Polytope of Type {6,156}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,156}*1872b
Also Known As : {6,156|2}. if this polytope has another 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 : 2
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}*936b
3-fold quotients : {6,52}*624a, {2,156}*624
6-fold quotients : {6,26}*312, {2,78}*312
9-fold quotients : {2,52}*208
12-fold quotients : {2,39}*156
13-fold quotients : {6,12}*144a
18-fold quotients : {2,26}*104
26-fold quotients : {6,6}*72a
36-fold quotients : {2,13}*52
39-fold quotients : {2,12}*48, {6,4}*48a
78-fold quotients : {2,6}*24, {6,2}*24
117-fold quotients : {2,4}*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 := ( 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)( 40,288)
( 41,287)( 42,299)( 43,298)( 44,297)( 45,296)( 46,295)( 47,294)( 48,293)
( 49,292)( 50,291)( 51,290)( 52,289)( 53,275)( 54,274)( 55,286)( 56,285)
( 57,284)( 58,283)( 59,282)( 60,281)( 61,280)( 62,279)( 63,278)( 64,277)
( 65,276)( 66,301)( 67,300)( 68,312)( 69,311)( 70,310)( 71,309)( 72,308)
( 73,307)( 74,306)( 75,305)( 76,304)( 77,303)( 78,302)( 79,327)( 80,326)
( 81,338)( 82,337)( 83,336)( 84,335)( 85,334)( 86,333)( 87,332)( 88,331)
( 89,330)( 90,329)( 91,328)( 92,314)( 93,313)( 94,325)( 95,324)( 96,323)
( 97,322)( 98,321)( 99,320)(100,319)(101,318)(102,317)(103,316)(104,315)
(105,340)(106,339)(107,351)(108,350)(109,349)(110,348)(111,347)(112,346)
(113,345)(114,344)(115,343)(116,342)(117,341)(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)(157,405)(158,404)(159,416)(160,415)
(161,414)(162,413)(163,412)(164,411)(165,410)(166,409)(167,408)(168,407)
(169,406)(170,392)(171,391)(172,403)(173,402)(174,401)(175,400)(176,399)
(177,398)(178,397)(179,396)(180,395)(181,394)(182,393)(183,418)(184,417)
(185,429)(186,428)(187,427)(188,426)(189,425)(190,424)(191,423)(192,422)
(193,421)(194,420)(195,419)(196,444)(197,443)(198,455)(199,454)(200,453)
(201,452)(202,451)(203,450)(204,449)(205,448)(206,447)(207,446)(208,445)
(209,431)(210,430)(211,442)(212,441)(213,440)(214,439)(215,438)(216,437)
(217,436)(218,435)(219,434)(220,433)(221,432)(222,457)(223,456)(224,468)
(225,467)(226,466)(227,465)(228,464)(229,463)(230,462)(231,461)(232,460)
(233,459)(234,458);;
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,
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)
( 40,288)( 41,287)( 42,299)( 43,298)( 44,297)( 45,296)( 46,295)( 47,294)
( 48,293)( 49,292)( 50,291)( 51,290)( 52,289)( 53,275)( 54,274)( 55,286)
( 56,285)( 57,284)( 58,283)( 59,282)( 60,281)( 61,280)( 62,279)( 63,278)
( 64,277)( 65,276)( 66,301)( 67,300)( 68,312)( 69,311)( 70,310)( 71,309)
( 72,308)( 73,307)( 74,306)( 75,305)( 76,304)( 77,303)( 78,302)( 79,327)
( 80,326)( 81,338)( 82,337)( 83,336)( 84,335)( 85,334)( 86,333)( 87,332)
( 88,331)( 89,330)( 90,329)( 91,328)( 92,314)( 93,313)( 94,325)( 95,324)
( 96,323)( 97,322)( 98,321)( 99,320)(100,319)(101,318)(102,317)(103,316)
(104,315)(105,340)(106,339)(107,351)(108,350)(109,349)(110,348)(111,347)
(112,346)(113,345)(114,344)(115,343)(116,342)(117,341)(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)(157,405)(158,404)(159,416)
(160,415)(161,414)(162,413)(163,412)(164,411)(165,410)(166,409)(167,408)
(168,407)(169,406)(170,392)(171,391)(172,403)(173,402)(174,401)(175,400)
(176,399)(177,398)(178,397)(179,396)(180,395)(181,394)(182,393)(183,418)
(184,417)(185,429)(186,428)(187,427)(188,426)(189,425)(190,424)(191,423)
(192,422)(193,421)(194,420)(195,419)(196,444)(197,443)(198,455)(199,454)
(200,453)(201,452)(202,451)(203,450)(204,449)(205,448)(206,447)(207,446)
(208,445)(209,431)(210,430)(211,442)(212,441)(213,440)(214,439)(215,438)
(216,437)(217,436)(218,435)(219,434)(220,433)(221,432)(222,457)(223,456)
(224,468)(225,467)(226,466)(227,465)(228,464)(229,463)(230,462)(231,461)
(232,460)(233,459)(234,458);
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,
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