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Polytope of Type {12,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,24}*1728a
if this polytope has a name.
Group : SmallGroup(1728,3511)
Rank : 3
Schlafli Type : {12,24}
Number of vertices, edges, etc : 36, 432, 72
Order of s0s1s2 : 24
Order of s0s1s2s1 : 12
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 : {12,12}*864a
   3-fold quotients : {12,24}*576a
   4-fold quotients : {12,6}*432a, {6,12}*432c
   6-fold quotients : {12,12}*288c
   8-fold quotients : {6,6}*216c
   9-fold quotients : {12,8}*192b
   12-fold quotients : {12,6}*144b, {6,12}*144c
   16-fold quotients : {3,6}*108
   18-fold quotients : {12,4}*96a
   24-fold quotients : {6,6}*72c
   27-fold quotients : {4,8}*64b
   36-fold quotients : {12,2}*48, {6,4}*48a
   48-fold quotients : {3,6}*36
   54-fold quotients : {4,4}*32
   72-fold quotients : {6,2}*24
   108-fold quotients : {2,4}*16, {4,2}*16
   144-fold quotients : {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)( 10, 19)( 11, 20)( 12, 21)( 13, 25)( 14, 26)
( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)( 37, 46)
( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)
( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)( 61, 85)( 62, 86)
( 63, 87)( 64,100)( 65,101)( 66,102)( 67,106)( 68,107)( 69,108)( 70,103)
( 71,104)( 72,105)( 73, 91)( 74, 92)( 75, 93)( 76, 97)( 77, 98)( 78, 99)
( 79, 94)( 80, 95)( 81, 96)(109,136)(110,137)(111,138)(112,142)(113,143)
(114,144)(115,139)(116,140)(117,141)(118,154)(119,155)(120,156)(121,160)
(122,161)(123,162)(124,157)(125,158)(126,159)(127,145)(128,146)(129,147)
(130,151)(131,152)(132,153)(133,148)(134,149)(135,150)(166,169)(167,170)
(168,171)(172,181)(173,182)(174,183)(175,187)(176,188)(177,189)(178,184)
(179,185)(180,186)(193,196)(194,197)(195,198)(199,208)(200,209)(201,210)
(202,214)(203,215)(204,216)(205,211)(206,212)(207,213)(217,379)(218,380)
(219,381)(220,385)(221,386)(222,387)(223,382)(224,383)(225,384)(226,397)
(227,398)(228,399)(229,403)(230,404)(231,405)(232,400)(233,401)(234,402)
(235,388)(236,389)(237,390)(238,394)(239,395)(240,396)(241,391)(242,392)
(243,393)(244,406)(245,407)(246,408)(247,412)(248,413)(249,414)(250,409)
(251,410)(252,411)(253,424)(254,425)(255,426)(256,430)(257,431)(258,432)
(259,427)(260,428)(261,429)(262,415)(263,416)(264,417)(265,421)(266,422)
(267,423)(268,418)(269,419)(270,420)(271,325)(272,326)(273,327)(274,331)
(275,332)(276,333)(277,328)(278,329)(279,330)(280,343)(281,344)(282,345)
(283,349)(284,350)(285,351)(286,346)(287,347)(288,348)(289,334)(290,335)
(291,336)(292,340)(293,341)(294,342)(295,337)(296,338)(297,339)(298,352)
(299,353)(300,354)(301,358)(302,359)(303,360)(304,355)(305,356)(306,357)
(307,370)(308,371)(309,372)(310,376)(311,377)(312,378)(313,373)(314,374)
(315,375)(316,361)(317,362)(318,363)(319,367)(320,368)(321,369)(322,364)
(323,365)(324,366);;
s1 := (  1,239)(  2,240)(  3,238)(  4,237)(  5,235)(  6,236)(  7,241)(  8,242)
(  9,243)( 10,230)( 11,231)( 12,229)( 13,228)( 14,226)( 15,227)( 16,232)
( 17,233)( 18,234)( 19,221)( 20,222)( 21,220)( 22,219)( 23,217)( 24,218)
( 25,223)( 26,224)( 27,225)( 28,266)( 29,267)( 30,265)( 31,264)( 32,262)
( 33,263)( 34,268)( 35,269)( 36,270)( 37,257)( 38,258)( 39,256)( 40,255)
( 41,253)( 42,254)( 43,259)( 44,260)( 45,261)( 46,248)( 47,249)( 48,247)
( 49,246)( 50,244)( 51,245)( 52,250)( 53,251)( 54,252)( 55,320)( 56,321)
( 57,319)( 58,318)( 59,316)( 60,317)( 61,322)( 62,323)( 63,324)( 64,311)
( 65,312)( 66,310)( 67,309)( 68,307)( 69,308)( 70,313)( 71,314)( 72,315)
( 73,302)( 74,303)( 75,301)( 76,300)( 77,298)( 78,299)( 79,304)( 80,305)
( 81,306)( 82,293)( 83,294)( 84,292)( 85,291)( 86,289)( 87,290)( 88,295)
( 89,296)( 90,297)( 91,284)( 92,285)( 93,283)( 94,282)( 95,280)( 96,281)
( 97,286)( 98,287)( 99,288)(100,275)(101,276)(102,274)(103,273)(104,271)
(105,272)(106,277)(107,278)(108,279)(109,374)(110,375)(111,373)(112,372)
(113,370)(114,371)(115,376)(116,377)(117,378)(118,365)(119,366)(120,364)
(121,363)(122,361)(123,362)(124,367)(125,368)(126,369)(127,356)(128,357)
(129,355)(130,354)(131,352)(132,353)(133,358)(134,359)(135,360)(136,347)
(137,348)(138,346)(139,345)(140,343)(141,344)(142,349)(143,350)(144,351)
(145,338)(146,339)(147,337)(148,336)(149,334)(150,335)(151,340)(152,341)
(153,342)(154,329)(155,330)(156,328)(157,327)(158,325)(159,326)(160,331)
(161,332)(162,333)(163,401)(164,402)(165,400)(166,399)(167,397)(168,398)
(169,403)(170,404)(171,405)(172,392)(173,393)(174,391)(175,390)(176,388)
(177,389)(178,394)(179,395)(180,396)(181,383)(182,384)(183,382)(184,381)
(185,379)(186,380)(187,385)(188,386)(189,387)(190,428)(191,429)(192,427)
(193,426)(194,424)(195,425)(196,430)(197,431)(198,432)(199,419)(200,420)
(201,418)(202,417)(203,415)(204,416)(205,421)(206,422)(207,423)(208,410)
(209,411)(210,409)(211,408)(212,406)(213,407)(214,412)(215,413)(216,414);;
s2 := (  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 59, 87)( 60, 86)( 61, 88)( 62, 90)
( 63, 89)( 64,100)( 65,102)( 66,101)( 67,103)( 68,105)( 69,104)( 70,106)
( 71,108)( 72,107)( 73, 91)( 74, 93)( 75, 92)( 76, 94)( 77, 96)( 78, 95)
( 79, 97)( 80, 99)( 81, 98)(110,111)(113,114)(116,117)(118,127)(119,129)
(120,128)(121,130)(122,132)(123,131)(124,133)(125,135)(126,134)(137,138)
(140,141)(143,144)(145,154)(146,156)(147,155)(148,157)(149,159)(150,158)
(151,160)(152,162)(153,161)(163,190)(164,192)(165,191)(166,193)(167,195)
(168,194)(169,196)(170,198)(171,197)(172,208)(173,210)(174,209)(175,211)
(176,213)(177,212)(178,214)(179,216)(180,215)(181,199)(182,201)(183,200)
(184,202)(185,204)(186,203)(187,205)(188,207)(189,206)(217,271)(218,273)
(219,272)(220,274)(221,276)(222,275)(223,277)(224,279)(225,278)(226,289)
(227,291)(228,290)(229,292)(230,294)(231,293)(232,295)(233,297)(234,296)
(235,280)(236,282)(237,281)(238,283)(239,285)(240,284)(241,286)(242,288)
(243,287)(244,298)(245,300)(246,299)(247,301)(248,303)(249,302)(250,304)
(251,306)(252,305)(253,316)(254,318)(255,317)(256,319)(257,321)(258,320)
(259,322)(260,324)(261,323)(262,307)(263,309)(264,308)(265,310)(266,312)
(267,311)(268,313)(269,315)(270,314)(325,379)(326,381)(327,380)(328,382)
(329,384)(330,383)(331,385)(332,387)(333,386)(334,397)(335,399)(336,398)
(337,400)(338,402)(339,401)(340,403)(341,405)(342,404)(343,388)(344,390)
(345,389)(346,391)(347,393)(348,392)(349,394)(350,396)(351,395)(352,406)
(353,408)(354,407)(355,409)(356,411)(357,410)(358,412)(359,414)(360,413)
(361,424)(362,426)(363,425)(364,427)(365,429)(366,428)(367,430)(368,432)
(369,431)(370,415)(371,417)(372,416)(373,418)(374,420)(375,419)(376,421)
(377,423)(378,422);;
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*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(432)!(  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)
( 14, 26)( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)
( 37, 46)( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)
( 45, 51)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)( 61, 85)
( 62, 86)( 63, 87)( 64,100)( 65,101)( 66,102)( 67,106)( 68,107)( 69,108)
( 70,103)( 71,104)( 72,105)( 73, 91)( 74, 92)( 75, 93)( 76, 97)( 77, 98)
( 78, 99)( 79, 94)( 80, 95)( 81, 96)(109,136)(110,137)(111,138)(112,142)
(113,143)(114,144)(115,139)(116,140)(117,141)(118,154)(119,155)(120,156)
(121,160)(122,161)(123,162)(124,157)(125,158)(126,159)(127,145)(128,146)
(129,147)(130,151)(131,152)(132,153)(133,148)(134,149)(135,150)(166,169)
(167,170)(168,171)(172,181)(173,182)(174,183)(175,187)(176,188)(177,189)
(178,184)(179,185)(180,186)(193,196)(194,197)(195,198)(199,208)(200,209)
(201,210)(202,214)(203,215)(204,216)(205,211)(206,212)(207,213)(217,379)
(218,380)(219,381)(220,385)(221,386)(222,387)(223,382)(224,383)(225,384)
(226,397)(227,398)(228,399)(229,403)(230,404)(231,405)(232,400)(233,401)
(234,402)(235,388)(236,389)(237,390)(238,394)(239,395)(240,396)(241,391)
(242,392)(243,393)(244,406)(245,407)(246,408)(247,412)(248,413)(249,414)
(250,409)(251,410)(252,411)(253,424)(254,425)(255,426)(256,430)(257,431)
(258,432)(259,427)(260,428)(261,429)(262,415)(263,416)(264,417)(265,421)
(266,422)(267,423)(268,418)(269,419)(270,420)(271,325)(272,326)(273,327)
(274,331)(275,332)(276,333)(277,328)(278,329)(279,330)(280,343)(281,344)
(282,345)(283,349)(284,350)(285,351)(286,346)(287,347)(288,348)(289,334)
(290,335)(291,336)(292,340)(293,341)(294,342)(295,337)(296,338)(297,339)
(298,352)(299,353)(300,354)(301,358)(302,359)(303,360)(304,355)(305,356)
(306,357)(307,370)(308,371)(309,372)(310,376)(311,377)(312,378)(313,373)
(314,374)(315,375)(316,361)(317,362)(318,363)(319,367)(320,368)(321,369)
(322,364)(323,365)(324,366);
s1 := Sym(432)!(  1,239)(  2,240)(  3,238)(  4,237)(  5,235)(  6,236)(  7,241)
(  8,242)(  9,243)( 10,230)( 11,231)( 12,229)( 13,228)( 14,226)( 15,227)
( 16,232)( 17,233)( 18,234)( 19,221)( 20,222)( 21,220)( 22,219)( 23,217)
( 24,218)( 25,223)( 26,224)( 27,225)( 28,266)( 29,267)( 30,265)( 31,264)
( 32,262)( 33,263)( 34,268)( 35,269)( 36,270)( 37,257)( 38,258)( 39,256)
( 40,255)( 41,253)( 42,254)( 43,259)( 44,260)( 45,261)( 46,248)( 47,249)
( 48,247)( 49,246)( 50,244)( 51,245)( 52,250)( 53,251)( 54,252)( 55,320)
( 56,321)( 57,319)( 58,318)( 59,316)( 60,317)( 61,322)( 62,323)( 63,324)
( 64,311)( 65,312)( 66,310)( 67,309)( 68,307)( 69,308)( 70,313)( 71,314)
( 72,315)( 73,302)( 74,303)( 75,301)( 76,300)( 77,298)( 78,299)( 79,304)
( 80,305)( 81,306)( 82,293)( 83,294)( 84,292)( 85,291)( 86,289)( 87,290)
( 88,295)( 89,296)( 90,297)( 91,284)( 92,285)( 93,283)( 94,282)( 95,280)
( 96,281)( 97,286)( 98,287)( 99,288)(100,275)(101,276)(102,274)(103,273)
(104,271)(105,272)(106,277)(107,278)(108,279)(109,374)(110,375)(111,373)
(112,372)(113,370)(114,371)(115,376)(116,377)(117,378)(118,365)(119,366)
(120,364)(121,363)(122,361)(123,362)(124,367)(125,368)(126,369)(127,356)
(128,357)(129,355)(130,354)(131,352)(132,353)(133,358)(134,359)(135,360)
(136,347)(137,348)(138,346)(139,345)(140,343)(141,344)(142,349)(143,350)
(144,351)(145,338)(146,339)(147,337)(148,336)(149,334)(150,335)(151,340)
(152,341)(153,342)(154,329)(155,330)(156,328)(157,327)(158,325)(159,326)
(160,331)(161,332)(162,333)(163,401)(164,402)(165,400)(166,399)(167,397)
(168,398)(169,403)(170,404)(171,405)(172,392)(173,393)(174,391)(175,390)
(176,388)(177,389)(178,394)(179,395)(180,396)(181,383)(182,384)(183,382)
(184,381)(185,379)(186,380)(187,385)(188,386)(189,387)(190,428)(191,429)
(192,427)(193,426)(194,424)(195,425)(196,430)(197,431)(198,432)(199,419)
(200,420)(201,418)(202,417)(203,415)(204,416)(205,421)(206,422)(207,423)
(208,410)(209,411)(210,409)(211,408)(212,406)(213,407)(214,412)(215,413)
(216,414);
s2 := Sym(432)!(  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)
( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)
( 45, 53)( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 59, 87)( 60, 86)( 61, 88)
( 62, 90)( 63, 89)( 64,100)( 65,102)( 66,101)( 67,103)( 68,105)( 69,104)
( 70,106)( 71,108)( 72,107)( 73, 91)( 74, 93)( 75, 92)( 76, 94)( 77, 96)
( 78, 95)( 79, 97)( 80, 99)( 81, 98)(110,111)(113,114)(116,117)(118,127)
(119,129)(120,128)(121,130)(122,132)(123,131)(124,133)(125,135)(126,134)
(137,138)(140,141)(143,144)(145,154)(146,156)(147,155)(148,157)(149,159)
(150,158)(151,160)(152,162)(153,161)(163,190)(164,192)(165,191)(166,193)
(167,195)(168,194)(169,196)(170,198)(171,197)(172,208)(173,210)(174,209)
(175,211)(176,213)(177,212)(178,214)(179,216)(180,215)(181,199)(182,201)
(183,200)(184,202)(185,204)(186,203)(187,205)(188,207)(189,206)(217,271)
(218,273)(219,272)(220,274)(221,276)(222,275)(223,277)(224,279)(225,278)
(226,289)(227,291)(228,290)(229,292)(230,294)(231,293)(232,295)(233,297)
(234,296)(235,280)(236,282)(237,281)(238,283)(239,285)(240,284)(241,286)
(242,288)(243,287)(244,298)(245,300)(246,299)(247,301)(248,303)(249,302)
(250,304)(251,306)(252,305)(253,316)(254,318)(255,317)(256,319)(257,321)
(258,320)(259,322)(260,324)(261,323)(262,307)(263,309)(264,308)(265,310)
(266,312)(267,311)(268,313)(269,315)(270,314)(325,379)(326,381)(327,380)
(328,382)(329,384)(330,383)(331,385)(332,387)(333,386)(334,397)(335,399)
(336,398)(337,400)(338,402)(339,401)(340,403)(341,405)(342,404)(343,388)
(344,390)(345,389)(346,391)(347,393)(348,392)(349,394)(350,396)(351,395)
(352,406)(353,408)(354,407)(355,409)(356,411)(357,410)(358,412)(359,414)
(360,413)(361,424)(362,426)(363,425)(364,427)(365,429)(366,428)(367,430)
(368,432)(369,431)(370,415)(371,417)(372,416)(373,418)(374,420)(375,419)
(376,421)(377,423)(378,422);
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*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*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 >; 
 
References : None.
to this polytope