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# Polytope of Type {36,12}

Atlas Canonical Name : {36,12}*1728h
if this polytope has a name.
Group : SmallGroup(1728,30313)
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 : {18,12}*864a
3-fold quotients : {36,4}*576c, {12,12}*576j
4-fold quotients : {18,12}*432c
6-fold quotients : {18,4}*288, {6,12}*288a
8-fold quotients : {18,6}*216a
9-fold quotients : {12,4}*192c
12-fold quotients : {9,4}*144, {18,4}*144b, {18,4}*144c, {6,12}*144d
18-fold quotients : {6,4}*96
24-fold quotients : {9,4}*72, {18,2}*72, {6,6}*72a
36-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
48-fold quotients : {9,2}*36
72-fold quotients : {3,4}*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 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 15, 16)( 17, 21)( 18, 22)
( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)( 37, 81)
( 38, 82)( 39, 84)( 40, 83)( 41, 77)( 42, 78)( 43, 80)( 44, 79)( 45, 73)
( 46, 74)( 47, 76)( 48, 75)( 49, 93)( 50, 94)( 51, 96)( 52, 95)( 53, 89)
( 54, 90)( 55, 92)( 56, 91)( 57, 85)( 58, 86)( 59, 88)( 60, 87)( 61,105)
( 62,106)( 63,108)( 64,107)( 65,101)( 66,102)( 67,104)( 68,103)( 69, 97)
( 70, 98)( 71,100)( 72, 99)(111,112)(113,117)(114,118)(115,120)(116,119)
(123,124)(125,129)(126,130)(127,132)(128,131)(135,136)(137,141)(138,142)
(139,144)(140,143)(145,189)(146,190)(147,192)(148,191)(149,185)(150,186)
(151,188)(152,187)(153,181)(154,182)(155,184)(156,183)(157,201)(158,202)
(159,204)(160,203)(161,197)(162,198)(163,200)(164,199)(165,193)(166,194)
(167,196)(168,195)(169,213)(170,214)(171,216)(172,215)(173,209)(174,210)
(175,212)(176,211)(177,205)(178,206)(179,208)(180,207)(217,325)(218,326)
(219,328)(220,327)(221,333)(222,334)(223,336)(224,335)(225,329)(226,330)
(227,332)(228,331)(229,337)(230,338)(231,340)(232,339)(233,345)(234,346)
(235,348)(236,347)(237,341)(238,342)(239,344)(240,343)(241,349)(242,350)
(243,352)(244,351)(245,357)(246,358)(247,360)(248,359)(249,353)(250,354)
(251,356)(252,355)(253,405)(254,406)(255,408)(256,407)(257,401)(258,402)
(259,404)(260,403)(261,397)(262,398)(263,400)(264,399)(265,417)(266,418)
(267,420)(268,419)(269,413)(270,414)(271,416)(272,415)(273,409)(274,410)
(275,412)(276,411)(277,429)(278,430)(279,432)(280,431)(281,425)(282,426)
(283,428)(284,427)(285,421)(286,422)(287,424)(288,423)(289,369)(290,370)
(291,372)(292,371)(293,365)(294,366)(295,368)(296,367)(297,361)(298,362)
(299,364)(300,363)(301,381)(302,382)(303,384)(304,383)(305,377)(306,378)
(307,380)(308,379)(309,373)(310,374)(311,376)(312,375)(313,393)(314,394)
(315,396)(316,395)(317,389)(318,390)(319,392)(320,391)(321,385)(322,386)
(323,388)(324,387);;
s1 := (  1,253)(  2,256)(  3,255)(  4,254)(  5,261)(  6,264)(  7,263)(  8,262)
(  9,257)( 10,260)( 11,259)( 12,258)( 13,277)( 14,280)( 15,279)( 16,278)
( 17,285)( 18,288)( 19,287)( 20,286)( 21,281)( 22,284)( 23,283)( 24,282)
( 25,265)( 26,268)( 27,267)( 28,266)( 29,273)( 30,276)( 31,275)( 32,274)
( 33,269)( 34,272)( 35,271)( 36,270)( 37,217)( 38,220)( 39,219)( 40,218)
( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)( 48,222)
( 49,241)( 50,244)( 51,243)( 52,242)( 53,249)( 54,252)( 55,251)( 56,250)
( 57,245)( 58,248)( 59,247)( 60,246)( 61,229)( 62,232)( 63,231)( 64,230)
( 65,237)( 66,240)( 67,239)( 68,238)( 69,233)( 70,236)( 71,235)( 72,234)
( 73,297)( 74,300)( 75,299)( 76,298)( 77,293)( 78,296)( 79,295)( 80,294)
( 81,289)( 82,292)( 83,291)( 84,290)( 85,321)( 86,324)( 87,323)( 88,322)
( 89,317)( 90,320)( 91,319)( 92,318)( 93,313)( 94,316)( 95,315)( 96,314)
( 97,309)( 98,312)( 99,311)(100,310)(101,305)(102,308)(103,307)(104,306)
(105,301)(106,304)(107,303)(108,302)(109,361)(110,364)(111,363)(112,362)
(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)(120,366)
(121,385)(122,388)(123,387)(124,386)(125,393)(126,396)(127,395)(128,394)
(129,389)(130,392)(131,391)(132,390)(133,373)(134,376)(135,375)(136,374)
(137,381)(138,384)(139,383)(140,382)(141,377)(142,380)(143,379)(144,378)
(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)(152,334)
(153,329)(154,332)(155,331)(156,330)(157,349)(158,352)(159,351)(160,350)
(161,357)(162,360)(163,359)(164,358)(165,353)(166,356)(167,355)(168,354)
(169,337)(170,340)(171,339)(172,338)(173,345)(174,348)(175,347)(176,346)
(177,341)(178,344)(179,343)(180,342)(181,405)(182,408)(183,407)(184,406)
(185,401)(186,404)(187,403)(188,402)(189,397)(190,400)(191,399)(192,398)
(193,429)(194,432)(195,431)(196,430)(197,425)(198,428)(199,427)(200,426)
(201,421)(202,424)(203,423)(204,422)(205,417)(206,420)(207,419)(208,418)
(209,413)(210,416)(211,415)(212,414)(213,409)(214,412)(215,411)(216,410);;
s2 := (  1, 14)(  2, 13)(  3, 16)(  4, 15)(  5, 18)(  6, 17)(  7, 20)(  8, 19)
(  9, 22)( 10, 21)( 11, 24)( 12, 23)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 50)( 38, 49)( 39, 52)( 40, 51)( 41, 54)( 42, 53)
( 43, 56)( 44, 55)( 45, 58)( 46, 57)( 47, 60)( 48, 59)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 86)( 74, 85)( 75, 88)( 76, 87)
( 77, 90)( 78, 89)( 79, 92)( 80, 91)( 81, 94)( 82, 93)( 83, 96)( 84, 95)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,122)(110,121)
(111,124)(112,123)(113,126)(114,125)(115,128)(116,127)(117,130)(118,129)
(119,132)(120,131)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)
(145,158)(146,157)(147,160)(148,159)(149,162)(150,161)(151,164)(152,163)
(153,166)(154,165)(155,168)(156,167)(169,170)(171,172)(173,174)(175,176)
(177,178)(179,180)(181,194)(182,193)(183,196)(184,195)(185,198)(186,197)
(187,200)(188,199)(189,202)(190,201)(191,204)(192,203)(205,206)(207,208)
(209,210)(211,212)(213,214)(215,216)(217,338)(218,337)(219,340)(220,339)
(221,342)(222,341)(223,344)(224,343)(225,346)(226,345)(227,348)(228,347)
(229,326)(230,325)(231,328)(232,327)(233,330)(234,329)(235,332)(236,331)
(237,334)(238,333)(239,336)(240,335)(241,350)(242,349)(243,352)(244,351)
(245,354)(246,353)(247,356)(248,355)(249,358)(250,357)(251,360)(252,359)
(253,374)(254,373)(255,376)(256,375)(257,378)(258,377)(259,380)(260,379)
(261,382)(262,381)(263,384)(264,383)(265,362)(266,361)(267,364)(268,363)
(269,366)(270,365)(271,368)(272,367)(273,370)(274,369)(275,372)(276,371)
(277,386)(278,385)(279,388)(280,387)(281,390)(282,389)(283,392)(284,391)
(285,394)(286,393)(287,396)(288,395)(289,410)(290,409)(291,412)(292,411)
(293,414)(294,413)(295,416)(296,415)(297,418)(298,417)(299,420)(300,419)
(301,398)(302,397)(303,400)(304,399)(305,402)(306,401)(307,404)(308,403)
(309,406)(310,405)(311,408)(312,407)(313,422)(314,421)(315,424)(316,423)
(317,426)(318,425)(319,428)(320,427)(321,430)(322,429)(323,432)(324,431);;
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*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,
s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 15, 16)( 17, 21)
( 18, 22)( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)
( 37, 81)( 38, 82)( 39, 84)( 40, 83)( 41, 77)( 42, 78)( 43, 80)( 44, 79)
( 45, 73)( 46, 74)( 47, 76)( 48, 75)( 49, 93)( 50, 94)( 51, 96)( 52, 95)
( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 85)( 58, 86)( 59, 88)( 60, 87)
( 61,105)( 62,106)( 63,108)( 64,107)( 65,101)( 66,102)( 67,104)( 68,103)
( 69, 97)( 70, 98)( 71,100)( 72, 99)(111,112)(113,117)(114,118)(115,120)
(116,119)(123,124)(125,129)(126,130)(127,132)(128,131)(135,136)(137,141)
(138,142)(139,144)(140,143)(145,189)(146,190)(147,192)(148,191)(149,185)
(150,186)(151,188)(152,187)(153,181)(154,182)(155,184)(156,183)(157,201)
(158,202)(159,204)(160,203)(161,197)(162,198)(163,200)(164,199)(165,193)
(166,194)(167,196)(168,195)(169,213)(170,214)(171,216)(172,215)(173,209)
(174,210)(175,212)(176,211)(177,205)(178,206)(179,208)(180,207)(217,325)
(218,326)(219,328)(220,327)(221,333)(222,334)(223,336)(224,335)(225,329)
(226,330)(227,332)(228,331)(229,337)(230,338)(231,340)(232,339)(233,345)
(234,346)(235,348)(236,347)(237,341)(238,342)(239,344)(240,343)(241,349)
(242,350)(243,352)(244,351)(245,357)(246,358)(247,360)(248,359)(249,353)
(250,354)(251,356)(252,355)(253,405)(254,406)(255,408)(256,407)(257,401)
(258,402)(259,404)(260,403)(261,397)(262,398)(263,400)(264,399)(265,417)
(266,418)(267,420)(268,419)(269,413)(270,414)(271,416)(272,415)(273,409)
(274,410)(275,412)(276,411)(277,429)(278,430)(279,432)(280,431)(281,425)
(282,426)(283,428)(284,427)(285,421)(286,422)(287,424)(288,423)(289,369)
(290,370)(291,372)(292,371)(293,365)(294,366)(295,368)(296,367)(297,361)
(298,362)(299,364)(300,363)(301,381)(302,382)(303,384)(304,383)(305,377)
(306,378)(307,380)(308,379)(309,373)(310,374)(311,376)(312,375)(313,393)
(314,394)(315,396)(316,395)(317,389)(318,390)(319,392)(320,391)(321,385)
(322,386)(323,388)(324,387);
s1 := Sym(432)!(  1,253)(  2,256)(  3,255)(  4,254)(  5,261)(  6,264)(  7,263)
(  8,262)(  9,257)( 10,260)( 11,259)( 12,258)( 13,277)( 14,280)( 15,279)
( 16,278)( 17,285)( 18,288)( 19,287)( 20,286)( 21,281)( 22,284)( 23,283)
( 24,282)( 25,265)( 26,268)( 27,267)( 28,266)( 29,273)( 30,276)( 31,275)
( 32,274)( 33,269)( 34,272)( 35,271)( 36,270)( 37,217)( 38,220)( 39,219)
( 40,218)( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)
( 48,222)( 49,241)( 50,244)( 51,243)( 52,242)( 53,249)( 54,252)( 55,251)
( 56,250)( 57,245)( 58,248)( 59,247)( 60,246)( 61,229)( 62,232)( 63,231)
( 64,230)( 65,237)( 66,240)( 67,239)( 68,238)( 69,233)( 70,236)( 71,235)
( 72,234)( 73,297)( 74,300)( 75,299)( 76,298)( 77,293)( 78,296)( 79,295)
( 80,294)( 81,289)( 82,292)( 83,291)( 84,290)( 85,321)( 86,324)( 87,323)
( 88,322)( 89,317)( 90,320)( 91,319)( 92,318)( 93,313)( 94,316)( 95,315)
( 96,314)( 97,309)( 98,312)( 99,311)(100,310)(101,305)(102,308)(103,307)
(104,306)(105,301)(106,304)(107,303)(108,302)(109,361)(110,364)(111,363)
(112,362)(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)
(120,366)(121,385)(122,388)(123,387)(124,386)(125,393)(126,396)(127,395)
(128,394)(129,389)(130,392)(131,391)(132,390)(133,373)(134,376)(135,375)
(136,374)(137,381)(138,384)(139,383)(140,382)(141,377)(142,380)(143,379)
(144,378)(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)
(152,334)(153,329)(154,332)(155,331)(156,330)(157,349)(158,352)(159,351)
(160,350)(161,357)(162,360)(163,359)(164,358)(165,353)(166,356)(167,355)
(168,354)(169,337)(170,340)(171,339)(172,338)(173,345)(174,348)(175,347)
(176,346)(177,341)(178,344)(179,343)(180,342)(181,405)(182,408)(183,407)
(184,406)(185,401)(186,404)(187,403)(188,402)(189,397)(190,400)(191,399)
(192,398)(193,429)(194,432)(195,431)(196,430)(197,425)(198,428)(199,427)
(200,426)(201,421)(202,424)(203,423)(204,422)(205,417)(206,420)(207,419)
(208,418)(209,413)(210,416)(211,415)(212,414)(213,409)(214,412)(215,411)
(216,410);
s2 := Sym(432)!(  1, 14)(  2, 13)(  3, 16)(  4, 15)(  5, 18)(  6, 17)(  7, 20)
(  8, 19)(  9, 22)( 10, 21)( 11, 24)( 12, 23)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 34)( 35, 36)( 37, 50)( 38, 49)( 39, 52)( 40, 51)( 41, 54)
( 42, 53)( 43, 56)( 44, 55)( 45, 58)( 46, 57)( 47, 60)( 48, 59)( 61, 62)
( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 86)( 74, 85)( 75, 88)
( 76, 87)( 77, 90)( 78, 89)( 79, 92)( 80, 91)( 81, 94)( 82, 93)( 83, 96)
( 84, 95)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,122)
(110,121)(111,124)(112,123)(113,126)(114,125)(115,128)(116,127)(117,130)
(118,129)(119,132)(120,131)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144)(145,158)(146,157)(147,160)(148,159)(149,162)(150,161)(151,164)
(152,163)(153,166)(154,165)(155,168)(156,167)(169,170)(171,172)(173,174)
(175,176)(177,178)(179,180)(181,194)(182,193)(183,196)(184,195)(185,198)
(186,197)(187,200)(188,199)(189,202)(190,201)(191,204)(192,203)(205,206)
(207,208)(209,210)(211,212)(213,214)(215,216)(217,338)(218,337)(219,340)
(220,339)(221,342)(222,341)(223,344)(224,343)(225,346)(226,345)(227,348)
(228,347)(229,326)(230,325)(231,328)(232,327)(233,330)(234,329)(235,332)
(236,331)(237,334)(238,333)(239,336)(240,335)(241,350)(242,349)(243,352)
(244,351)(245,354)(246,353)(247,356)(248,355)(249,358)(250,357)(251,360)
(252,359)(253,374)(254,373)(255,376)(256,375)(257,378)(258,377)(259,380)
(260,379)(261,382)(262,381)(263,384)(264,383)(265,362)(266,361)(267,364)
(268,363)(269,366)(270,365)(271,368)(272,367)(273,370)(274,369)(275,372)
(276,371)(277,386)(278,385)(279,388)(280,387)(281,390)(282,389)(283,392)
(284,391)(285,394)(286,393)(287,396)(288,395)(289,410)(290,409)(291,412)
(292,411)(293,414)(294,413)(295,416)(296,415)(297,418)(298,417)(299,420)
(300,419)(301,398)(302,397)(303,400)(304,399)(305,402)(306,401)(307,404)
(308,403)(309,406)(310,405)(311,408)(312,407)(313,422)(314,421)(315,424)
(316,423)(317,426)(318,425)(319,428)(320,427)(321,430)(322,429)(323,432)
(324,431);
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*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,
s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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