Polytope of Type {156,6}

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

s0 := (  2, 13)(  3, 12)(  4, 11)(  5, 10)(  6,  9)(  7,  8)( 14, 27)( 15, 39)
( 16, 38)( 17, 37)( 18, 36)( 19, 35)( 20, 34)( 21, 33)( 22, 32)( 23, 31)
( 24, 30)( 25, 29)( 26, 28)( 41, 52)( 42, 51)( 43, 50)( 44, 49)( 45, 48)
( 46, 47)( 53, 66)( 54, 78)( 55, 77)( 56, 76)( 57, 75)( 58, 74)( 59, 73)
( 60, 72)( 61, 71)( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 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)(119,130)(120,129)(121,128)(122,127)(123,126)(124,125)(131,144)
(132,156)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)
(140,148)(141,147)(142,146)(143,145)(158,169)(159,168)(160,167)(161,166)
(162,165)(163,164)(170,183)(171,195)(172,194)(173,193)(174,192)(175,191)
(176,190)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184)(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)(235,352)(236,364)(237,363)(238,362)(239,361)(240,360)
(241,359)(242,358)(243,357)(244,356)(245,355)(246,354)(247,353)(248,378)
(249,390)(250,389)(251,388)(252,387)(253,386)(254,385)(255,384)(256,383)
(257,382)(258,381)(259,380)(260,379)(261,365)(262,377)(263,376)(264,375)
(265,374)(266,373)(267,372)(268,371)(269,370)(270,369)(271,368)(272,367)
(273,366)(274,391)(275,403)(276,402)(277,401)(278,400)(279,399)(280,398)
(281,397)(282,396)(283,395)(284,394)(285,393)(286,392)(287,417)(288,429)
(289,428)(290,427)(291,426)(292,425)(293,424)(294,423)(295,422)(296,421)
(297,420)(298,419)(299,418)(300,404)(301,416)(302,415)(303,414)(304,413)
(305,412)(306,411)(307,410)(308,409)(309,408)(310,407)(311,406)(312,405)
(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);;
s1 := (  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);;
s2 := (  1, 40)(  2, 41)(  3, 42)(  4, 43)(  5, 44)(  6, 45)(  7, 46)(  8, 47)
(  9, 48)( 10, 49)( 11, 50)( 12, 51)( 13, 52)( 14, 53)( 15, 54)( 16, 55)
( 17, 56)( 18, 57)( 19, 58)( 20, 59)( 21, 60)( 22, 61)( 23, 62)( 24, 63)
( 25, 64)( 26, 65)( 27, 66)( 28, 67)( 29, 68)( 30, 69)( 31, 70)( 32, 71)
( 33, 72)( 34, 73)( 35, 74)( 36, 75)( 37, 76)( 38, 77)( 39, 78)(118,157)
(119,158)(120,159)(121,160)(122,161)(123,162)(124,163)(125,164)(126,165)
(127,166)(128,167)(129,168)(130,169)(131,170)(132,171)(133,172)(134,173)
(135,174)(136,175)(137,176)(138,177)(139,178)(140,179)(141,180)(142,181)
(143,182)(144,183)(145,184)(146,185)(147,186)(148,187)(149,188)(150,189)
(151,190)(152,191)(153,192)(154,193)(155,194)(156,195)(235,274)(236,275)
(237,276)(238,277)(239,278)(240,279)(241,280)(242,281)(243,282)(244,283)
(245,284)(246,285)(247,286)(248,287)(249,288)(250,289)(251,290)(252,291)
(253,292)(254,293)(255,294)(256,295)(257,296)(258,297)(259,298)(260,299)
(261,300)(262,301)(263,302)(264,303)(265,304)(266,305)(267,306)(268,307)
(269,308)(270,309)(271,310)(272,311)(273,312)(352,391)(353,392)(354,393)
(355,394)(356,395)(357,396)(358,397)(359,398)(360,399)(361,400)(362,401)
(363,402)(364,403)(365,404)(366,405)(367,406)(368,407)(369,408)(370,409)
(371,410)(372,411)(373,412)(374,413)(375,414)(376,415)(377,416)(378,417)
(379,418)(380,419)(381,420)(382,421)(383,422)(384,423)(385,424)(386,425)
(387,426)(388,427)(389,428)(390,429);;
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*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(468)!(  2, 13)(  3, 12)(  4, 11)(  5, 10)(  6,  9)(  7,  8)( 14, 27)
( 15, 39)( 16, 38)( 17, 37)( 18, 36)( 19, 35)( 20, 34)( 21, 33)( 22, 32)
( 23, 31)( 24, 30)( 25, 29)( 26, 28)( 41, 52)( 42, 51)( 43, 50)( 44, 49)
( 45, 48)( 46, 47)( 53, 66)( 54, 78)( 55, 77)( 56, 76)( 57, 75)( 58, 74)
( 59, 73)( 60, 72)( 61, 71)( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 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)(119,130)(120,129)(121,128)(122,127)(123,126)(124,125)
(131,144)(132,156)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)
(139,149)(140,148)(141,147)(142,146)(143,145)(158,169)(159,168)(160,167)
(161,166)(162,165)(163,164)(170,183)(171,195)(172,194)(173,193)(174,192)
(175,191)(176,190)(177,189)(178,188)(179,187)(180,186)(181,185)(182,184)
(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)(235,352)(236,364)(237,363)(238,362)(239,361)
(240,360)(241,359)(242,358)(243,357)(244,356)(245,355)(246,354)(247,353)
(248,378)(249,390)(250,389)(251,388)(252,387)(253,386)(254,385)(255,384)
(256,383)(257,382)(258,381)(259,380)(260,379)(261,365)(262,377)(263,376)
(264,375)(265,374)(266,373)(267,372)(268,371)(269,370)(270,369)(271,368)
(272,367)(273,366)(274,391)(275,403)(276,402)(277,401)(278,400)(279,399)
(280,398)(281,397)(282,396)(283,395)(284,394)(285,393)(286,392)(287,417)
(288,429)(289,428)(290,427)(291,426)(292,425)(293,424)(294,423)(295,422)
(296,421)(297,420)(298,419)(299,418)(300,404)(301,416)(302,415)(303,414)
(304,413)(305,412)(306,411)(307,410)(308,409)(309,408)(310,407)(311,406)
(312,405)(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);
s1 := 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);
s2 := Sym(468)!(  1, 40)(  2, 41)(  3, 42)(  4, 43)(  5, 44)(  6, 45)(  7, 46)
(  8, 47)(  9, 48)( 10, 49)( 11, 50)( 12, 51)( 13, 52)( 14, 53)( 15, 54)
( 16, 55)( 17, 56)( 18, 57)( 19, 58)( 20, 59)( 21, 60)( 22, 61)( 23, 62)
( 24, 63)( 25, 64)( 26, 65)( 27, 66)( 28, 67)( 29, 68)( 30, 69)( 31, 70)
( 32, 71)( 33, 72)( 34, 73)( 35, 74)( 36, 75)( 37, 76)( 38, 77)( 39, 78)
(118,157)(119,158)(120,159)(121,160)(122,161)(123,162)(124,163)(125,164)
(126,165)(127,166)(128,167)(129,168)(130,169)(131,170)(132,171)(133,172)
(134,173)(135,174)(136,175)(137,176)(138,177)(139,178)(140,179)(141,180)
(142,181)(143,182)(144,183)(145,184)(146,185)(147,186)(148,187)(149,188)
(150,189)(151,190)(152,191)(153,192)(154,193)(155,194)(156,195)(235,274)
(236,275)(237,276)(238,277)(239,278)(240,279)(241,280)(242,281)(243,282)
(244,283)(245,284)(246,285)(247,286)(248,287)(249,288)(250,289)(251,290)
(252,291)(253,292)(254,293)(255,294)(256,295)(257,296)(258,297)(259,298)
(260,299)(261,300)(262,301)(263,302)(264,303)(265,304)(266,305)(267,306)
(268,307)(269,308)(270,309)(271,310)(272,311)(273,312)(352,391)(353,392)
(354,393)(355,394)(356,395)(357,396)(358,397)(359,398)(360,399)(361,400)
(362,401)(363,402)(364,403)(365,404)(366,405)(367,406)(368,407)(369,408)
(370,409)(371,410)(372,411)(373,412)(374,413)(375,414)(376,415)(377,416)
(378,417)(379,418)(380,419)(381,420)(382,421)(383,422)(384,423)(385,424)
(386,425)(387,426)(388,427)(389,428)(390,429);
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, 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*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