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# Polytope of Type {12,36}

Atlas Canonical Name : {12,36}*1728h
if this polytope has a name.
Group : SmallGroup(1728,30313)
Rank : 3
Schlafli Type : {12,36}
Number of vertices, edges, etc : 24, 432, 72
Order of s0s1s2 : 18
Order of s0s1s2s1 : 4
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}*864a
3-fold quotients : {4,36}*576c, {12,12}*576k
4-fold quotients : {12,18}*432c
6-fold quotients : {4,18}*288, {12,6}*288a
8-fold quotients : {6,18}*216a
9-fold quotients : {4,12}*192c
12-fold quotients : {4,9}*144, {4,18}*144b, {4,18}*144c, {12,6}*144d
18-fold quotients : {4,6}*96
24-fold quotients : {4,9}*72, {2,18}*72, {6,6}*72a
36-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
48-fold quotients : {2,9}*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,  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, 49)( 14, 52)( 15, 51)( 16, 50)
( 17, 57)( 18, 60)( 19, 59)( 20, 58)( 21, 53)( 22, 56)( 23, 55)( 24, 54)
( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 29, 69)( 30, 72)( 31, 71)( 32, 70)
( 33, 65)( 34, 68)( 35, 67)( 36, 66)( 73, 81)( 74, 84)( 75, 83)( 76, 82)
( 78, 80)( 85, 93)( 86, 96)( 87, 95)( 88, 94)( 90, 92)( 97,105)( 98,108)
( 99,107)(100,106)(102,104)(109,145)(110,148)(111,147)(112,146)(113,153)
(114,156)(115,155)(116,154)(117,149)(118,152)(119,151)(120,150)(121,157)
(122,160)(123,159)(124,158)(125,165)(126,168)(127,167)(128,166)(129,161)
(130,164)(131,163)(132,162)(133,169)(134,172)(135,171)(136,170)(137,177)
(138,180)(139,179)(140,178)(141,173)(142,176)(143,175)(144,174)(181,189)
(182,192)(183,191)(184,190)(186,188)(193,201)(194,204)(195,203)(196,202)
(198,200)(205,213)(206,216)(207,215)(208,214)(210,212)(217,361)(218,364)
(219,363)(220,362)(221,369)(222,372)(223,371)(224,370)(225,365)(226,368)
(227,367)(228,366)(229,373)(230,376)(231,375)(232,374)(233,381)(234,384)
(235,383)(236,382)(237,377)(238,380)(239,379)(240,378)(241,385)(242,388)
(243,387)(244,386)(245,393)(246,396)(247,395)(248,394)(249,389)(250,392)
(251,391)(252,390)(253,325)(254,328)(255,327)(256,326)(257,333)(258,336)
(259,335)(260,334)(261,329)(262,332)(263,331)(264,330)(265,337)(266,340)
(267,339)(268,338)(269,345)(270,348)(271,347)(272,346)(273,341)(274,344)
(275,343)(276,342)(277,349)(278,352)(279,351)(280,350)(281,357)(282,360)
(283,359)(284,358)(285,353)(286,356)(287,355)(288,354)(289,405)(290,408)
(291,407)(292,406)(293,401)(294,404)(295,403)(296,402)(297,397)(298,400)
(299,399)(300,398)(301,417)(302,420)(303,419)(304,418)(305,413)(306,416)
(307,415)(308,414)(309,409)(310,412)(311,411)(312,410)(313,429)(314,432)
(315,431)(316,430)(317,425)(318,428)(319,427)(320,426)(321,421)(322,424)
(323,423)(324,422);;
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,
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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*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, 49)( 14, 52)( 15, 51)
( 16, 50)( 17, 57)( 18, 60)( 19, 59)( 20, 58)( 21, 53)( 22, 56)( 23, 55)
( 24, 54)( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 29, 69)( 30, 72)( 31, 71)
( 32, 70)( 33, 65)( 34, 68)( 35, 67)( 36, 66)( 73, 81)( 74, 84)( 75, 83)
( 76, 82)( 78, 80)( 85, 93)( 86, 96)( 87, 95)( 88, 94)( 90, 92)( 97,105)
( 98,108)( 99,107)(100,106)(102,104)(109,145)(110,148)(111,147)(112,146)
(113,153)(114,156)(115,155)(116,154)(117,149)(118,152)(119,151)(120,150)
(121,157)(122,160)(123,159)(124,158)(125,165)(126,168)(127,167)(128,166)
(129,161)(130,164)(131,163)(132,162)(133,169)(134,172)(135,171)(136,170)
(137,177)(138,180)(139,179)(140,178)(141,173)(142,176)(143,175)(144,174)
(181,189)(182,192)(183,191)(184,190)(186,188)(193,201)(194,204)(195,203)
(196,202)(198,200)(205,213)(206,216)(207,215)(208,214)(210,212)(217,361)
(218,364)(219,363)(220,362)(221,369)(222,372)(223,371)(224,370)(225,365)
(226,368)(227,367)(228,366)(229,373)(230,376)(231,375)(232,374)(233,381)
(234,384)(235,383)(236,382)(237,377)(238,380)(239,379)(240,378)(241,385)
(242,388)(243,387)(244,386)(245,393)(246,396)(247,395)(248,394)(249,389)
(250,392)(251,391)(252,390)(253,325)(254,328)(255,327)(256,326)(257,333)
(258,336)(259,335)(260,334)(261,329)(262,332)(263,331)(264,330)(265,337)
(266,340)(267,339)(268,338)(269,345)(270,348)(271,347)(272,346)(273,341)
(274,344)(275,343)(276,342)(277,349)(278,352)(279,351)(280,350)(281,357)
(282,360)(283,359)(284,358)(285,353)(286,356)(287,355)(288,354)(289,405)
(290,408)(291,407)(292,406)(293,401)(294,404)(295,403)(296,402)(297,397)
(298,400)(299,399)(300,398)(301,417)(302,420)(303,419)(304,418)(305,413)
(306,416)(307,415)(308,414)(309,409)(310,412)(311,411)(312,410)(313,429)
(314,432)(315,431)(316,430)(317,425)(318,428)(319,427)(320,426)(321,421)
(322,424)(323,423)(324,422);
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,
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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

```
References : None.
to this polytope