Polytope of Type {12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*1728f
if this polytope has a name.
Group : SmallGroup(1728,30326)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 144, 432, 72
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 12
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 : {12,6}*864a
   3-fold quotients : {12,6}*576f
   4-fold quotients : {12,6}*432c, {12,3}*432
   6-fold quotients : {12,6}*288b
   8-fold quotients : {6,6}*216a
   9-fold quotients : {4,6}*192b
   12-fold quotients : {12,6}*144c, {12,3}*144
   16-fold quotients : {6,3}*108
   18-fold quotients : {4,6}*96
   24-fold quotients : {6,6}*72b
   36-fold quotients : {4,6}*48a, {4,3}*48, {4,6}*48b, {4,6}*48c
   48-fold quotients : {6,3}*36
   72-fold quotients : {4,3}*24, {2,6}*24
   108-fold quotients : {4,2}*16
   144-fold quotients : {2,3}*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, 11)(  6, 12)(  7,  9)(  8, 10)( 13, 27)( 14, 28)
( 15, 25)( 16, 26)( 17, 35)( 18, 36)( 19, 33)( 20, 34)( 21, 31)( 22, 32)
( 23, 29)( 24, 30)( 37, 39)( 38, 40)( 41, 47)( 42, 48)( 43, 45)( 44, 46)
( 49, 63)( 50, 64)( 51, 61)( 52, 62)( 53, 71)( 54, 72)( 55, 69)( 56, 70)
( 57, 67)( 58, 68)( 59, 65)( 60, 66)( 73, 75)( 74, 76)( 77, 83)( 78, 84)
( 79, 81)( 80, 82)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,107)( 90,108)
( 91,105)( 92,106)( 93,103)( 94,104)( 95,101)( 96,102)(109,111)(110,112)
(113,119)(114,120)(115,117)(116,118)(121,135)(122,136)(123,133)(124,134)
(125,143)(126,144)(127,141)(128,142)(129,139)(130,140)(131,137)(132,138)
(145,147)(146,148)(149,155)(150,156)(151,153)(152,154)(157,171)(158,172)
(159,169)(160,170)(161,179)(162,180)(163,177)(164,178)(165,175)(166,176)
(167,173)(168,174)(181,183)(182,184)(185,191)(186,192)(187,189)(188,190)
(193,207)(194,208)(195,205)(196,206)(197,215)(198,216)(199,213)(200,214)
(201,211)(202,212)(203,209)(204,210)(217,327)(218,328)(219,325)(220,326)
(221,335)(222,336)(223,333)(224,334)(225,331)(226,332)(227,329)(228,330)
(229,351)(230,352)(231,349)(232,350)(233,359)(234,360)(235,357)(236,358)
(237,355)(238,356)(239,353)(240,354)(241,339)(242,340)(243,337)(244,338)
(245,347)(246,348)(247,345)(248,346)(249,343)(250,344)(251,341)(252,342)
(253,363)(254,364)(255,361)(256,362)(257,371)(258,372)(259,369)(260,370)
(261,367)(262,368)(263,365)(264,366)(265,387)(266,388)(267,385)(268,386)
(269,395)(270,396)(271,393)(272,394)(273,391)(274,392)(275,389)(276,390)
(277,375)(278,376)(279,373)(280,374)(281,383)(282,384)(283,381)(284,382)
(285,379)(286,380)(287,377)(288,378)(289,399)(290,400)(291,397)(292,398)
(293,407)(294,408)(295,405)(296,406)(297,403)(298,404)(299,401)(300,402)
(301,423)(302,424)(303,421)(304,422)(305,431)(306,432)(307,429)(308,430)
(309,427)(310,428)(311,425)(312,426)(313,411)(314,412)(315,409)(316,410)
(317,419)(318,420)(319,417)(320,418)(321,415)(322,416)(323,413)(324,414);;
s1 := (  1,229)(  2,230)(  3,232)(  4,231)(  5,233)(  6,234)(  7,236)(  8,235)
(  9,237)( 10,238)( 11,240)( 12,239)( 13,217)( 14,218)( 15,220)( 16,219)
( 17,221)( 18,222)( 19,224)( 20,223)( 21,225)( 22,226)( 23,228)( 24,227)
( 25,241)( 26,242)( 27,244)( 28,243)( 29,245)( 30,246)( 31,248)( 32,247)
( 33,249)( 34,250)( 35,252)( 36,251)( 37,301)( 38,302)( 39,304)( 40,303)
( 41,305)( 42,306)( 43,308)( 44,307)( 45,309)( 46,310)( 47,312)( 48,311)
( 49,289)( 50,290)( 51,292)( 52,291)( 53,293)( 54,294)( 55,296)( 56,295)
( 57,297)( 58,298)( 59,300)( 60,299)( 61,313)( 62,314)( 63,316)( 64,315)
( 65,317)( 66,318)( 67,320)( 68,319)( 69,321)( 70,322)( 71,324)( 72,323)
( 73,265)( 74,266)( 75,268)( 76,267)( 77,269)( 78,270)( 79,272)( 80,271)
( 81,273)( 82,274)( 83,276)( 84,275)( 85,253)( 86,254)( 87,256)( 88,255)
( 89,257)( 90,258)( 91,260)( 92,259)( 93,261)( 94,262)( 95,264)( 96,263)
( 97,277)( 98,278)( 99,280)(100,279)(101,281)(102,282)(103,284)(104,283)
(105,285)(106,286)(107,288)(108,287)(109,337)(110,338)(111,340)(112,339)
(113,341)(114,342)(115,344)(116,343)(117,345)(118,346)(119,348)(120,347)
(121,325)(122,326)(123,328)(124,327)(125,329)(126,330)(127,332)(128,331)
(129,333)(130,334)(131,336)(132,335)(133,349)(134,350)(135,352)(136,351)
(137,353)(138,354)(139,356)(140,355)(141,357)(142,358)(143,360)(144,359)
(145,409)(146,410)(147,412)(148,411)(149,413)(150,414)(151,416)(152,415)
(153,417)(154,418)(155,420)(156,419)(157,397)(158,398)(159,400)(160,399)
(161,401)(162,402)(163,404)(164,403)(165,405)(166,406)(167,408)(168,407)
(169,421)(170,422)(171,424)(172,423)(173,425)(174,426)(175,428)(176,427)
(177,429)(178,430)(179,432)(180,431)(181,373)(182,374)(183,376)(184,375)
(185,377)(186,378)(187,380)(188,379)(189,381)(190,382)(191,384)(192,383)
(193,361)(194,362)(195,364)(196,363)(197,365)(198,366)(199,368)(200,367)
(201,369)(202,370)(203,372)(204,371)(205,385)(206,386)(207,388)(208,387)
(209,389)(210,390)(211,392)(212,391)(213,393)(214,394)(215,396)(216,395);;
s2 := (  1, 37)(  2, 40)(  3, 39)(  4, 38)(  5, 41)(  6, 44)(  7, 43)(  8, 42)
(  9, 45)( 10, 48)( 11, 47)( 12, 46)( 13, 69)( 14, 72)( 15, 71)( 16, 70)
( 17, 61)( 18, 64)( 19, 63)( 20, 62)( 21, 65)( 22, 68)( 23, 67)( 24, 66)
( 25, 53)( 26, 56)( 27, 55)( 28, 54)( 29, 57)( 30, 60)( 31, 59)( 32, 58)
( 33, 49)( 34, 52)( 35, 51)( 36, 50)( 74, 76)( 78, 80)( 82, 84)( 85,105)
( 86,108)( 87,107)( 88,106)( 89, 97)( 90,100)( 91, 99)( 92, 98)( 93,101)
( 94,104)( 95,103)( 96,102)(109,145)(110,148)(111,147)(112,146)(113,149)
(114,152)(115,151)(116,150)(117,153)(118,156)(119,155)(120,154)(121,177)
(122,180)(123,179)(124,178)(125,169)(126,172)(127,171)(128,170)(129,173)
(130,176)(131,175)(132,174)(133,161)(134,164)(135,163)(136,162)(137,165)
(138,168)(139,167)(140,166)(141,157)(142,160)(143,159)(144,158)(182,184)
(186,188)(190,192)(193,213)(194,216)(195,215)(196,214)(197,205)(198,208)
(199,207)(200,206)(201,209)(202,212)(203,211)(204,210)(217,253)(218,256)
(219,255)(220,254)(221,257)(222,260)(223,259)(224,258)(225,261)(226,264)
(227,263)(228,262)(229,285)(230,288)(231,287)(232,286)(233,277)(234,280)
(235,279)(236,278)(237,281)(238,284)(239,283)(240,282)(241,269)(242,272)
(243,271)(244,270)(245,273)(246,276)(247,275)(248,274)(249,265)(250,268)
(251,267)(252,266)(290,292)(294,296)(298,300)(301,321)(302,324)(303,323)
(304,322)(305,313)(306,316)(307,315)(308,314)(309,317)(310,320)(311,319)
(312,318)(325,361)(326,364)(327,363)(328,362)(329,365)(330,368)(331,367)
(332,366)(333,369)(334,372)(335,371)(336,370)(337,393)(338,396)(339,395)
(340,394)(341,385)(342,388)(343,387)(344,386)(345,389)(346,392)(347,391)
(348,390)(349,377)(350,380)(351,379)(352,378)(353,381)(354,384)(355,383)
(356,382)(357,373)(358,376)(359,375)(360,374)(398,400)(402,404)(406,408)
(409,429)(410,432)(411,431)(412,430)(413,421)(414,424)(415,423)(416,422)
(417,425)(418,428)(419,427)(420,426);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope