Polytope of Type {138,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {138,6}*1656b
Also Known As : {138,6|2}. if this polytope has another name.
Group : SmallGroup(1656,129)
Rank : 3
Schlafli Type : {138,6}
Number of vertices, edges, etc : 138, 414, 6
Order of s0s1s2 : 138
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) :
3-fold quotients : {46,6}*552, {138,2}*552
6-fold quotients : {69,2}*276
9-fold quotients : {46,2}*184
18-fold quotients : {23,2}*92
23-fold quotients : {6,6}*72a
69-fold quotients : {2,6}*24, {6,2}*24
138-fold quotients : {2,3}*12, {3,2}*12
207-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 23)( 3, 22)( 4, 21)( 5, 20)( 6, 19)( 7, 18)( 8, 17)( 9, 16)
( 10, 15)( 11, 14)( 12, 13)( 24, 47)( 25, 69)( 26, 68)( 27, 67)( 28, 66)
( 29, 65)( 30, 64)( 31, 63)( 32, 62)( 33, 61)( 34, 60)( 35, 59)( 36, 58)
( 37, 57)( 38, 56)( 39, 55)( 40, 54)( 41, 53)( 42, 52)( 43, 51)( 44, 50)
( 45, 49)( 46, 48)( 71, 92)( 72, 91)( 73, 90)( 74, 89)( 75, 88)( 76, 87)
( 77, 86)( 78, 85)( 79, 84)( 80, 83)( 81, 82)( 93,116)( 94,138)( 95,137)
( 96,136)( 97,135)( 98,134)( 99,133)(100,132)(101,131)(102,130)(103,129)
(104,128)(105,127)(106,126)(107,125)(108,124)(109,123)(110,122)(111,121)
(112,120)(113,119)(114,118)(115,117)(140,161)(141,160)(142,159)(143,158)
(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151)(162,185)
(163,207)(164,206)(165,205)(166,204)(167,203)(168,202)(169,201)(170,200)
(171,199)(172,198)(173,197)(174,196)(175,195)(176,194)(177,193)(178,192)
(179,191)(180,190)(181,189)(182,188)(183,187)(184,186)(209,230)(210,229)
(211,228)(212,227)(213,226)(214,225)(215,224)(216,223)(217,222)(218,221)
(219,220)(231,254)(232,276)(233,275)(234,274)(235,273)(236,272)(237,271)
(238,270)(239,269)(240,268)(241,267)(242,266)(243,265)(244,264)(245,263)
(246,262)(247,261)(248,260)(249,259)(250,258)(251,257)(252,256)(253,255)
(278,299)(279,298)(280,297)(281,296)(282,295)(283,294)(284,293)(285,292)
(286,291)(287,290)(288,289)(300,323)(301,345)(302,344)(303,343)(304,342)
(305,341)(306,340)(307,339)(308,338)(309,337)(310,336)(311,335)(312,334)
(313,333)(314,332)(315,331)(316,330)(317,329)(318,328)(319,327)(320,326)
(321,325)(322,324)(347,368)(348,367)(349,366)(350,365)(351,364)(352,363)
(353,362)(354,361)(355,360)(356,359)(357,358)(369,392)(370,414)(371,413)
(372,412)(373,411)(374,410)(375,409)(376,408)(377,407)(378,406)(379,405)
(380,404)(381,403)(382,402)(383,401)(384,400)(385,399)(386,398)(387,397)
(388,396)(389,395)(390,394)(391,393);;
s1 := ( 1, 25)( 2, 24)( 3, 46)( 4, 45)( 5, 44)( 6, 43)( 7, 42)( 8, 41)
( 9, 40)( 10, 39)( 11, 38)( 12, 37)( 13, 36)( 14, 35)( 15, 34)( 16, 33)
( 17, 32)( 18, 31)( 19, 30)( 20, 29)( 21, 28)( 22, 27)( 23, 26)( 47, 48)
( 49, 69)( 50, 68)( 51, 67)( 52, 66)( 53, 65)( 54, 64)( 55, 63)( 56, 62)
( 57, 61)( 58, 60)( 70,163)( 71,162)( 72,184)( 73,183)( 74,182)( 75,181)
( 76,180)( 77,179)( 78,178)( 79,177)( 80,176)( 81,175)( 82,174)( 83,173)
( 84,172)( 85,171)( 86,170)( 87,169)( 88,168)( 89,167)( 90,166)( 91,165)
( 92,164)( 93,140)( 94,139)( 95,161)( 96,160)( 97,159)( 98,158)( 99,157)
(100,156)(101,155)(102,154)(103,153)(104,152)(105,151)(106,150)(107,149)
(108,148)(109,147)(110,146)(111,145)(112,144)(113,143)(114,142)(115,141)
(116,186)(117,185)(118,207)(119,206)(120,205)(121,204)(122,203)(123,202)
(124,201)(125,200)(126,199)(127,198)(128,197)(129,196)(130,195)(131,194)
(132,193)(133,192)(134,191)(135,190)(136,189)(137,188)(138,187)(208,232)
(209,231)(210,253)(211,252)(212,251)(213,250)(214,249)(215,248)(216,247)
(217,246)(218,245)(219,244)(220,243)(221,242)(222,241)(223,240)(224,239)
(225,238)(226,237)(227,236)(228,235)(229,234)(230,233)(254,255)(256,276)
(257,275)(258,274)(259,273)(260,272)(261,271)(262,270)(263,269)(264,268)
(265,267)(277,370)(278,369)(279,391)(280,390)(281,389)(282,388)(283,387)
(284,386)(285,385)(286,384)(287,383)(288,382)(289,381)(290,380)(291,379)
(292,378)(293,377)(294,376)(295,375)(296,374)(297,373)(298,372)(299,371)
(300,347)(301,346)(302,368)(303,367)(304,366)(305,365)(306,364)(307,363)
(308,362)(309,361)(310,360)(311,359)(312,358)(313,357)(314,356)(315,355)
(316,354)(317,353)(318,352)(319,351)(320,350)(321,349)(322,348)(323,393)
(324,392)(325,414)(326,413)(327,412)(328,411)(329,410)(330,409)(331,408)
(332,407)(333,406)(334,405)(335,404)(336,403)(337,402)(338,401)(339,400)
(340,399)(341,398)(342,397)(343,396)(344,395)(345,394);;
s2 := ( 1,277)( 2,278)( 3,279)( 4,280)( 5,281)( 6,282)( 7,283)( 8,284)
( 9,285)( 10,286)( 11,287)( 12,288)( 13,289)( 14,290)( 15,291)( 16,292)
( 17,293)( 18,294)( 19,295)( 20,296)( 21,297)( 22,298)( 23,299)( 24,300)
( 25,301)( 26,302)( 27,303)( 28,304)( 29,305)( 30,306)( 31,307)( 32,308)
( 33,309)( 34,310)( 35,311)( 36,312)( 37,313)( 38,314)( 39,315)( 40,316)
( 41,317)( 42,318)( 43,319)( 44,320)( 45,321)( 46,322)( 47,323)( 48,324)
( 49,325)( 50,326)( 51,327)( 52,328)( 53,329)( 54,330)( 55,331)( 56,332)
( 57,333)( 58,334)( 59,335)( 60,336)( 61,337)( 62,338)( 63,339)( 64,340)
( 65,341)( 66,342)( 67,343)( 68,344)( 69,345)( 70,208)( 71,209)( 72,210)
( 73,211)( 74,212)( 75,213)( 76,214)( 77,215)( 78,216)( 79,217)( 80,218)
( 81,219)( 82,220)( 83,221)( 84,222)( 85,223)( 86,224)( 87,225)( 88,226)
( 89,227)( 90,228)( 91,229)( 92,230)( 93,231)( 94,232)( 95,233)( 96,234)
( 97,235)( 98,236)( 99,237)(100,238)(101,239)(102,240)(103,241)(104,242)
(105,243)(106,244)(107,245)(108,246)(109,247)(110,248)(111,249)(112,250)
(113,251)(114,252)(115,253)(116,254)(117,255)(118,256)(119,257)(120,258)
(121,259)(122,260)(123,261)(124,262)(125,263)(126,264)(127,265)(128,266)
(129,267)(130,268)(131,269)(132,270)(133,271)(134,272)(135,273)(136,274)
(137,275)(138,276)(139,346)(140,347)(141,348)(142,349)(143,350)(144,351)
(145,352)(146,353)(147,354)(148,355)(149,356)(150,357)(151,358)(152,359)
(153,360)(154,361)(155,362)(156,363)(157,364)(158,365)(159,366)(160,367)
(161,368)(162,369)(163,370)(164,371)(165,372)(166,373)(167,374)(168,375)
(169,376)(170,377)(171,378)(172,379)(173,380)(174,381)(175,382)(176,383)
(177,384)(178,385)(179,386)(180,387)(181,388)(182,389)(183,390)(184,391)
(185,392)(186,393)(187,394)(188,395)(189,396)(190,397)(191,398)(192,399)
(193,400)(194,401)(195,402)(196,403)(197,404)(198,405)(199,406)(200,407)
(201,408)(202,409)(203,410)(204,411)(205,412)(206,413)(207,414);;
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(414)!( 2, 23)( 3, 22)( 4, 21)( 5, 20)( 6, 19)( 7, 18)( 8, 17)
( 9, 16)( 10, 15)( 11, 14)( 12, 13)( 24, 47)( 25, 69)( 26, 68)( 27, 67)
( 28, 66)( 29, 65)( 30, 64)( 31, 63)( 32, 62)( 33, 61)( 34, 60)( 35, 59)
( 36, 58)( 37, 57)( 38, 56)( 39, 55)( 40, 54)( 41, 53)( 42, 52)( 43, 51)
( 44, 50)( 45, 49)( 46, 48)( 71, 92)( 72, 91)( 73, 90)( 74, 89)( 75, 88)
( 76, 87)( 77, 86)( 78, 85)( 79, 84)( 80, 83)( 81, 82)( 93,116)( 94,138)
( 95,137)( 96,136)( 97,135)( 98,134)( 99,133)(100,132)(101,131)(102,130)
(103,129)(104,128)(105,127)(106,126)(107,125)(108,124)(109,123)(110,122)
(111,121)(112,120)(113,119)(114,118)(115,117)(140,161)(141,160)(142,159)
(143,158)(144,157)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151)
(162,185)(163,207)(164,206)(165,205)(166,204)(167,203)(168,202)(169,201)
(170,200)(171,199)(172,198)(173,197)(174,196)(175,195)(176,194)(177,193)
(178,192)(179,191)(180,190)(181,189)(182,188)(183,187)(184,186)(209,230)
(210,229)(211,228)(212,227)(213,226)(214,225)(215,224)(216,223)(217,222)
(218,221)(219,220)(231,254)(232,276)(233,275)(234,274)(235,273)(236,272)
(237,271)(238,270)(239,269)(240,268)(241,267)(242,266)(243,265)(244,264)
(245,263)(246,262)(247,261)(248,260)(249,259)(250,258)(251,257)(252,256)
(253,255)(278,299)(279,298)(280,297)(281,296)(282,295)(283,294)(284,293)
(285,292)(286,291)(287,290)(288,289)(300,323)(301,345)(302,344)(303,343)
(304,342)(305,341)(306,340)(307,339)(308,338)(309,337)(310,336)(311,335)
(312,334)(313,333)(314,332)(315,331)(316,330)(317,329)(318,328)(319,327)
(320,326)(321,325)(322,324)(347,368)(348,367)(349,366)(350,365)(351,364)
(352,363)(353,362)(354,361)(355,360)(356,359)(357,358)(369,392)(370,414)
(371,413)(372,412)(373,411)(374,410)(375,409)(376,408)(377,407)(378,406)
(379,405)(380,404)(381,403)(382,402)(383,401)(384,400)(385,399)(386,398)
(387,397)(388,396)(389,395)(390,394)(391,393);
s1 := Sym(414)!( 1, 25)( 2, 24)( 3, 46)( 4, 45)( 5, 44)( 6, 43)( 7, 42)
( 8, 41)( 9, 40)( 10, 39)( 11, 38)( 12, 37)( 13, 36)( 14, 35)( 15, 34)
( 16, 33)( 17, 32)( 18, 31)( 19, 30)( 20, 29)( 21, 28)( 22, 27)( 23, 26)
( 47, 48)( 49, 69)( 50, 68)( 51, 67)( 52, 66)( 53, 65)( 54, 64)( 55, 63)
( 56, 62)( 57, 61)( 58, 60)( 70,163)( 71,162)( 72,184)( 73,183)( 74,182)
( 75,181)( 76,180)( 77,179)( 78,178)( 79,177)( 80,176)( 81,175)( 82,174)
( 83,173)( 84,172)( 85,171)( 86,170)( 87,169)( 88,168)( 89,167)( 90,166)
( 91,165)( 92,164)( 93,140)( 94,139)( 95,161)( 96,160)( 97,159)( 98,158)
( 99,157)(100,156)(101,155)(102,154)(103,153)(104,152)(105,151)(106,150)
(107,149)(108,148)(109,147)(110,146)(111,145)(112,144)(113,143)(114,142)
(115,141)(116,186)(117,185)(118,207)(119,206)(120,205)(121,204)(122,203)
(123,202)(124,201)(125,200)(126,199)(127,198)(128,197)(129,196)(130,195)
(131,194)(132,193)(133,192)(134,191)(135,190)(136,189)(137,188)(138,187)
(208,232)(209,231)(210,253)(211,252)(212,251)(213,250)(214,249)(215,248)
(216,247)(217,246)(218,245)(219,244)(220,243)(221,242)(222,241)(223,240)
(224,239)(225,238)(226,237)(227,236)(228,235)(229,234)(230,233)(254,255)
(256,276)(257,275)(258,274)(259,273)(260,272)(261,271)(262,270)(263,269)
(264,268)(265,267)(277,370)(278,369)(279,391)(280,390)(281,389)(282,388)
(283,387)(284,386)(285,385)(286,384)(287,383)(288,382)(289,381)(290,380)
(291,379)(292,378)(293,377)(294,376)(295,375)(296,374)(297,373)(298,372)
(299,371)(300,347)(301,346)(302,368)(303,367)(304,366)(305,365)(306,364)
(307,363)(308,362)(309,361)(310,360)(311,359)(312,358)(313,357)(314,356)
(315,355)(316,354)(317,353)(318,352)(319,351)(320,350)(321,349)(322,348)
(323,393)(324,392)(325,414)(326,413)(327,412)(328,411)(329,410)(330,409)
(331,408)(332,407)(333,406)(334,405)(335,404)(336,403)(337,402)(338,401)
(339,400)(340,399)(341,398)(342,397)(343,396)(344,395)(345,394);
s2 := Sym(414)!( 1,277)( 2,278)( 3,279)( 4,280)( 5,281)( 6,282)( 7,283)
( 8,284)( 9,285)( 10,286)( 11,287)( 12,288)( 13,289)( 14,290)( 15,291)
( 16,292)( 17,293)( 18,294)( 19,295)( 20,296)( 21,297)( 22,298)( 23,299)
( 24,300)( 25,301)( 26,302)( 27,303)( 28,304)( 29,305)( 30,306)( 31,307)
( 32,308)( 33,309)( 34,310)( 35,311)( 36,312)( 37,313)( 38,314)( 39,315)
( 40,316)( 41,317)( 42,318)( 43,319)( 44,320)( 45,321)( 46,322)( 47,323)
( 48,324)( 49,325)( 50,326)( 51,327)( 52,328)( 53,329)( 54,330)( 55,331)
( 56,332)( 57,333)( 58,334)( 59,335)( 60,336)( 61,337)( 62,338)( 63,339)
( 64,340)( 65,341)( 66,342)( 67,343)( 68,344)( 69,345)( 70,208)( 71,209)
( 72,210)( 73,211)( 74,212)( 75,213)( 76,214)( 77,215)( 78,216)( 79,217)
( 80,218)( 81,219)( 82,220)( 83,221)( 84,222)( 85,223)( 86,224)( 87,225)
( 88,226)( 89,227)( 90,228)( 91,229)( 92,230)( 93,231)( 94,232)( 95,233)
( 96,234)( 97,235)( 98,236)( 99,237)(100,238)(101,239)(102,240)(103,241)
(104,242)(105,243)(106,244)(107,245)(108,246)(109,247)(110,248)(111,249)
(112,250)(113,251)(114,252)(115,253)(116,254)(117,255)(118,256)(119,257)
(120,258)(121,259)(122,260)(123,261)(124,262)(125,263)(126,264)(127,265)
(128,266)(129,267)(130,268)(131,269)(132,270)(133,271)(134,272)(135,273)
(136,274)(137,275)(138,276)(139,346)(140,347)(141,348)(142,349)(143,350)
(144,351)(145,352)(146,353)(147,354)(148,355)(149,356)(150,357)(151,358)
(152,359)(153,360)(154,361)(155,362)(156,363)(157,364)(158,365)(159,366)
(160,367)(161,368)(162,369)(163,370)(164,371)(165,372)(166,373)(167,374)
(168,375)(169,376)(170,377)(171,378)(172,379)(173,380)(174,381)(175,382)
(176,383)(177,384)(178,385)(179,386)(180,387)(181,388)(182,389)(183,390)
(184,391)(185,392)(186,393)(187,394)(188,395)(189,396)(190,397)(191,398)
(192,399)(193,400)(194,401)(195,402)(196,403)(197,404)(198,405)(199,406)
(200,407)(201,408)(202,409)(203,410)(204,411)(205,412)(206,413)(207,414);
poly := sub<Sym(414)|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 >;
References : None.
to this polytope