)( 82,216)( 83,215)( 84,214)( 85,209)( 86,212)( 87,211)( 88,210)
( 89,221)( 90,224)( 91,223)( 92,222)( 93,217)( 94,220)( 95,219)( 96,218)
( 97,245)( 98,248)( 99,247)(100,246)(101,241)(102,244)(103,243)(104,242)
(105,253)(106,256)(107,255)(108,254)(109,249)(110,252)(111,251)(112,250)
(113,277)(114,280)(115,279)(116,278)(117,273)(118,276)(119,275)(120,274)
(121,285)(122,288)(123,287)(124,286)(125,281)(126,284)(127,283)(128,282)
(129,261)(130,264)(131,263)(132,262)(133,257)(134,260)(135,259)(136,258)
(137,269)(138,272)(139,271)(140,270)(141,265)(142,268)(143,267)(144,266)
(289,325)(290,328)(291,327)(292,326)(293,321)(294,324)(295,323)(296,322)
(297,333)(298,336)(299,335)(300,334)(301,329)(302,332)(303,331)(304,330)
(305,309)(306,312)(307,311)(308,310)(313,317)(314,320)(315,319)(316,318)
(337,373)(338,376)(339,375)(340,374)(341,369)(342,372)(343,371)(344,370)
(345,381)(346,384)(347,383)(348,382)(349,377)(350,380)(351,379)(352,378)
(353,357)(354,360)(355,359)(356,358)(361,365)(362,368)(363,367)(364,366)
(385,421)(386,424)(387,423)(388,422)(389,417)(390,420)(391,419)(392,418)
(393,429)(394,432)(395,431)(396,430)(397,425)(398,428)(399,427)(400,426)
(401,405)(402,408)(403,407)(404,406)(409,413)(410,416)(411,415)(412,414);;
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*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*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 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(432)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 15)( 10, 16)( 11, 13)
( 12, 14)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 31)( 26, 32)( 27, 29)
( 28, 30)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 47)( 42, 48)( 43, 45)
( 44, 46)( 49, 99)( 50,100)( 51, 97)( 52, 98)( 53,103)( 54,104)( 55,101)
( 56,102)( 57,111)( 58,112)( 59,109)( 60,110)( 61,107)( 62,108)( 63,105)
( 64,106)( 65,115)( 66,116)( 67,113)( 68,114)( 69,119)( 70,120)( 71,117)
( 72,118)( 73,127)( 74,128)( 75,125)( 76,126)( 77,123)( 78,124)( 79,121)
( 80,122)( 81,131)( 82,132)( 83,129)( 84,130)( 85,135)( 86,136)( 87,133)
( 88,134)( 89,143)( 90,144)( 91,141)( 92,142)( 93,139)( 94,140)( 95,137)
( 96,138)(145,147)(146,148)(149,151)(150,152)(153,159)(154,160)(155,157)
(156,158)(161,163)(162,164)(165,167)(166,168)(169,175)(170,176)(171,173)
(172,174)(177,179)(178,180)(181,183)(182,184)(185,191)(186,192)(187,189)
(188,190)(193,243)(194,244)(195,241)(196,242)(197,247)(198,248)(199,245)
(200,246)(201,255)(202,256)(203,253)(204,254)(205,251)(206,252)(207,249)
(208,250)(209,259)(210,260)(211,257)(212,258)(213,263)(214,264)(215,261)
(216,262)(217,271)(218,272)(219,269)(220,270)(221,267)(222,268)(223,265)
(224,266)(225,275)(226,276)(227,273)(228,274)(229,279)(230,280)(231,277)
(232,278)(233,287)(234,288)(235,285)(236,286)(237,283)(238,284)(239,281)
(240,282)(289,291)(290,292)(293,295)(294,296)(297,303)(298,304)(299,301)
(300,302)(305,307)(306,308)(309,311)(310,312)(313,319)(314,320)(315,317)
(316,318)(321,323)(322,324)(325,327)(326,328)(329,335)(330,336)(331,333)
(332,334)(337,387)(338,388)(339,385)(340,386)(341,391)(342,392)(343,389)
(344,390)(345,399)(346,400)(347,397)(348,398)(349,395)(350,396)(351,393)
(352,394)(353,403)(354,404)(355,401)(356,402)(357,407)(358,408)(359,405)
(360,406)(361,415)(362,416)(363,413)(364,414)(365,411)(366,412)(367,409)
(368,410)(369,419)(370,420)(371,417)(372,418)(373,423)(374,424)(375,421)
(376,422)(377,431)(378,432)(379,429)(380,430)(381,427)(382,428)(383,425)
(384,426);
s1 := Sym(432)!(  1, 49)(  2, 50)(  3, 52)(  4, 51)(  5, 61)(  6, 62)(  7, 64)
(  8, 63)(  9, 57)( 10, 58)( 11, 60)( 12, 59)( 13, 53)( 14, 54)( 15, 56)
( 16, 55)( 17, 81)( 18, 82)( 19, 84)( 20, 83)( 21, 93)( 22, 94)( 23, 96)
( 24, 95)( 25, 89)( 26, 90)( 27, 92)( 28, 91)( 29, 85)( 30, 86)( 31, 88)
( 32, 87)( 33, 65)( 34, 66)( 35, 68)( 36, 67)( 37, 77)( 38, 78)( 39, 80)
( 40, 79)( 41, 73)( 42, 74)( 43, 76)( 44, 75)( 45, 69)( 46, 70)( 47, 72)
( 48, 71)( 99,100)(101,109)(102,110)(103,112)(104,111)(107,108)(113,129)
(114,130)(115,132)(116,131)(117,141)(118,142)(119,144)(120,143)(121,137)
(122,138)(123,140)(124,139)(125,133)(126,134)(127,136)(128,135)(145,369)
(146,370)(147,372)(148,371)(149,381)(150,382)(151,384)(152,383)(153,377)
(154,378)(155,380)(156,379)(157,373)(158,374)(159,376)(160,375)(161,353)
(162,354)(163,356)(164,355)(165,365)(166,366)(167,368)(168,367)(169,361)
(170,362)(171,364)(172,363)(173,357)(174,358)(175,360)(176,359)(177,337)
(178,338)(179,340)(180,339)(181,349)(182,350)(183,352)(184,351)(185,345)
(186,346)(187,348)(188,347)(189,341)(190,342)(191,344)(192,343)(193,321)
(194,322)(195,324)(196,323)(197,333)(198,334)(199,336)(200,335)(201,329)
(202,330)(203,332)(204,331)(205,325)(206,326)(207,328)(208,327)(209,305)
(210,306)(211,308)(212,307)(213,317)(214,318)(215,320)(216,319)(217,313)
(218,314)(219,316)(220,315)(221,309)(222,310)(223,312)(224,311)(225,289)
(226,290)(227,292)(228,291)(229,301)(230,302)(231,304)(232,303)(233,297)
(234,298)(235,300)(236,299)(237,293)(238,294)(239,296)(240,295)(241,417)
(242,418)(243,420)(244,419)(245,429)(246,430)(247,432)(248,431)(249,425)
(250,426)(251,428)(252,427)(253,421)(254,422)(255,424)(256,423)(257,401)
(258,402)(259,404)(260,403)(261,413)(262,414)(263,416)(264,415)(265,409)
(266,410)(267,412)(268,411)(269,405)(270,406)(271,408)(272,407)(273,385)
(274,386)(275,388)(276,387)(277,397)(278,398)(279,400)(280,399)(281,393)
(282,394)(283,396)(284,395)(285,389)(286,390)(287,392)(288,391);
s2 := Sym(432)!(  1,149)(  2,152)(  3,151)(  4,150)(  5,145)(  6,148)(  7,147)
(  8,146)(  9,157)( 10,160)( 11,159)( 12,158)( 13,153)( 14,156)( 15,155)
( 16,154)( 17,181)( 18,184)( 19,183)( 20,182)( 21,177)( 22,180)( 23,179)
( 24,178)( 25,189)( 26,192)( 27,191)( 28,190)( 29,185)( 30,188)( 31,187)
( 32,186)( 33,165)( 34,168)( 35,167)( 36,166)( 37,161)( 38,164)( 39,163)
( 40,162)( 41,173)( 42,176)( 43,175)( 44,174)( 45,169)( 46,172)( 47,171)
( 48,170)( 49,197)( 50,200)( 51,199)( 52,198)( 53,193)( 54,196)( 55,195)
( 56,194)( 57,205)( 58,208)( 59,207)( 60,206)( 61,201)( 62,204)( 63,203)
( 64,202)( 65,229)( 66,232)( 67,231)( 68,230)( 69,225)( 70,228)( 71,227)
( 72,226)( 73,237)( 74,240)( 75,239)( 76,238)( 77,233)( 78,236)( 79,235)
( 80,234)( 81,213)( 82,216)( 83,215)( 84,214)( 85,209)( 86,212)( 87,211)
( 88,210)( 89,221)( 90,224)( 91,223)( 92,222)( 93,217)( 94,220)( 95,219)
( 96,218)( 97,245)( 98,248)( 99,247)(100,246)(101,241)(102,244)(103,243)
(104,242)(105,253)(106,256)(107,255)(108,254)(109,249)(110,252)(111,251)
(112,250)(113,277)(114,280)(115,279)(116,278)(117,273)(118,276)(119,275)
(120,274)(121,285)(122,288)(123,287)(124,286)(125,281)(126,284)(127,283)
(128,282)(129,261)(130,264)(131,263)(132,262)(133,257)(134,260)(135,259)
(136,258)(137,269)(138,272)(139,271)(140,270)(141,265)(142,268)(143,267)
(144,266)(289,325)(290,328)(291,327)(292,326)(293,321)(294,324)(295,323)
(296,322)(297,333)(298,336)(299,335)(300,334)(301,329)(302,332)(303,331)
(304,330)(305,309)(306,312)(307,311)(308,310)(313,317)(314,320)(315,319)
(316,318)(337,373)(338,376)(339,375)(340,374)(341,369)(342,372)(343,371)
(344,370)(345,381)(346,384)(347,383)(348,382)(349,377)(350,380)(351,379)
(352,378)(353,357)(354,360)(355,359)(356,358)(361,365)(362,368)(363,367)
(364,366)(385,421)(386,424)(387,423)(388,422)(389,417)(390,420)(391,419)
(392,418)(393,429)(394,432)(395,431)(396,430)(397,425)(398,428)(399,427)
(400,426)(401,405)(402,408)(403,407)(404,406)(409,413)(410,416)(411,415)
(412,414);
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*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*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 >;

```
References : None.
to this polytope