Polytope of Type {138,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {138,6}*1656c
if this polytope has a 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 : 6
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 : {69,6}*828
   3-fold quotients : {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}*72c
   46-fold quotients : {3,6}*36
   69-fold quotients : {6,2}*24
   138-fold quotients : {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)( 70,139)( 71,161)( 72,160)( 73,159)( 74,158)( 75,157)
( 76,156)( 77,155)( 78,154)( 79,153)( 80,152)( 81,151)( 82,150)( 83,149)
( 84,148)( 85,147)( 86,146)( 87,145)( 88,144)( 89,143)( 90,142)( 91,141)
( 92,140)( 93,185)( 94,207)( 95,206)( 96,205)( 97,204)( 98,203)( 99,202)
(100,201)(101,200)(102,199)(103,198)(104,197)(105,196)(106,195)(107,194)
(108,193)(109,192)(110,191)(111,190)(112,189)(113,188)(114,187)(115,186)
(116,162)(117,184)(118,183)(119,182)(120,181)(121,180)(122,179)(123,178)
(124,177)(125,176)(126,175)(127,174)(128,173)(129,172)(130,171)(131,170)
(132,169)(133,168)(134,167)(135,166)(136,165)(137,164)(138,163)(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)(277,346)(278,368)(279,367)(280,366)(281,365)(282,364)(283,363)
(284,362)(285,361)(286,360)(287,359)(288,358)(289,357)(290,356)(291,355)
(292,354)(293,353)(294,352)(295,351)(296,350)(297,349)(298,348)(299,347)
(300,392)(301,414)(302,413)(303,412)(304,411)(305,410)(306,409)(307,408)
(308,407)(309,406)(310,405)(311,404)(312,403)(313,402)(314,401)(315,400)
(316,399)(317,398)(318,397)(319,396)(320,395)(321,394)(322,393)(323,369)
(324,391)(325,390)(326,389)(327,388)(328,387)(329,386)(330,385)(331,384)
(332,383)(333,382)(334,381)(335,380)(336,379)(337,378)(338,377)(339,376)
(340,375)(341,374)(342,373)(343,372)(344,371)(345,370);;
s1 := (  1,301)(  2,300)(  3,322)(  4,321)(  5,320)(  6,319)(  7,318)(  8,317)
(  9,316)( 10,315)( 11,314)( 12,313)( 13,312)( 14,311)( 15,310)( 16,309)
( 17,308)( 18,307)( 19,306)( 20,305)( 21,304)( 22,303)( 23,302)( 24,278)
( 25,277)( 26,299)( 27,298)( 28,297)( 29,296)( 30,295)( 31,294)( 32,293)
( 33,292)( 34,291)( 35,290)( 36,289)( 37,288)( 38,287)( 39,286)( 40,285)
( 41,284)( 42,283)( 43,282)( 44,281)( 45,280)( 46,279)( 47,324)( 48,323)
( 49,345)( 50,344)( 51,343)( 52,342)( 53,341)( 54,340)( 55,339)( 56,338)
( 57,337)( 58,336)( 59,335)( 60,334)( 61,333)( 62,332)( 63,331)( 64,330)
( 65,329)( 66,328)( 67,327)( 68,326)( 69,325)( 70,232)( 71,231)( 72,253)
( 73,252)( 74,251)( 75,250)( 76,249)( 77,248)( 78,247)( 79,246)( 80,245)
( 81,244)( 82,243)( 83,242)( 84,241)( 85,240)( 86,239)( 87,238)( 88,237)
( 89,236)( 90,235)( 91,234)( 92,233)( 93,209)( 94,208)( 95,230)( 96,229)
( 97,228)( 98,227)( 99,226)(100,225)(101,224)(102,223)(103,222)(104,221)
(105,220)(106,219)(107,218)(108,217)(109,216)(110,215)(111,214)(112,213)
(113,212)(114,211)(115,210)(116,255)(117,254)(118,276)(119,275)(120,274)
(121,273)(122,272)(123,271)(124,270)(125,269)(126,268)(127,267)(128,266)
(129,265)(130,264)(131,263)(132,262)(133,261)(134,260)(135,259)(136,258)
(137,257)(138,256)(139,370)(140,369)(141,391)(142,390)(143,389)(144,388)
(145,387)(146,386)(147,385)(148,384)(149,383)(150,382)(151,381)(152,380)
(153,379)(154,378)(155,377)(156,376)(157,375)(158,374)(159,373)(160,372)
(161,371)(162,347)(163,346)(164,368)(165,367)(166,366)(167,365)(168,364)
(169,363)(170,362)(171,361)(172,360)(173,359)(174,358)(175,357)(176,356)
(177,355)(178,354)(179,353)(180,352)(181,351)(182,350)(183,349)(184,348)
(185,393)(186,392)(187,414)(188,413)(189,412)(190,411)(191,410)(192,409)
(193,408)(194,407)(195,406)(196,405)(197,404)(198,403)(199,402)(200,401)
(201,400)(202,399)(203,398)(204,397)(205,396)(206,395)(207,394);;
s2 := ( 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);;
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*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)( 70,139)( 71,161)( 72,160)( 73,159)( 74,158)
( 75,157)( 76,156)( 77,155)( 78,154)( 79,153)( 80,152)( 81,151)( 82,150)
( 83,149)( 84,148)( 85,147)( 86,146)( 87,145)( 88,144)( 89,143)( 90,142)
( 91,141)( 92,140)( 93,185)( 94,207)( 95,206)( 96,205)( 97,204)( 98,203)
( 99,202)(100,201)(101,200)(102,199)(103,198)(104,197)(105,196)(106,195)
(107,194)(108,193)(109,192)(110,191)(111,190)(112,189)(113,188)(114,187)
(115,186)(116,162)(117,184)(118,183)(119,182)(120,181)(121,180)(122,179)
(123,178)(124,177)(125,176)(126,175)(127,174)(128,173)(129,172)(130,171)
(131,170)(132,169)(133,168)(134,167)(135,166)(136,165)(137,164)(138,163)
(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)(277,346)(278,368)(279,367)(280,366)(281,365)(282,364)
(283,363)(284,362)(285,361)(286,360)(287,359)(288,358)(289,357)(290,356)
(291,355)(292,354)(293,353)(294,352)(295,351)(296,350)(297,349)(298,348)
(299,347)(300,392)(301,414)(302,413)(303,412)(304,411)(305,410)(306,409)
(307,408)(308,407)(309,406)(310,405)(311,404)(312,403)(313,402)(314,401)
(315,400)(316,399)(317,398)(318,397)(319,396)(320,395)(321,394)(322,393)
(323,369)(324,391)(325,390)(326,389)(327,388)(328,387)(329,386)(330,385)
(331,384)(332,383)(333,382)(334,381)(335,380)(336,379)(337,378)(338,377)
(339,376)(340,375)(341,374)(342,373)(343,372)(344,371)(345,370);
s1 := Sym(414)!(  1,301)(  2,300)(  3,322)(  4,321)(  5,320)(  6,319)(  7,318)
(  8,317)(  9,316)( 10,315)( 11,314)( 12,313)( 13,312)( 14,311)( 15,310)
( 16,309)( 17,308)( 18,307)( 19,306)( 20,305)( 21,304)( 22,303)( 23,302)
( 24,278)( 25,277)( 26,299)( 27,298)( 28,297)( 29,296)( 30,295)( 31,294)
( 32,293)( 33,292)( 34,291)( 35,290)( 36,289)( 37,288)( 38,287)( 39,286)
( 40,285)( 41,284)( 42,283)( 43,282)( 44,281)( 45,280)( 46,279)( 47,324)
( 48,323)( 49,345)( 50,344)( 51,343)( 52,342)( 53,341)( 54,340)( 55,339)
( 56,338)( 57,337)( 58,336)( 59,335)( 60,334)( 61,333)( 62,332)( 63,331)
( 64,330)( 65,329)( 66,328)( 67,327)( 68,326)( 69,325)( 70,232)( 71,231)
( 72,253)( 73,252)( 74,251)( 75,250)( 76,249)( 77,248)( 78,247)( 79,246)
( 80,245)( 81,244)( 82,243)( 83,242)( 84,241)( 85,240)( 86,239)( 87,238)
( 88,237)( 89,236)( 90,235)( 91,234)( 92,233)( 93,209)( 94,208)( 95,230)
( 96,229)( 97,228)( 98,227)( 99,226)(100,225)(101,224)(102,223)(103,222)
(104,221)(105,220)(106,219)(107,218)(108,217)(109,216)(110,215)(111,214)
(112,213)(113,212)(114,211)(115,210)(116,255)(117,254)(118,276)(119,275)
(120,274)(121,273)(122,272)(123,271)(124,270)(125,269)(126,268)(127,267)
(128,266)(129,265)(130,264)(131,263)(132,262)(133,261)(134,260)(135,259)
(136,258)(137,257)(138,256)(139,370)(140,369)(141,391)(142,390)(143,389)
(144,388)(145,387)(146,386)(147,385)(148,384)(149,383)(150,382)(151,381)
(152,380)(153,379)(154,378)(155,377)(156,376)(157,375)(158,374)(159,373)
(160,372)(161,371)(162,347)(163,346)(164,368)(165,367)(166,366)(167,365)
(168,364)(169,363)(170,362)(171,361)(172,360)(173,359)(174,358)(175,357)
(176,356)(177,355)(178,354)(179,353)(180,352)(181,351)(182,350)(183,349)
(184,348)(185,393)(186,392)(187,414)(188,413)(189,412)(190,411)(191,410)
(192,409)(193,408)(194,407)(195,406)(196,405)(197,404)(198,403)(199,402)
(200,401)(201,400)(202,399)(203,398)(204,397)(205,396)(206,395)(207,394);
s2 := 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);
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*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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