Polytope of Type {6,156}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,156}*1872c
if this polytope has a name.
Group : SmallGroup(1872,908)
Rank : 3
Schlafli Type : {6,156}
Number of vertices, edges, etc : 6, 468, 156
Order of s₀s₁s₂ : 156
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
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 : {6,78}*936c
   3-fold quotients : {2,156}*624
   4-fold quotients : {6,39}*468
   6-fold quotients : {2,78}*312
   9-fold quotients : {2,52}*208
   12-fold quotients : {2,39}*156
   13-fold quotients : {6,12}*144b
   18-fold quotients : {2,26}*104
   26-fold quotients : {6,6}*72b
   36-fold quotients : {2,13}*52
   39-fold quotients : {2,12}*48
   52-fold quotients : {6,3}*36
   78-fold quotients : {2,6}*24
   117-fold quotients : {2,4}*16
   156-fold quotients : {2,3}*12
   234-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 40, 79)( 41, 80)( 42, 81)( 43, 82)( 44, 83)( 45, 84)( 46, 85)( 47, 86)
( 48, 87)( 49, 88)( 50, 89)( 51, 90)( 52, 91)( 53, 92)( 54, 93)( 55, 94)
( 56, 95)( 57, 96)( 58, 97)( 59, 98)( 60, 99)( 61,100)( 62,101)( 63,102)
( 64,103)( 65,104)( 66,105)( 67,106)( 68,107)( 69,108)( 70,109)( 71,110)
( 72,111)( 73,112)( 74,113)( 75,114)( 76,115)( 77,116)( 78,117)(157,196)
(158,197)(159,198)(160,199)(161,200)(162,201)(163,202)(164,203)(165,204)
(166,205)(167,206)(168,207)(169,208)(170,209)(171,210)(172,211)(173,212)
(174,213)(175,214)(176,215)(177,216)(178,217)(179,218)(180,219)(181,220)
(182,221)(183,222)(184,223)(185,224)(186,225)(187,226)(188,227)(189,228)
(190,229)(191,230)(192,231)(193,232)(194,233)(195,234)(274,313)(275,314)
(276,315)(277,316)(278,317)(279,318)(280,319)(281,320)(282,321)(283,322)
(284,323)(285,324)(286,325)(287,326)(288,327)(289,328)(290,329)(291,330)
(292,331)(293,332)(294,333)(295,334)(296,335)(297,336)(298,337)(299,338)
(300,339)(301,340)(302,341)(303,342)(304,343)(305,344)(306,345)(307,346)
(308,347)(309,348)(310,349)(311,350)(312,351)(391,430)(392,431)(393,432)
(394,433)(395,434)(396,435)(397,436)(398,437)(399,438)(400,439)(401,440)
(402,441)(403,442)(404,443)(405,444)(406,445)(407,446)(408,447)(409,448)
(410,449)(411,450)(412,451)(413,452)(414,453)(415,454)(416,455)(417,456)
(418,457)(419,458)(420,459)(421,460)(422,461)(423,462)(424,463)(425,464)
(426,465)(427,466)(428,467)(429,468);;
s1 := (  1, 40)(  2, 52)(  3, 51)(  4, 50)(  5, 49)(  6, 48)(  7, 47)(  8, 46)
(  9, 45)( 10, 44)( 11, 43)( 12, 42)( 13, 41)( 14, 66)( 15, 78)( 16, 77)
( 17, 76)( 18, 75)( 19, 74)( 20, 73)( 21, 72)( 22, 71)( 23, 70)( 24, 69)
( 25, 68)( 26, 67)( 27, 53)( 28, 65)( 29, 64)( 30, 63)( 31, 62)( 32, 61)
( 33, 60)( 34, 59)( 35, 58)( 36, 57)( 37, 56)( 38, 55)( 39, 54)( 80, 91)
( 81, 90)( 82, 89)( 83, 88)( 84, 87)( 85, 86)( 92,105)( 93,117)( 94,116)
( 95,115)( 96,114)( 97,113)( 98,112)( 99,111)(100,110)(101,109)(102,108)
(103,107)(104,106)(118,157)(119,169)(120,168)(121,167)(122,166)(123,165)
(124,164)(125,163)(126,162)(127,161)(128,160)(129,159)(130,158)(131,183)
(132,195)(133,194)(134,193)(135,192)(136,191)(137,190)(138,189)(139,188)
(140,187)(141,186)(142,185)(143,184)(144,170)(145,182)(146,181)(147,180)
(148,179)(149,178)(150,177)(151,176)(152,175)(153,174)(154,173)(155,172)
(156,171)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(209,222)
(210,234)(211,233)(212,232)(213,231)(214,230)(215,229)(216,228)(217,227)
(218,226)(219,225)(220,224)(221,223)(235,391)(236,403)(237,402)(238,401)
(239,400)(240,399)(241,398)(242,397)(243,396)(244,395)(245,394)(246,393)
(247,392)(248,417)(249,429)(250,428)(251,427)(252,426)(253,425)(254,424)
(255,423)(256,422)(257,421)(258,420)(259,419)(260,418)(261,404)(262,416)
(263,415)(264,414)(265,413)(266,412)(267,411)(268,410)(269,409)(270,408)
(271,407)(272,406)(273,405)(274,352)(275,364)(276,363)(277,362)(278,361)
(279,360)(280,359)(281,358)(282,357)(283,356)(284,355)(285,354)(286,353)
(287,378)(288,390)(289,389)(290,388)(291,387)(292,386)(293,385)(294,384)
(295,383)(296,382)(297,381)(298,380)(299,379)(300,365)(301,377)(302,376)
(303,375)(304,374)(305,373)(306,372)(307,371)(308,370)(309,369)(310,368)
(311,367)(312,366)(313,430)(314,442)(315,441)(316,440)(317,439)(318,438)
(319,437)(320,436)(321,435)(322,434)(323,433)(324,432)(325,431)(326,456)
(327,468)(328,467)(329,466)(330,465)(331,464)(332,463)(333,462)(334,461)
(335,460)(336,459)(337,458)(338,457)(339,443)(340,455)(341,454)(342,453)
(343,452)(344,451)(345,450)(346,449)(347,448)(348,447)(349,446)(350,445)
(351,444);;
s2 := (  1,249)(  2,248)(  3,260)(  4,259)(  5,258)(  6,257)(  7,256)(  8,255)
(  9,254)( 10,253)( 11,252)( 12,251)( 13,250)( 14,236)( 15,235)( 16,247)
( 17,246)( 18,245)( 19,244)( 20,243)( 21,242)( 22,241)( 23,240)( 24,239)
( 25,238)( 26,237)( 27,262)( 28,261)( 29,273)( 30,272)( 31,271)( 32,270)
( 33,269)( 34,268)( 35,267)( 36,266)( 37,265)( 38,264)( 39,263)( 40,327)
( 41,326)( 42,338)( 43,337)( 44,336)( 45,335)( 46,334)( 47,333)( 48,332)
( 49,331)( 50,330)( 51,329)( 52,328)( 53,314)( 54,313)( 55,325)( 56,324)
( 57,323)( 58,322)( 59,321)( 60,320)( 61,319)( 62,318)( 63,317)( 64,316)
( 65,315)( 66,340)( 67,339)( 68,351)( 69,350)( 70,349)( 71,348)( 72,347)
( 73,346)( 74,345)( 75,344)( 76,343)( 77,342)( 78,341)( 79,288)( 80,287)
( 81,299)( 82,298)( 83,297)( 84,296)( 85,295)( 86,294)( 87,293)( 88,292)
( 89,291)( 90,290)( 91,289)( 92,275)( 93,274)( 94,286)( 95,285)( 96,284)
( 97,283)( 98,282)( 99,281)(100,280)(101,279)(102,278)(103,277)(104,276)
(105,301)(106,300)(107,312)(108,311)(109,310)(110,309)(111,308)(112,307)
(113,306)(114,305)(115,304)(116,303)(117,302)(118,366)(119,365)(120,377)
(121,376)(122,375)(123,374)(124,373)(125,372)(126,371)(127,370)(128,369)
(129,368)(130,367)(131,353)(132,352)(133,364)(134,363)(135,362)(136,361)
(137,360)(138,359)(139,358)(140,357)(141,356)(142,355)(143,354)(144,379)
(145,378)(146,390)(147,389)(148,388)(149,387)(150,386)(151,385)(152,384)
(153,383)(154,382)(155,381)(156,380)(157,444)(158,443)(159,455)(160,454)
(161,453)(162,452)(163,451)(164,450)(165,449)(166,448)(167,447)(168,446)
(169,445)(170,431)(171,430)(172,442)(173,441)(174,440)(175,439)(176,438)
(177,437)(178,436)(179,435)(180,434)(181,433)(182,432)(183,457)(184,456)
(185,468)(186,467)(187,466)(188,465)(189,464)(190,463)(191,462)(192,461)
(193,460)(194,459)(195,458)(196,405)(197,404)(198,416)(199,415)(200,414)
(201,413)(202,412)(203,411)(204,410)(205,409)(206,408)(207,407)(208,406)
(209,392)(210,391)(211,403)(212,402)(213,401)(214,400)(215,399)(216,398)
(217,397)(218,396)(219,395)(220,394)(221,393)(222,418)(223,417)(224,429)
(225,428)(226,427)(227,426)(228,425)(229,424)(230,423)(231,422)(232,421)
(233,420)(234,419);;
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*s0*s1*s2*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*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*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*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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(468)!( 40, 79)( 41, 80)( 42, 81)( 43, 82)( 44, 83)( 45, 84)( 46, 85)
( 47, 86)( 48, 87)( 49, 88)( 50, 89)( 51, 90)( 52, 91)( 53, 92)( 54, 93)
( 55, 94)( 56, 95)( 57, 96)( 58, 97)( 59, 98)( 60, 99)( 61,100)( 62,101)
( 63,102)( 64,103)( 65,104)( 66,105)( 67,106)( 68,107)( 69,108)( 70,109)
( 71,110)( 72,111)( 73,112)( 74,113)( 75,114)( 76,115)( 77,116)( 78,117)
(157,196)(158,197)(159,198)(160,199)(161,200)(162,201)(163,202)(164,203)
(165,204)(166,205)(167,206)(168,207)(169,208)(170,209)(171,210)(172,211)
(173,212)(174,213)(175,214)(176,215)(177,216)(178,217)(179,218)(180,219)
(181,220)(182,221)(183,222)(184,223)(185,224)(186,225)(187,226)(188,227)
(189,228)(190,229)(191,230)(192,231)(193,232)(194,233)(195,234)(274,313)
(275,314)(276,315)(277,316)(278,317)(279,318)(280,319)(281,320)(282,321)
(283,322)(284,323)(285,324)(286,325)(287,326)(288,327)(289,328)(290,329)
(291,330)(292,331)(293,332)(294,333)(295,334)(296,335)(297,336)(298,337)
(299,338)(300,339)(301,340)(302,341)(303,342)(304,343)(305,344)(306,345)
(307,346)(308,347)(309,348)(310,349)(311,350)(312,351)(391,430)(392,431)
(393,432)(394,433)(395,434)(396,435)(397,436)(398,437)(399,438)(400,439)
(401,440)(402,441)(403,442)(404,443)(405,444)(406,445)(407,446)(408,447)
(409,448)(410,449)(411,450)(412,451)(413,452)(414,453)(415,454)(416,455)
(417,456)(418,457)(419,458)(420,459)(421,460)(422,461)(423,462)(424,463)
(425,464)(426,465)(427,466)(428,467)(429,468);
s1 := Sym(468)!(  1, 40)(  2, 52)(  3, 51)(  4, 50)(  5, 49)(  6, 48)(  7, 47)
(  8, 46)(  9, 45)( 10, 44)( 11, 43)( 12, 42)( 13, 41)( 14, 66)( 15, 78)
( 16, 77)( 17, 76)( 18, 75)( 19, 74)( 20, 73)( 21, 72)( 22, 71)( 23, 70)
( 24, 69)( 25, 68)( 26, 67)( 27, 53)( 28, 65)( 29, 64)( 30, 63)( 31, 62)
( 32, 61)( 33, 60)( 34, 59)( 35, 58)( 36, 57)( 37, 56)( 38, 55)( 39, 54)
( 80, 91)( 81, 90)( 82, 89)( 83, 88)( 84, 87)( 85, 86)( 92,105)( 93,117)
( 94,116)( 95,115)( 96,114)( 97,113)( 98,112)( 99,111)(100,110)(101,109)
(102,108)(103,107)(104,106)(118,157)(119,169)(120,168)(121,167)(122,166)
(123,165)(124,164)(125,163)(126,162)(127,161)(128,160)(129,159)(130,158)
(131,183)(132,195)(133,194)(134,193)(135,192)(136,191)(137,190)(138,189)
(139,188)(140,187)(141,186)(142,185)(143,184)(144,170)(145,182)(146,181)
(147,180)(148,179)(149,178)(150,177)(151,176)(152,175)(153,174)(154,173)
(155,172)(156,171)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)
(209,222)(210,234)(211,233)(212,232)(213,231)(214,230)(215,229)(216,228)
(217,227)(218,226)(219,225)(220,224)(221,223)(235,391)(236,403)(237,402)
(238,401)(239,400)(240,399)(241,398)(242,397)(243,396)(244,395)(245,394)
(246,393)(247,392)(248,417)(249,429)(250,428)(251,427)(252,426)(253,425)
(254,424)(255,423)(256,422)(257,421)(258,420)(259,419)(260,418)(261,404)
(262,416)(263,415)(264,414)(265,413)(266,412)(267,411)(268,410)(269,409)
(270,408)(271,407)(272,406)(273,405)(274,352)(275,364)(276,363)(277,362)
(278,361)(279,360)(280,359)(281,358)(282,357)(283,356)(284,355)(285,354)
(286,353)(287,378)(288,390)(289,389)(290,388)(291,387)(292,386)(293,385)
(294,384)(295,383)(296,382)(297,381)(298,380)(299,379)(300,365)(301,377)
(302,376)(303,375)(304,374)(305,373)(306,372)(307,371)(308,370)(309,369)
(310,368)(311,367)(312,366)(313,430)(314,442)(315,441)(316,440)(317,439)
(318,438)(319,437)(320,436)(321,435)(322,434)(323,433)(324,432)(325,431)
(326,456)(327,468)(328,467)(329,466)(330,465)(331,464)(332,463)(333,462)
(334,461)(335,460)(336,459)(337,458)(338,457)(339,443)(340,455)(341,454)
(342,453)(343,452)(344,451)(345,450)(346,449)(347,448)(348,447)(349,446)
(350,445)(351,444);
s2 := Sym(468)!(  1,249)(  2,248)(  3,260)(  4,259)(  5,258)(  6,257)(  7,256)
(  8,255)(  9,254)( 10,253)( 11,252)( 12,251)( 13,250)( 14,236)( 15,235)
( 16,247)( 17,246)( 18,245)( 19,244)( 20,243)( 21,242)( 22,241)( 23,240)
( 24,239)( 25,238)( 26,237)( 27,262)( 28,261)( 29,273)( 30,272)( 31,271)
( 32,270)( 33,269)( 34,268)( 35,267)( 36,266)( 37,265)( 38,264)( 39,263)
( 40,327)( 41,326)( 42,338)( 43,337)( 44,336)( 45,335)( 46,334)( 47,333)
( 48,332)( 49,331)( 50,330)( 51,329)( 52,328)( 53,314)( 54,313)( 55,325)
( 56,324)( 57,323)( 58,322)( 59,321)( 60,320)( 61,319)( 62,318)( 63,317)
( 64,316)( 65,315)( 66,340)( 67,339)( 68,351)( 69,350)( 70,349)( 71,348)
( 72,347)( 73,346)( 74,345)( 75,344)( 76,343)( 77,342)( 78,341)( 79,288)
( 80,287)( 81,299)( 82,298)( 83,297)( 84,296)( 85,295)( 86,294)( 87,293)
( 88,292)( 89,291)( 90,290)( 91,289)( 92,275)( 93,274)( 94,286)( 95,285)
( 96,284)( 97,283)( 98,282)( 99,281)(100,280)(101,279)(102,278)(103,277)
(104,276)(105,301)(106,300)(107,312)(108,311)(109,310)(110,309)(111,308)
(112,307)(113,306)(114,305)(115,304)(116,303)(117,302)(118,366)(119,365)
(120,377)(121,376)(122,375)(123,374)(124,373)(125,372)(126,371)(127,370)
(128,369)(129,368)(130,367)(131,353)(132,352)(133,364)(134,363)(135,362)
(136,361)(137,360)(138,359)(139,358)(140,357)(141,356)(142,355)(143,354)
(144,379)(145,378)(146,390)(147,389)(148,388)(149,387)(150,386)(151,385)
(152,384)(153,383)(154,382)(155,381)(156,380)(157,444)(158,443)(159,455)
(160,454)(161,453)(162,452)(163,451)(164,450)(165,449)(166,448)(167,447)
(168,446)(169,445)(170,431)(171,430)(172,442)(173,441)(174,440)(175,439)
(176,438)(177,437)(178,436)(179,435)(180,434)(181,433)(182,432)(183,457)
(184,456)(185,468)(186,467)(187,466)(188,465)(189,464)(190,463)(191,462)
(192,461)(193,460)(194,459)(195,458)(196,405)(197,404)(198,416)(199,415)
(200,414)(201,413)(202,412)(203,411)(204,410)(205,409)(206,408)(207,407)
(208,406)(209,392)(210,391)(211,403)(212,402)(213,401)(214,400)(215,399)
(216,398)(217,397)(218,396)(219,395)(220,394)(221,393)(222,418)(223,417)
(224,429)(225,428)(226,427)(227,426)(228,425)(229,424)(230,423)(231,422)
(232,421)(233,420)(234,419);
poly := sub<Sym(468)|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*s0*s1*s2*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*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*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*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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope