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# Polytope of Type {36,12}

Atlas Canonical Name : {36,12}*1728d
if this polytope has a name.
Group : SmallGroup(1728,30216)
Rank : 3
Schlafli Type : {36,12}
Number of vertices, edges, etc : 72, 432, 24
Order of s0s1s2 : 18
Order of s0s1s2s1 : 4
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 : {36,6}*864
3-fold quotients : {12,12}*576k
4-fold quotients : {36,6}*432c
6-fold quotients : {12,6}*288a
8-fold quotients : {18,6}*216a
9-fold quotients : {4,12}*192c
12-fold quotients : {12,6}*144d
18-fold quotients : {4,6}*96
24-fold quotients : {18,2}*72, {6,6}*72a
36-fold quotients : {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
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)(217,327)(218,328)(219,325)(220,326)
(221,335)(222,336)(223,333)(224,334)(225,331)(226,332)(227,329)(228,330)
(229,359)(230,360)(231,357)(232,358)(233,355)(234,356)(235,353)(236,354)
(237,351)(238,352)(239,349)(240,350)(241,347)(242,348)(243,345)(244,346)
(245,343)(246,344)(247,341)(248,342)(249,339)(250,340)(251,337)(252,338)
(253,363)(254,364)(255,361)(256,362)(257,371)(258,372)(259,369)(260,370)
(261,367)(262,368)(263,365)(264,366)(265,395)(266,396)(267,393)(268,394)
(269,391)(270,392)(271,389)(272,390)(273,387)(274,388)(275,385)(276,386)
(277,383)(278,384)(279,381)(280,382)(281,379)(282,380)(283,377)(284,378)
(285,375)(286,376)(287,373)(288,374)(289,399)(290,400)(291,397)(292,398)
(293,407)(294,408)(295,405)(296,406)(297,403)(298,404)(299,401)(300,402)
(301,431)(302,432)(303,429)(304,430)(305,427)(306,428)(307,425)(308,426)
(309,423)(310,424)(311,421)(312,422)(313,419)(314,420)(315,417)(316,418)
(317,415)(318,416)(319,413)(320,414)(321,411)(322,412)(323,409)(324,410);;
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)
( 25,249)( 26,250)( 27,252)( 28,251)( 29,245)( 30,246)( 31,248)( 32,247)
( 33,241)( 34,242)( 35,244)( 36,243)( 37,301)( 38,302)( 39,304)( 40,303)
( 41,309)( 42,310)( 43,312)( 44,311)( 45,305)( 46,306)( 47,308)( 48,307)
( 49,289)( 50,290)( 51,292)( 52,291)( 53,297)( 54,298)( 55,300)( 56,299)
( 57,293)( 58,294)( 59,296)( 60,295)( 61,321)( 62,322)( 63,324)( 64,323)
( 65,317)( 66,318)( 67,320)( 68,319)( 69,313)( 70,314)( 71,316)( 72,315)
( 73,265)( 74,266)( 75,268)( 76,267)( 77,273)( 78,274)( 79,276)( 80,275)
( 81,269)( 82,270)( 83,272)( 84,271)( 85,253)( 86,254)( 87,256)( 88,255)
( 89,261)( 90,262)( 91,264)( 92,263)( 93,257)( 94,258)( 95,260)( 96,259)
( 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)(133,357)(134,358)(135,360)(136,359)
(137,353)(138,354)(139,356)(140,355)(141,349)(142,350)(143,352)(144,351)
(145,409)(146,410)(147,412)(148,411)(149,417)(150,418)(151,420)(152,419)
(153,413)(154,414)(155,416)(156,415)(157,397)(158,398)(159,400)(160,399)
(161,405)(162,406)(163,408)(164,407)(165,401)(166,402)(167,404)(168,403)
(169,429)(170,430)(171,432)(172,431)(173,425)(174,426)(175,428)(176,427)
(177,421)(178,422)(179,424)(180,423)(181,373)(182,374)(183,376)(184,375)
(185,381)(186,382)(187,384)(188,383)(189,377)(190,378)(191,380)(192,379)
(193,361)(194,362)(195,364)(196,363)(197,369)(198,370)(199,372)(200,371)
(201,365)(202,366)(203,368)(204,367)(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, 41)(  6, 44)(  7, 43)(  8, 42)
(  9, 45)( 10, 48)( 11, 47)( 12, 46)( 13, 49)( 14, 52)( 15, 51)( 16, 50)
( 17, 53)( 18, 56)( 19, 55)( 20, 54)( 21, 57)( 22, 60)( 23, 59)( 24, 58)
( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 29, 65)( 30, 68)( 31, 67)( 32, 66)
( 33, 69)( 34, 72)( 35, 71)( 36, 70)( 74, 76)( 78, 80)( 82, 84)( 86, 88)
( 90, 92)( 94, 96)( 98,100)(102,104)(106,108)(109,145)(110,148)(111,147)
(112,146)(113,149)(114,152)(115,151)(116,150)(117,153)(118,156)(119,155)
(120,154)(121,157)(122,160)(123,159)(124,158)(125,161)(126,164)(127,163)
(128,162)(129,165)(130,168)(131,167)(132,166)(133,169)(134,172)(135,171)
(136,170)(137,173)(138,176)(139,175)(140,174)(141,177)(142,180)(143,179)
(144,178)(182,184)(186,188)(190,192)(194,196)(198,200)(202,204)(206,208)
(210,212)(214,216)(217,361)(218,364)(219,363)(220,362)(221,365)(222,368)
(223,367)(224,366)(225,369)(226,372)(227,371)(228,370)(229,373)(230,376)
(231,375)(232,374)(233,377)(234,380)(235,379)(236,378)(237,381)(238,384)
(239,383)(240,382)(241,385)(242,388)(243,387)(244,386)(245,389)(246,392)
(247,391)(248,390)(249,393)(250,396)(251,395)(252,394)(253,325)(254,328)
(255,327)(256,326)(257,329)(258,332)(259,331)(260,330)(261,333)(262,336)
(263,335)(264,334)(265,337)(266,340)(267,339)(268,338)(269,341)(270,344)
(271,343)(272,342)(273,345)(274,348)(275,347)(276,346)(277,349)(278,352)
(279,351)(280,350)(281,353)(282,356)(283,355)(284,354)(285,357)(286,360)
(287,359)(288,358)(289,397)(290,400)(291,399)(292,398)(293,401)(294,404)
(295,403)(296,402)(297,405)(298,408)(299,407)(300,406)(301,409)(302,412)
(303,411)(304,410)(305,413)(306,416)(307,415)(308,414)(309,417)(310,420)
(311,419)(312,418)(313,421)(314,424)(315,423)(316,422)(317,425)(318,428)
(319,427)(320,426)(321,429)(322,432)(323,431)(324,430);;
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,
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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s2*s1*s0 ];;
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)(217,327)(218,328)(219,325)
(220,326)(221,335)(222,336)(223,333)(224,334)(225,331)(226,332)(227,329)
(228,330)(229,359)(230,360)(231,357)(232,358)(233,355)(234,356)(235,353)
(236,354)(237,351)(238,352)(239,349)(240,350)(241,347)(242,348)(243,345)
(244,346)(245,343)(246,344)(247,341)(248,342)(249,339)(250,340)(251,337)
(252,338)(253,363)(254,364)(255,361)(256,362)(257,371)(258,372)(259,369)
(260,370)(261,367)(262,368)(263,365)(264,366)(265,395)(266,396)(267,393)
(268,394)(269,391)(270,392)(271,389)(272,390)(273,387)(274,388)(275,385)
(276,386)(277,383)(278,384)(279,381)(280,382)(281,379)(282,380)(283,377)
(284,378)(285,375)(286,376)(287,373)(288,374)(289,399)(290,400)(291,397)
(292,398)(293,407)(294,408)(295,405)(296,406)(297,403)(298,404)(299,401)
(300,402)(301,431)(302,432)(303,429)(304,430)(305,427)(306,428)(307,425)
(308,426)(309,423)(310,424)(311,421)(312,422)(313,419)(314,420)(315,417)
(316,418)(317,415)(318,416)(319,413)(320,414)(321,411)(322,412)(323,409)
(324,410);
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)( 25,249)( 26,250)( 27,252)( 28,251)( 29,245)( 30,246)( 31,248)
( 32,247)( 33,241)( 34,242)( 35,244)( 36,243)( 37,301)( 38,302)( 39,304)
( 40,303)( 41,309)( 42,310)( 43,312)( 44,311)( 45,305)( 46,306)( 47,308)
( 48,307)( 49,289)( 50,290)( 51,292)( 52,291)( 53,297)( 54,298)( 55,300)
( 56,299)( 57,293)( 58,294)( 59,296)( 60,295)( 61,321)( 62,322)( 63,324)
( 64,323)( 65,317)( 66,318)( 67,320)( 68,319)( 69,313)( 70,314)( 71,316)
( 72,315)( 73,265)( 74,266)( 75,268)( 76,267)( 77,273)( 78,274)( 79,276)
( 80,275)( 81,269)( 82,270)( 83,272)( 84,271)( 85,253)( 86,254)( 87,256)
( 88,255)( 89,261)( 90,262)( 91,264)( 92,263)( 93,257)( 94,258)( 95,260)
( 96,259)( 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)(133,357)(134,358)(135,360)
(136,359)(137,353)(138,354)(139,356)(140,355)(141,349)(142,350)(143,352)
(144,351)(145,409)(146,410)(147,412)(148,411)(149,417)(150,418)(151,420)
(152,419)(153,413)(154,414)(155,416)(156,415)(157,397)(158,398)(159,400)
(160,399)(161,405)(162,406)(163,408)(164,407)(165,401)(166,402)(167,404)
(168,403)(169,429)(170,430)(171,432)(172,431)(173,425)(174,426)(175,428)
(176,427)(177,421)(178,422)(179,424)(180,423)(181,373)(182,374)(183,376)
(184,375)(185,381)(186,382)(187,384)(188,383)(189,377)(190,378)(191,380)
(192,379)(193,361)(194,362)(195,364)(196,363)(197,369)(198,370)(199,372)
(200,371)(201,365)(202,366)(203,368)(204,367)(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, 41)(  6, 44)(  7, 43)
(  8, 42)(  9, 45)( 10, 48)( 11, 47)( 12, 46)( 13, 49)( 14, 52)( 15, 51)
( 16, 50)( 17, 53)( 18, 56)( 19, 55)( 20, 54)( 21, 57)( 22, 60)( 23, 59)
( 24, 58)( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 29, 65)( 30, 68)( 31, 67)
( 32, 66)( 33, 69)( 34, 72)( 35, 71)( 36, 70)( 74, 76)( 78, 80)( 82, 84)
( 86, 88)( 90, 92)( 94, 96)( 98,100)(102,104)(106,108)(109,145)(110,148)
(111,147)(112,146)(113,149)(114,152)(115,151)(116,150)(117,153)(118,156)
(119,155)(120,154)(121,157)(122,160)(123,159)(124,158)(125,161)(126,164)
(127,163)(128,162)(129,165)(130,168)(131,167)(132,166)(133,169)(134,172)
(135,171)(136,170)(137,173)(138,176)(139,175)(140,174)(141,177)(142,180)
(143,179)(144,178)(182,184)(186,188)(190,192)(194,196)(198,200)(202,204)
(206,208)(210,212)(214,216)(217,361)(218,364)(219,363)(220,362)(221,365)
(222,368)(223,367)(224,366)(225,369)(226,372)(227,371)(228,370)(229,373)
(230,376)(231,375)(232,374)(233,377)(234,380)(235,379)(236,378)(237,381)
(238,384)(239,383)(240,382)(241,385)(242,388)(243,387)(244,386)(245,389)
(246,392)(247,391)(248,390)(249,393)(250,396)(251,395)(252,394)(253,325)
(254,328)(255,327)(256,326)(257,329)(258,332)(259,331)(260,330)(261,333)
(262,336)(263,335)(264,334)(265,337)(266,340)(267,339)(268,338)(269,341)
(270,344)(271,343)(272,342)(273,345)(274,348)(275,347)(276,346)(277,349)
(278,352)(279,351)(280,350)(281,353)(282,356)(283,355)(284,354)(285,357)
(286,360)(287,359)(288,358)(289,397)(290,400)(291,399)(292,398)(293,401)
(294,404)(295,403)(296,402)(297,405)(298,408)(299,407)(300,406)(301,409)
(302,412)(303,411)(304,410)(305,413)(306,416)(307,415)(308,414)(309,417)
(310,420)(311,419)(312,418)(313,421)(314,424)(315,423)(316,422)(317,425)
(318,428)(319,427)(320,426)(321,429)(322,432)(323,431)(324,430);
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,
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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s2*s1*s0 >;

```
References : None.
to this polytope