Polytope of Type {144,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {144,6}*1728a
Also Known As : {144,6|2}. if this polytope has another name.
Group : SmallGroup(1728,3030)
Rank : 3
Schlafli Type : {144,6}
Number of vertices, edges, etc : 144, 432, 6
Order of s₀s₁s₂ : 144
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
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 : {72,6}*864a
   3-fold quotients : {144,2}*576, {48,6}*576a
   4-fold quotients : {36,6}*432a
   6-fold quotients : {72,2}*288, {24,6}*288a
   8-fold quotients : {18,6}*216a
   9-fold quotients : {48,2}*192, {16,6}*192
   12-fold quotients : {36,2}*144, {12,6}*144a
   18-fold quotients : {24,2}*96, {8,6}*96
   24-fold quotients : {18,2}*72, {6,6}*72a
   27-fold quotients : {16,2}*64
   36-fold quotients : {12,2}*48, {4,6}*48a
   48-fold quotients : {9,2}*36
   54-fold quotients : {8,2}*32
   72-fold quotients : {2,6}*24, {6,2}*24
   108-fold quotients : {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 := (  2,  3)(  4,  9)(  5,  8)(  6,  7)( 11, 12)( 13, 18)( 14, 17)( 15, 16)
( 20, 21)( 22, 27)( 23, 26)( 24, 25)( 29, 30)( 31, 36)( 32, 35)( 33, 34)
( 38, 39)( 40, 45)( 41, 44)( 42, 43)( 47, 48)( 49, 54)( 50, 53)( 51, 52)
( 55, 82)( 56, 84)( 57, 83)( 58, 90)( 59, 89)( 60, 88)( 61, 87)( 62, 86)
( 63, 85)( 64, 91)( 65, 93)( 66, 92)( 67, 99)( 68, 98)( 69, 97)( 70, 96)
( 71, 95)( 72, 94)( 73,100)( 74,102)( 75,101)( 76,108)( 77,107)( 78,106)
( 79,105)( 80,104)( 81,103)(109,163)(110,165)(111,164)(112,171)(113,170)
(114,169)(115,168)(116,167)(117,166)(118,172)(119,174)(120,173)(121,180)
(122,179)(123,178)(124,177)(125,176)(126,175)(127,181)(128,183)(129,182)
(130,189)(131,188)(132,187)(133,186)(134,185)(135,184)(136,190)(137,192)
(138,191)(139,198)(140,197)(141,196)(142,195)(143,194)(144,193)(145,199)
(146,201)(147,200)(148,207)(149,206)(150,205)(151,204)(152,203)(153,202)
(154,208)(155,210)(156,209)(157,216)(158,215)(159,214)(160,213)(161,212)
(162,211)(217,325)(218,327)(219,326)(220,333)(221,332)(222,331)(223,330)
(224,329)(225,328)(226,334)(227,336)(228,335)(229,342)(230,341)(231,340)
(232,339)(233,338)(234,337)(235,343)(236,345)(237,344)(238,351)(239,350)
(240,349)(241,348)(242,347)(243,346)(244,352)(245,354)(246,353)(247,360)
(248,359)(249,358)(250,357)(251,356)(252,355)(253,361)(254,363)(255,362)
(256,369)(257,368)(258,367)(259,366)(260,365)(261,364)(262,370)(263,372)
(264,371)(265,378)(266,377)(267,376)(268,375)(269,374)(270,373)(271,406)
(272,408)(273,407)(274,414)(275,413)(276,412)(277,411)(278,410)(279,409)
(280,415)(281,417)(282,416)(283,423)(284,422)(285,421)(286,420)(287,419)
(288,418)(289,424)(290,426)(291,425)(292,432)(293,431)(294,430)(295,429)
(296,428)(297,427)(298,379)(299,381)(300,380)(301,387)(302,386)(303,385)
(304,384)(305,383)(306,382)(307,388)(308,390)(309,389)(310,396)(311,395)
(312,394)(313,393)(314,392)(315,391)(316,397)(317,399)(318,398)(319,405)
(320,404)(321,403)(322,402)(323,401)(324,400);;
s1 := (  1,220)(  2,222)(  3,221)(  4,217)(  5,219)(  6,218)(  7,225)(  8,224)
(  9,223)( 10,238)( 11,240)( 12,239)( 13,235)( 14,237)( 15,236)( 16,243)
( 17,242)( 18,241)( 19,229)( 20,231)( 21,230)( 22,226)( 23,228)( 24,227)
( 25,234)( 26,233)( 27,232)( 28,247)( 29,249)( 30,248)( 31,244)( 32,246)
( 33,245)( 34,252)( 35,251)( 36,250)( 37,265)( 38,267)( 39,266)( 40,262)
( 41,264)( 42,263)( 43,270)( 44,269)( 45,268)( 46,256)( 47,258)( 48,257)
( 49,253)( 50,255)( 51,254)( 52,261)( 53,260)( 54,259)( 55,301)( 56,303)
( 57,302)( 58,298)( 59,300)( 60,299)( 61,306)( 62,305)( 63,304)( 64,319)
( 65,321)( 66,320)( 67,316)( 68,318)( 69,317)( 70,324)( 71,323)( 72,322)
( 73,310)( 74,312)( 75,311)( 76,307)( 77,309)( 78,308)( 79,315)( 80,314)
( 81,313)( 82,274)( 83,276)( 84,275)( 85,271)( 86,273)( 87,272)( 88,279)
( 89,278)( 90,277)( 91,292)( 92,294)( 93,293)( 94,289)( 95,291)( 96,290)
( 97,297)( 98,296)( 99,295)(100,283)(101,285)(102,284)(103,280)(104,282)
(105,281)(106,288)(107,287)(108,286)(109,382)(110,384)(111,383)(112,379)
(113,381)(114,380)(115,387)(116,386)(117,385)(118,400)(119,402)(120,401)
(121,397)(122,399)(123,398)(124,405)(125,404)(126,403)(127,391)(128,393)
(129,392)(130,388)(131,390)(132,389)(133,396)(134,395)(135,394)(136,409)
(137,411)(138,410)(139,406)(140,408)(141,407)(142,414)(143,413)(144,412)
(145,427)(146,429)(147,428)(148,424)(149,426)(150,425)(151,432)(152,431)
(153,430)(154,418)(155,420)(156,419)(157,415)(158,417)(159,416)(160,423)
(161,422)(162,421)(163,328)(164,330)(165,329)(166,325)(167,327)(168,326)
(169,333)(170,332)(171,331)(172,346)(173,348)(174,347)(175,343)(176,345)
(177,344)(178,351)(179,350)(180,349)(181,337)(182,339)(183,338)(184,334)
(185,336)(186,335)(187,342)(188,341)(189,340)(190,355)(191,357)(192,356)
(193,352)(194,354)(195,353)(196,360)(197,359)(198,358)(199,373)(200,375)
(201,374)(202,370)(203,372)(204,371)(205,378)(206,377)(207,376)(208,364)
(209,366)(210,365)(211,361)(212,363)(213,362)(214,369)(215,368)(216,367);;
s2 := (  1, 10)(  2, 11)(  3, 12)(  4, 13)(  5, 14)(  6, 15)(  7, 16)(  8, 17)
(  9, 18)( 28, 37)( 29, 38)( 30, 39)( 31, 40)( 32, 41)( 33, 42)( 34, 43)
( 35, 44)( 36, 45)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)( 60, 69)
( 61, 70)( 62, 71)( 63, 72)( 82, 91)( 83, 92)( 84, 93)( 85, 94)( 86, 95)
( 87, 96)( 88, 97)( 89, 98)( 90, 99)(109,118)(110,119)(111,120)(112,121)
(113,122)(114,123)(115,124)(116,125)(117,126)(136,145)(137,146)(138,147)
(139,148)(140,149)(141,150)(142,151)(143,152)(144,153)(163,172)(164,173)
(165,174)(166,175)(167,176)(168,177)(169,178)(170,179)(171,180)(190,199)
(191,200)(192,201)(193,202)(194,203)(195,204)(196,205)(197,206)(198,207)
(217,226)(218,227)(219,228)(220,229)(221,230)(222,231)(223,232)(224,233)
(225,234)(244,253)(245,254)(246,255)(247,256)(248,257)(249,258)(250,259)
(251,260)(252,261)(271,280)(272,281)(273,282)(274,283)(275,284)(276,285)
(277,286)(278,287)(279,288)(298,307)(299,308)(300,309)(301,310)(302,311)
(303,312)(304,313)(305,314)(306,315)(325,334)(326,335)(327,336)(328,337)
(329,338)(330,339)(331,340)(332,341)(333,342)(352,361)(353,362)(354,363)
(355,364)(356,365)(357,366)(358,367)(359,368)(360,369)(379,388)(380,389)
(381,390)(382,391)(383,392)(384,393)(385,394)(386,395)(387,396)(406,415)
(407,416)(408,417)(409,418)(410,419)(411,420)(412,421)(413,422)(414,423);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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*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*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*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*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*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*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*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*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*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*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope