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# Polytope of Type {12,36}

Atlas Canonical Name : {12,36}*1728e
if this polytope has a name.
Group : SmallGroup(1728,30228)
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 : {12,18}*864a
3-fold quotients : {4,36}*576b, {12,12}*576f
4-fold quotients : {6,36}*432a, {12,18}*432c
6-fold quotients : {4,36}*288b, {4,36}*288c, {4,18}*288, {12,6}*288a
8-fold quotients : {6,18}*216a
9-fold quotients : {4,12}*192b
12-fold quotients : {2,36}*144, {4,9}*144, {4,18}*144b, {4,18}*144c, {6,12}*144a, {12,6}*144d
18-fold quotients : {4,12}*96b, {4,12}*96c, {4,6}*96
24-fold quotients : {4,9}*72, {2,18}*72, {6,6}*72a
36-fold quotients : {2,12}*48, {6,4}*48a, {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
108-fold quotients : {2,4}*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 := (  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)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,243)(230,244)(231,241)(232,242)(233,247)(234,248)
(235,245)(236,246)(237,251)(238,252)(239,249)(240,250)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,279)(266,280)(267,277)(268,278)
(269,283)(270,284)(271,281)(272,282)(273,287)(274,288)(275,285)(276,286)
(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,315)(302,316)
(303,313)(304,314)(305,319)(306,320)(307,317)(308,318)(309,323)(310,324)
(311,321)(312,322)(325,327)(326,328)(329,331)(330,332)(333,335)(334,336)
(337,351)(338,352)(339,349)(340,350)(341,355)(342,356)(343,353)(344,354)
(345,359)(346,360)(347,357)(348,358)(361,363)(362,364)(365,367)(366,368)
(369,371)(370,372)(373,387)(374,388)(375,385)(376,386)(377,391)(378,392)
(379,389)(380,390)(381,395)(382,396)(383,393)(384,394)(397,399)(398,400)
(401,403)(402,404)(405,407)(406,408)(409,423)(410,424)(411,421)(412,422)
(413,427)(414,428)(415,425)(416,426)(417,431)(418,432)(419,429)(420,430);;
s1 := (  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 21)(  6, 22)(  7, 24)(  8, 23)
(  9, 17)( 10, 18)( 11, 20)( 12, 19)( 27, 28)( 29, 33)( 30, 34)( 31, 36)
( 32, 35)( 37, 93)( 38, 94)( 39, 96)( 40, 95)( 41, 89)( 42, 90)( 43, 92)
( 44, 91)( 45, 85)( 46, 86)( 47, 88)( 48, 87)( 49, 81)( 50, 82)( 51, 84)
( 52, 83)( 53, 77)( 54, 78)( 55, 80)( 56, 79)( 57, 73)( 58, 74)( 59, 76)
( 60, 75)( 61,105)( 62,106)( 63,108)( 64,107)( 65,101)( 66,102)( 67,104)
( 68,103)( 69, 97)( 70, 98)( 71,100)( 72, 99)(109,121)(110,122)(111,124)
(112,123)(113,129)(114,130)(115,132)(116,131)(117,125)(118,126)(119,128)
(120,127)(135,136)(137,141)(138,142)(139,144)(140,143)(145,201)(146,202)
(147,204)(148,203)(149,197)(150,198)(151,200)(152,199)(153,193)(154,194)
(155,196)(156,195)(157,189)(158,190)(159,192)(160,191)(161,185)(162,186)
(163,188)(164,187)(165,181)(166,182)(167,184)(168,183)(169,213)(170,214)
(171,216)(172,215)(173,209)(174,210)(175,212)(176,211)(177,205)(178,206)
(179,208)(180,207)(217,337)(218,338)(219,340)(220,339)(221,345)(222,346)
(223,348)(224,347)(225,341)(226,342)(227,344)(228,343)(229,325)(230,326)
(231,328)(232,327)(233,333)(234,334)(235,336)(236,335)(237,329)(238,330)
(239,332)(240,331)(241,349)(242,350)(243,352)(244,351)(245,357)(246,358)
(247,360)(248,359)(249,353)(250,354)(251,356)(252,355)(253,417)(254,418)
(255,420)(256,419)(257,413)(258,414)(259,416)(260,415)(261,409)(262,410)
(263,412)(264,411)(265,405)(266,406)(267,408)(268,407)(269,401)(270,402)
(271,404)(272,403)(273,397)(274,398)(275,400)(276,399)(277,429)(278,430)
(279,432)(280,431)(281,425)(282,426)(283,428)(284,427)(285,421)(286,422)
(287,424)(288,423)(289,381)(290,382)(291,384)(292,383)(293,377)(294,378)
(295,380)(296,379)(297,373)(298,374)(299,376)(300,375)(301,369)(302,370)
(303,372)(304,371)(305,365)(306,366)(307,368)(308,367)(309,361)(310,362)
(311,364)(312,363)(313,393)(314,394)(315,396)(316,395)(317,389)(318,390)
(319,392)(320,391)(321,385)(322,386)(323,388)(324,387);;
s2 := (  1,253)(  2,256)(  3,255)(  4,254)(  5,261)(  6,264)(  7,263)(  8,262)
(  9,257)( 10,260)( 11,259)( 12,258)( 13,265)( 14,268)( 15,267)( 16,266)
( 17,273)( 18,276)( 19,275)( 20,274)( 21,269)( 22,272)( 23,271)( 24,270)
( 25,277)( 26,280)( 27,279)( 28,278)( 29,285)( 30,288)( 31,287)( 32,286)
( 33,281)( 34,284)( 35,283)( 36,282)( 37,217)( 38,220)( 39,219)( 40,218)
( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)( 48,222)
( 49,229)( 50,232)( 51,231)( 52,230)( 53,237)( 54,240)( 55,239)( 56,238)
( 57,233)( 58,236)( 59,235)( 60,234)( 61,241)( 62,244)( 63,243)( 64,242)
( 65,249)( 66,252)( 67,251)( 68,250)( 69,245)( 70,248)( 71,247)( 72,246)
( 73,297)( 74,300)( 75,299)( 76,298)( 77,293)( 78,296)( 79,295)( 80,294)
( 81,289)( 82,292)( 83,291)( 84,290)( 85,309)( 86,312)( 87,311)( 88,310)
( 89,305)( 90,308)( 91,307)( 92,306)( 93,301)( 94,304)( 95,303)( 96,302)
( 97,321)( 98,324)( 99,323)(100,322)(101,317)(102,320)(103,319)(104,318)
(105,313)(106,316)(107,315)(108,314)(109,361)(110,364)(111,363)(112,362)
(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)(120,366)
(121,373)(122,376)(123,375)(124,374)(125,381)(126,384)(127,383)(128,382)
(129,377)(130,380)(131,379)(132,378)(133,385)(134,388)(135,387)(136,386)
(137,393)(138,396)(139,395)(140,394)(141,389)(142,392)(143,391)(144,390)
(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)(152,334)
(153,329)(154,332)(155,331)(156,330)(157,337)(158,340)(159,339)(160,338)
(161,345)(162,348)(163,347)(164,346)(165,341)(166,344)(167,343)(168,342)
(169,349)(170,352)(171,351)(172,350)(173,357)(174,360)(175,359)(176,358)
(177,353)(178,356)(179,355)(180,354)(181,405)(182,408)(183,407)(184,406)
(185,401)(186,404)(187,403)(188,402)(189,397)(190,400)(191,399)(192,398)
(193,417)(194,420)(195,419)(196,418)(197,413)(198,416)(199,415)(200,414)
(201,409)(202,412)(203,411)(204,410)(205,429)(206,432)(207,431)(208,430)
(209,425)(210,428)(211,427)(212,426)(213,421)(214,424)(215,423)(216,422);;
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, s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
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,
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)!(  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)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,243)(230,244)(231,241)(232,242)(233,247)
(234,248)(235,245)(236,246)(237,251)(238,252)(239,249)(240,250)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,279)(266,280)(267,277)
(268,278)(269,283)(270,284)(271,281)(272,282)(273,287)(274,288)(275,285)
(276,286)(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,315)
(302,316)(303,313)(304,314)(305,319)(306,320)(307,317)(308,318)(309,323)
(310,324)(311,321)(312,322)(325,327)(326,328)(329,331)(330,332)(333,335)
(334,336)(337,351)(338,352)(339,349)(340,350)(341,355)(342,356)(343,353)
(344,354)(345,359)(346,360)(347,357)(348,358)(361,363)(362,364)(365,367)
(366,368)(369,371)(370,372)(373,387)(374,388)(375,385)(376,386)(377,391)
(378,392)(379,389)(380,390)(381,395)(382,396)(383,393)(384,394)(397,399)
(398,400)(401,403)(402,404)(405,407)(406,408)(409,423)(410,424)(411,421)
(412,422)(413,427)(414,428)(415,425)(416,426)(417,431)(418,432)(419,429)
(420,430);
s1 := Sym(432)!(  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 21)(  6, 22)(  7, 24)
(  8, 23)(  9, 17)( 10, 18)( 11, 20)( 12, 19)( 27, 28)( 29, 33)( 30, 34)
( 31, 36)( 32, 35)( 37, 93)( 38, 94)( 39, 96)( 40, 95)( 41, 89)( 42, 90)
( 43, 92)( 44, 91)( 45, 85)( 46, 86)( 47, 88)( 48, 87)( 49, 81)( 50, 82)
( 51, 84)( 52, 83)( 53, 77)( 54, 78)( 55, 80)( 56, 79)( 57, 73)( 58, 74)
( 59, 76)( 60, 75)( 61,105)( 62,106)( 63,108)( 64,107)( 65,101)( 66,102)
( 67,104)( 68,103)( 69, 97)( 70, 98)( 71,100)( 72, 99)(109,121)(110,122)
(111,124)(112,123)(113,129)(114,130)(115,132)(116,131)(117,125)(118,126)
(119,128)(120,127)(135,136)(137,141)(138,142)(139,144)(140,143)(145,201)
(146,202)(147,204)(148,203)(149,197)(150,198)(151,200)(152,199)(153,193)
(154,194)(155,196)(156,195)(157,189)(158,190)(159,192)(160,191)(161,185)
(162,186)(163,188)(164,187)(165,181)(166,182)(167,184)(168,183)(169,213)
(170,214)(171,216)(172,215)(173,209)(174,210)(175,212)(176,211)(177,205)
(178,206)(179,208)(180,207)(217,337)(218,338)(219,340)(220,339)(221,345)
(222,346)(223,348)(224,347)(225,341)(226,342)(227,344)(228,343)(229,325)
(230,326)(231,328)(232,327)(233,333)(234,334)(235,336)(236,335)(237,329)
(238,330)(239,332)(240,331)(241,349)(242,350)(243,352)(244,351)(245,357)
(246,358)(247,360)(248,359)(249,353)(250,354)(251,356)(252,355)(253,417)
(254,418)(255,420)(256,419)(257,413)(258,414)(259,416)(260,415)(261,409)
(262,410)(263,412)(264,411)(265,405)(266,406)(267,408)(268,407)(269,401)
(270,402)(271,404)(272,403)(273,397)(274,398)(275,400)(276,399)(277,429)
(278,430)(279,432)(280,431)(281,425)(282,426)(283,428)(284,427)(285,421)
(286,422)(287,424)(288,423)(289,381)(290,382)(291,384)(292,383)(293,377)
(294,378)(295,380)(296,379)(297,373)(298,374)(299,376)(300,375)(301,369)
(302,370)(303,372)(304,371)(305,365)(306,366)(307,368)(308,367)(309,361)
(310,362)(311,364)(312,363)(313,393)(314,394)(315,396)(316,395)(317,389)
(318,390)(319,392)(320,391)(321,385)(322,386)(323,388)(324,387);
s2 := 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)( 13,265)( 14,268)( 15,267)
( 16,266)( 17,273)( 18,276)( 19,275)( 20,274)( 21,269)( 22,272)( 23,271)
( 24,270)( 25,277)( 26,280)( 27,279)( 28,278)( 29,285)( 30,288)( 31,287)
( 32,286)( 33,281)( 34,284)( 35,283)( 36,282)( 37,217)( 38,220)( 39,219)
( 40,218)( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)
( 48,222)( 49,229)( 50,232)( 51,231)( 52,230)( 53,237)( 54,240)( 55,239)
( 56,238)( 57,233)( 58,236)( 59,235)( 60,234)( 61,241)( 62,244)( 63,243)
( 64,242)( 65,249)( 66,252)( 67,251)( 68,250)( 69,245)( 70,248)( 71,247)
( 72,246)( 73,297)( 74,300)( 75,299)( 76,298)( 77,293)( 78,296)( 79,295)
( 80,294)( 81,289)( 82,292)( 83,291)( 84,290)( 85,309)( 86,312)( 87,311)
( 88,310)( 89,305)( 90,308)( 91,307)( 92,306)( 93,301)( 94,304)( 95,303)
( 96,302)( 97,321)( 98,324)( 99,323)(100,322)(101,317)(102,320)(103,319)
(104,318)(105,313)(106,316)(107,315)(108,314)(109,361)(110,364)(111,363)
(112,362)(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)
(120,366)(121,373)(122,376)(123,375)(124,374)(125,381)(126,384)(127,383)
(128,382)(129,377)(130,380)(131,379)(132,378)(133,385)(134,388)(135,387)
(136,386)(137,393)(138,396)(139,395)(140,394)(141,389)(142,392)(143,391)
(144,390)(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)
(152,334)(153,329)(154,332)(155,331)(156,330)(157,337)(158,340)(159,339)
(160,338)(161,345)(162,348)(163,347)(164,346)(165,341)(166,344)(167,343)
(168,342)(169,349)(170,352)(171,351)(172,350)(173,357)(174,360)(175,359)
(176,358)(177,353)(178,356)(179,355)(180,354)(181,405)(182,408)(183,407)
(184,406)(185,401)(186,404)(187,403)(188,402)(189,397)(190,400)(191,399)
(192,398)(193,417)(194,420)(195,419)(196,418)(197,413)(198,416)(199,415)
(200,414)(201,409)(202,412)(203,411)(204,410)(205,429)(206,432)(207,431)
(208,430)(209,425)(210,428)(211,427)(212,426)(213,421)(214,424)(215,423)
(216,422);
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, s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
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,
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