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# Polytope of Type {12,36}

Atlas Canonical Name : {12,36}*1728a
if this polytope has a name.
Group : SmallGroup(1728,3530)
Rank : 3
Schlafli Type : {12,36}
Number of vertices, edges, etc : 24, 432, 72
Order of s0s1s2 : 36
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 : {12,36}*864a
3-fold quotients : {4,36}*576a, {12,12}*576a
4-fold quotients : {6,36}*432a, {12,18}*432a
6-fold quotients : {4,36}*288a, {12,12}*288a
8-fold quotients : {6,18}*216a
9-fold quotients : {4,12}*192a, {12,4}*192a
12-fold quotients : {2,36}*144, {4,18}*144a, {6,12}*144a, {12,6}*144a
18-fold quotients : {4,12}*96a, {12,4}*96a
24-fold quotients : {2,18}*72, {6,6}*72a
27-fold quotients : {4,4}*64
36-fold quotients : {2,12}*48, {12,2}*48, {4,6}*48a, {6,4}*48a
48-fold quotients : {2,9}*36
54-fold quotients : {4,4}*32
72-fold quotients : {2,6}*24, {6,2}*24
108-fold quotients : {2,4}*16, {4,2}*16
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 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)
( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)( 68, 98)
( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)( 76,106)
( 77,107)( 78,108)( 79,103)( 80,104)( 81,105)(112,115)(113,116)(114,117)
(121,124)(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)
(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(163,190)
(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)(171,195)
(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)(179,203)
(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)(187,211)
(188,212)(189,213)(217,325)(218,326)(219,327)(220,331)(221,332)(222,333)
(223,328)(224,329)(225,330)(226,334)(227,335)(228,336)(229,340)(230,341)
(231,342)(232,337)(233,338)(234,339)(235,343)(236,344)(237,345)(238,349)
(239,350)(240,351)(241,346)(242,347)(243,348)(244,352)(245,353)(246,354)
(247,358)(248,359)(249,360)(250,355)(251,356)(252,357)(253,361)(254,362)
(255,363)(256,367)(257,368)(258,369)(259,364)(260,365)(261,366)(262,370)
(263,371)(264,372)(265,376)(266,377)(267,378)(268,373)(269,374)(270,375)
(271,406)(272,407)(273,408)(274,412)(275,413)(276,414)(277,409)(278,410)
(279,411)(280,415)(281,416)(282,417)(283,421)(284,422)(285,423)(286,418)
(287,419)(288,420)(289,424)(290,425)(291,426)(292,430)(293,431)(294,432)
(295,427)(296,428)(297,429)(298,379)(299,380)(300,381)(301,385)(302,386)
(303,387)(304,382)(305,383)(306,384)(307,388)(308,389)(309,390)(310,394)
(311,395)(312,396)(313,391)(314,392)(315,393)(316,397)(317,398)(318,399)
(319,403)(320,404)(321,405)(322,400)(323,401)(324,402);;
s1 := (  1,220)(  2,222)(  3,221)(  4,217)(  5,219)(  6,218)(  7,223)(  8,225)
(  9,224)( 10,240)( 11,239)( 12,238)( 13,237)( 14,236)( 15,235)( 16,243)
( 17,242)( 18,241)( 19,231)( 20,230)( 21,229)( 22,228)( 23,227)( 24,226)
( 25,234)( 26,233)( 27,232)( 28,247)( 29,249)( 30,248)( 31,244)( 32,246)
( 33,245)( 34,250)( 35,252)( 36,251)( 37,267)( 38,266)( 39,265)( 40,264)
( 41,263)( 42,262)( 43,270)( 44,269)( 45,268)( 46,258)( 47,257)( 48,256)
( 49,255)( 50,254)( 51,253)( 52,261)( 53,260)( 54,259)( 55,274)( 56,276)
( 57,275)( 58,271)( 59,273)( 60,272)( 61,277)( 62,279)( 63,278)( 64,294)
( 65,293)( 66,292)( 67,291)( 68,290)( 69,289)( 70,297)( 71,296)( 72,295)
( 73,285)( 74,284)( 75,283)( 76,282)( 77,281)( 78,280)( 79,288)( 80,287)
( 81,286)( 82,301)( 83,303)( 84,302)( 85,298)( 86,300)( 87,299)( 88,304)
( 89,306)( 90,305)( 91,321)( 92,320)( 93,319)( 94,318)( 95,317)( 96,316)
( 97,324)( 98,323)( 99,322)(100,312)(101,311)(102,310)(103,309)(104,308)
(105,307)(106,315)(107,314)(108,313)(109,328)(110,330)(111,329)(112,325)
(113,327)(114,326)(115,331)(116,333)(117,332)(118,348)(119,347)(120,346)
(121,345)(122,344)(123,343)(124,351)(125,350)(126,349)(127,339)(128,338)
(129,337)(130,336)(131,335)(132,334)(133,342)(134,341)(135,340)(136,355)
(137,357)(138,356)(139,352)(140,354)(141,353)(142,358)(143,360)(144,359)
(145,375)(146,374)(147,373)(148,372)(149,371)(150,370)(151,378)(152,377)
(153,376)(154,366)(155,365)(156,364)(157,363)(158,362)(159,361)(160,369)
(161,368)(162,367)(163,382)(164,384)(165,383)(166,379)(167,381)(168,380)
(169,385)(170,387)(171,386)(172,402)(173,401)(174,400)(175,399)(176,398)
(177,397)(178,405)(179,404)(180,403)(181,393)(182,392)(183,391)(184,390)
(185,389)(186,388)(187,396)(188,395)(189,394)(190,409)(191,411)(192,410)
(193,406)(194,408)(195,407)(196,412)(197,414)(198,413)(199,429)(200,428)
(201,427)(202,426)(203,425)(204,424)(205,432)(206,431)(207,430)(208,420)
(209,419)(210,418)(211,417)(212,416)(213,415)(214,423)(215,422)(216,421);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 13)(  5, 15)(  6, 14)(  7, 16)(  8, 18)
(  9, 17)( 19, 21)( 22, 24)( 25, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 40)
( 32, 42)( 33, 41)( 34, 43)( 35, 45)( 36, 44)( 46, 48)( 49, 51)( 52, 54)
( 55, 64)( 56, 66)( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)( 62, 72)
( 63, 71)( 73, 75)( 76, 78)( 79, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 94)
( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)(100,102)(103,105)(106,108)
(109,145)(110,147)(111,146)(112,148)(113,150)(114,149)(115,151)(116,153)
(117,152)(118,136)(119,138)(120,137)(121,139)(122,141)(123,140)(124,142)
(125,144)(126,143)(127,156)(128,155)(129,154)(130,159)(131,158)(132,157)
(133,162)(134,161)(135,160)(163,199)(164,201)(165,200)(166,202)(167,204)
(168,203)(169,205)(170,207)(171,206)(172,190)(173,192)(174,191)(175,193)
(176,195)(177,194)(178,196)(179,198)(180,197)(181,210)(182,209)(183,208)
(184,213)(185,212)(186,211)(187,216)(188,215)(189,214)(217,280)(218,282)
(219,281)(220,283)(221,285)(222,284)(223,286)(224,288)(225,287)(226,271)
(227,273)(228,272)(229,274)(230,276)(231,275)(232,277)(233,279)(234,278)
(235,291)(236,290)(237,289)(238,294)(239,293)(240,292)(241,297)(242,296)
(243,295)(244,307)(245,309)(246,308)(247,310)(248,312)(249,311)(250,313)
(251,315)(252,314)(253,298)(254,300)(255,299)(256,301)(257,303)(258,302)
(259,304)(260,306)(261,305)(262,318)(263,317)(264,316)(265,321)(266,320)
(267,319)(268,324)(269,323)(270,322)(325,415)(326,417)(327,416)(328,418)
(329,420)(330,419)(331,421)(332,423)(333,422)(334,406)(335,408)(336,407)
(337,409)(338,411)(339,410)(340,412)(341,414)(342,413)(343,426)(344,425)
(345,424)(346,429)(347,428)(348,427)(349,432)(350,431)(351,430)(352,388)
(353,390)(354,389)(355,391)(356,393)(357,392)(358,394)(359,396)(360,395)
(361,379)(362,381)(363,380)(364,382)(365,384)(366,383)(367,385)(368,387)
(369,386)(370,399)(371,398)(372,397)(373,402)(374,401)(375,400)(376,405)
(377,404)(378,403);;
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*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*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*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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope