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Polytope of Type {12,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,36}*1728d
if this polytope has a name.
Group : SmallGroup(1728,30216)
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 : {6,36}*864
   3-fold quotients : {12,12}*576j
   4-fold quotients : {6,36}*432c
   6-fold quotients : {6,12}*288a
   8-fold quotients : {6,18}*216a
   9-fold quotients : {12,4}*192c
   12-fold quotients : {6,12}*144d
   18-fold quotients : {6,4}*96
   24-fold quotients : {2,18}*72, {6,6}*72a
   36-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   48-fold quotients : {2,9}*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)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)
( 35, 36)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)( 43, 80)
( 44, 79)( 45, 81)( 46, 82)( 47, 84)( 48, 83)( 49, 85)( 50, 86)( 51, 88)
( 52, 87)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 93)( 58, 94)( 59, 96)
( 60, 95)( 61, 97)( 62, 98)( 63,100)( 64, 99)( 65,101)( 66,102)( 67,104)
( 68,103)( 69,105)( 70,106)( 71,108)( 72,107)(111,112)(115,116)(119,120)
(123,124)(127,128)(131,132)(135,136)(139,140)(143,144)(145,181)(146,182)
(147,184)(148,183)(149,185)(150,186)(151,188)(152,187)(153,189)(154,190)
(155,192)(156,191)(157,193)(158,194)(159,196)(160,195)(161,197)(162,198)
(163,200)(164,199)(165,201)(166,202)(167,204)(168,203)(169,205)(170,206)
(171,208)(172,207)(173,209)(174,210)(175,212)(176,211)(177,213)(178,214)
(179,216)(180,215)(217,325)(218,326)(219,328)(220,327)(221,329)(222,330)
(223,332)(224,331)(225,333)(226,334)(227,336)(228,335)(229,337)(230,338)
(231,340)(232,339)(233,341)(234,342)(235,344)(236,343)(237,345)(238,346)
(239,348)(240,347)(241,349)(242,350)(243,352)(244,351)(245,353)(246,354)
(247,356)(248,355)(249,357)(250,358)(251,360)(252,359)(253,397)(254,398)
(255,400)(256,399)(257,401)(258,402)(259,404)(260,403)(261,405)(262,406)
(263,408)(264,407)(265,409)(266,410)(267,412)(268,411)(269,413)(270,414)
(271,416)(272,415)(273,417)(274,418)(275,420)(276,419)(277,421)(278,422)
(279,424)(280,423)(281,425)(282,426)(283,428)(284,427)(285,429)(286,430)
(287,432)(288,431)(289,361)(290,362)(291,364)(292,363)(293,365)(294,366)
(295,368)(296,367)(297,369)(298,370)(299,372)(300,371)(301,373)(302,374)
(303,376)(304,375)(305,377)(306,378)(307,380)(308,379)(309,381)(310,382)
(311,384)(312,383)(313,385)(314,386)(315,388)(316,387)(317,389)(318,390)
(319,392)(320,391)(321,393)(322,394)(323,396)(324,395);;
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)( 13,285)( 14,288)( 15,287)( 16,286)
( 17,281)( 18,284)( 19,283)( 20,282)( 21,277)( 22,280)( 23,279)( 24,278)
( 25,273)( 26,276)( 27,275)( 28,274)( 29,269)( 30,272)( 31,271)( 32,270)
( 33,265)( 34,268)( 35,267)( 36,266)( 37,217)( 38,220)( 39,219)( 40,218)
( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)( 48,222)
( 49,249)( 50,252)( 51,251)( 52,250)( 53,245)( 54,248)( 55,247)( 56,246)
( 57,241)( 58,244)( 59,243)( 60,242)( 61,237)( 62,240)( 63,239)( 64,238)
( 65,233)( 66,236)( 67,235)( 68,234)( 69,229)( 70,232)( 71,231)( 72,230)
( 73,289)( 74,292)( 75,291)( 76,290)( 77,297)( 78,300)( 79,299)( 80,298)
( 81,293)( 82,296)( 83,295)( 84,294)( 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)
(121,393)(122,396)(123,395)(124,394)(125,389)(126,392)(127,391)(128,390)
(129,385)(130,388)(131,387)(132,386)(133,381)(134,384)(135,383)(136,382)
(137,377)(138,380)(139,379)(140,378)(141,373)(142,376)(143,375)(144,374)
(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)(152,334)
(153,329)(154,332)(155,331)(156,330)(157,357)(158,360)(159,359)(160,358)
(161,353)(162,356)(163,355)(164,354)(165,349)(166,352)(167,351)(168,350)
(169,345)(170,348)(171,347)(172,346)(173,341)(174,344)(175,343)(176,342)
(177,337)(178,340)(179,339)(180,338)(181,397)(182,400)(183,399)(184,398)
(185,405)(186,408)(187,407)(188,406)(189,401)(190,404)(191,403)(192,402)
(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, 22)(  6, 21)(  7, 24)(  8, 23)
(  9, 18)( 10, 17)( 11, 20)( 12, 19)( 25, 34)( 26, 33)( 27, 36)( 28, 35)
( 29, 30)( 31, 32)( 37, 50)( 38, 49)( 39, 52)( 40, 51)( 41, 58)( 42, 57)
( 43, 60)( 44, 59)( 45, 54)( 46, 53)( 47, 56)( 48, 55)( 61, 70)( 62, 69)
( 63, 72)( 64, 71)( 65, 66)( 67, 68)( 73, 86)( 74, 85)( 75, 88)( 76, 87)
( 77, 94)( 78, 93)( 79, 96)( 80, 95)( 81, 90)( 82, 89)( 83, 92)( 84, 91)
( 97,106)( 98,105)( 99,108)(100,107)(101,102)(103,104)(109,122)(110,121)
(111,124)(112,123)(113,130)(114,129)(115,132)(116,131)(117,126)(118,125)
(119,128)(120,127)(133,142)(134,141)(135,144)(136,143)(137,138)(139,140)
(145,158)(146,157)(147,160)(148,159)(149,166)(150,165)(151,168)(152,167)
(153,162)(154,161)(155,164)(156,163)(169,178)(170,177)(171,180)(172,179)
(173,174)(175,176)(181,194)(182,193)(183,196)(184,195)(185,202)(186,201)
(187,204)(188,203)(189,198)(190,197)(191,200)(192,199)(205,214)(206,213)
(207,216)(208,215)(209,210)(211,212)(217,338)(218,337)(219,340)(220,339)
(221,346)(222,345)(223,348)(224,347)(225,342)(226,341)(227,344)(228,343)
(229,326)(230,325)(231,328)(232,327)(233,334)(234,333)(235,336)(236,335)
(237,330)(238,329)(239,332)(240,331)(241,358)(242,357)(243,360)(244,359)
(245,354)(246,353)(247,356)(248,355)(249,350)(250,349)(251,352)(252,351)
(253,374)(254,373)(255,376)(256,375)(257,382)(258,381)(259,384)(260,383)
(261,378)(262,377)(263,380)(264,379)(265,362)(266,361)(267,364)(268,363)
(269,370)(270,369)(271,372)(272,371)(273,366)(274,365)(275,368)(276,367)
(277,394)(278,393)(279,396)(280,395)(281,390)(282,389)(283,392)(284,391)
(285,386)(286,385)(287,388)(288,387)(289,410)(290,409)(291,412)(292,411)
(293,418)(294,417)(295,420)(296,419)(297,414)(298,413)(299,416)(300,415)
(301,398)(302,397)(303,400)(304,399)(305,406)(306,405)(307,408)(308,407)
(309,402)(310,401)(311,404)(312,403)(313,430)(314,429)(315,432)(316,431)
(317,426)(318,425)(319,428)(320,427)(321,422)(322,421)(323,424)(324,423);;
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, 
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*s0*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*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(432)!(  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)
( 31, 32)( 35, 36)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)
( 43, 80)( 44, 79)( 45, 81)( 46, 82)( 47, 84)( 48, 83)( 49, 85)( 50, 86)
( 51, 88)( 52, 87)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 93)( 58, 94)
( 59, 96)( 60, 95)( 61, 97)( 62, 98)( 63,100)( 64, 99)( 65,101)( 66,102)
( 67,104)( 68,103)( 69,105)( 70,106)( 71,108)( 72,107)(111,112)(115,116)
(119,120)(123,124)(127,128)(131,132)(135,136)(139,140)(143,144)(145,181)
(146,182)(147,184)(148,183)(149,185)(150,186)(151,188)(152,187)(153,189)
(154,190)(155,192)(156,191)(157,193)(158,194)(159,196)(160,195)(161,197)
(162,198)(163,200)(164,199)(165,201)(166,202)(167,204)(168,203)(169,205)
(170,206)(171,208)(172,207)(173,209)(174,210)(175,212)(176,211)(177,213)
(178,214)(179,216)(180,215)(217,325)(218,326)(219,328)(220,327)(221,329)
(222,330)(223,332)(224,331)(225,333)(226,334)(227,336)(228,335)(229,337)
(230,338)(231,340)(232,339)(233,341)(234,342)(235,344)(236,343)(237,345)
(238,346)(239,348)(240,347)(241,349)(242,350)(243,352)(244,351)(245,353)
(246,354)(247,356)(248,355)(249,357)(250,358)(251,360)(252,359)(253,397)
(254,398)(255,400)(256,399)(257,401)(258,402)(259,404)(260,403)(261,405)
(262,406)(263,408)(264,407)(265,409)(266,410)(267,412)(268,411)(269,413)
(270,414)(271,416)(272,415)(273,417)(274,418)(275,420)(276,419)(277,421)
(278,422)(279,424)(280,423)(281,425)(282,426)(283,428)(284,427)(285,429)
(286,430)(287,432)(288,431)(289,361)(290,362)(291,364)(292,363)(293,365)
(294,366)(295,368)(296,367)(297,369)(298,370)(299,372)(300,371)(301,373)
(302,374)(303,376)(304,375)(305,377)(306,378)(307,380)(308,379)(309,381)
(310,382)(311,384)(312,383)(313,385)(314,386)(315,388)(316,387)(317,389)
(318,390)(319,392)(320,391)(321,393)(322,394)(323,396)(324,395);
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)( 13,285)( 14,288)( 15,287)
( 16,286)( 17,281)( 18,284)( 19,283)( 20,282)( 21,277)( 22,280)( 23,279)
( 24,278)( 25,273)( 26,276)( 27,275)( 28,274)( 29,269)( 30,272)( 31,271)
( 32,270)( 33,265)( 34,268)( 35,267)( 36,266)( 37,217)( 38,220)( 39,219)
( 40,218)( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)
( 48,222)( 49,249)( 50,252)( 51,251)( 52,250)( 53,245)( 54,248)( 55,247)
( 56,246)( 57,241)( 58,244)( 59,243)( 60,242)( 61,237)( 62,240)( 63,239)
( 64,238)( 65,233)( 66,236)( 67,235)( 68,234)( 69,229)( 70,232)( 71,231)
( 72,230)( 73,289)( 74,292)( 75,291)( 76,290)( 77,297)( 78,300)( 79,299)
( 80,298)( 81,293)( 82,296)( 83,295)( 84,294)( 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)(121,393)(122,396)(123,395)(124,394)(125,389)(126,392)(127,391)
(128,390)(129,385)(130,388)(131,387)(132,386)(133,381)(134,384)(135,383)
(136,382)(137,377)(138,380)(139,379)(140,378)(141,373)(142,376)(143,375)
(144,374)(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)
(152,334)(153,329)(154,332)(155,331)(156,330)(157,357)(158,360)(159,359)
(160,358)(161,353)(162,356)(163,355)(164,354)(165,349)(166,352)(167,351)
(168,350)(169,345)(170,348)(171,347)(172,346)(173,341)(174,344)(175,343)
(176,342)(177,337)(178,340)(179,339)(180,338)(181,397)(182,400)(183,399)
(184,398)(185,405)(186,408)(187,407)(188,406)(189,401)(190,404)(191,403)
(192,402)(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, 22)(  6, 21)(  7, 24)
(  8, 23)(  9, 18)( 10, 17)( 11, 20)( 12, 19)( 25, 34)( 26, 33)( 27, 36)
( 28, 35)( 29, 30)( 31, 32)( 37, 50)( 38, 49)( 39, 52)( 40, 51)( 41, 58)
( 42, 57)( 43, 60)( 44, 59)( 45, 54)( 46, 53)( 47, 56)( 48, 55)( 61, 70)
( 62, 69)( 63, 72)( 64, 71)( 65, 66)( 67, 68)( 73, 86)( 74, 85)( 75, 88)
( 76, 87)( 77, 94)( 78, 93)( 79, 96)( 80, 95)( 81, 90)( 82, 89)( 83, 92)
( 84, 91)( 97,106)( 98,105)( 99,108)(100,107)(101,102)(103,104)(109,122)
(110,121)(111,124)(112,123)(113,130)(114,129)(115,132)(116,131)(117,126)
(118,125)(119,128)(120,127)(133,142)(134,141)(135,144)(136,143)(137,138)
(139,140)(145,158)(146,157)(147,160)(148,159)(149,166)(150,165)(151,168)
(152,167)(153,162)(154,161)(155,164)(156,163)(169,178)(170,177)(171,180)
(172,179)(173,174)(175,176)(181,194)(182,193)(183,196)(184,195)(185,202)
(186,201)(187,204)(188,203)(189,198)(190,197)(191,200)(192,199)(205,214)
(206,213)(207,216)(208,215)(209,210)(211,212)(217,338)(218,337)(219,340)
(220,339)(221,346)(222,345)(223,348)(224,347)(225,342)(226,341)(227,344)
(228,343)(229,326)(230,325)(231,328)(232,327)(233,334)(234,333)(235,336)
(236,335)(237,330)(238,329)(239,332)(240,331)(241,358)(242,357)(243,360)
(244,359)(245,354)(246,353)(247,356)(248,355)(249,350)(250,349)(251,352)
(252,351)(253,374)(254,373)(255,376)(256,375)(257,382)(258,381)(259,384)
(260,383)(261,378)(262,377)(263,380)(264,379)(265,362)(266,361)(267,364)
(268,363)(269,370)(270,369)(271,372)(272,371)(273,366)(274,365)(275,368)
(276,367)(277,394)(278,393)(279,396)(280,395)(281,390)(282,389)(283,392)
(284,391)(285,386)(286,385)(287,388)(288,387)(289,410)(290,409)(291,412)
(292,411)(293,418)(294,417)(295,420)(296,419)(297,414)(298,413)(299,416)
(300,415)(301,398)(302,397)(303,400)(304,399)(305,406)(306,405)(307,408)
(308,407)(309,402)(310,401)(311,404)(312,403)(313,430)(314,429)(315,432)
(316,431)(317,426)(318,425)(319,428)(320,427)(321,422)(322,421)(323,424)
(324,423);
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, 
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*s0*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*s2*s1*s2 >; 
 
References : None.
to this polytope