Polytope of Type {117,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {117,6}*1404
if this polytope has a name.
Group : SmallGroup(1404,120)
Rank : 3
Schlafli Type : {117,6}
Number of vertices, edges, etc : 117, 351, 6
Order of s₀s₁s₂ : 234
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) :
   3-fold quotients : {117,2}*468, {39,6}*468
   9-fold quotients : {39,2}*156
   13-fold quotients : {9,6}*108
   27-fold quotients : {13,2}*52
   39-fold quotients : {9,2}*36, {3,6}*36
   117-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope