Polytope of Type {6,138}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,138}*1656c
if this polytope has a name.
Group : SmallGroup(1656,129)
Rank : 3
Schlafli Type : {6,138}
Number of vertices, edges, etc : 6, 414, 138
Order of s₀s₁s₂ : 138
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 : {6,69}*828
   3-fold quotients : {2,138}*552
   6-fold quotients : {2,69}*276
   9-fold quotients : {2,46}*184
   18-fold quotients : {2,23}*92
   23-fold quotients : {6,6}*72b
   46-fold quotients : {6,3}*36
   69-fold quotients : {2,6}*24
   138-fold quotients : {2,3}*12
   207-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope