Polytope of Type {6,117}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,117}*1404
if this polytope has a name.
Group : SmallGroup(1404,120)
Rank : 3
Schlafli Type : {6,117}
Number of vertices, edges, etc : 6, 351, 117
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 : {2,117}*468, {6,39}*468
   9-fold quotients : {2,39}*156
   13-fold quotients : {6,9}*108
   27-fold quotients : {2,13}*52
   39-fold quotients : {2,9}*36, {6,3}*36
   117-fold quotients : {2,3}*12
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);;
s1 := (  1, 40)(  2, 42)(  3, 41)(  4, 76)(  5, 78)(  6, 77)(  7, 73)(  8, 75)
(  9, 74)( 10, 70)( 11, 72)( 12, 71)( 13, 67)( 14, 69)( 15, 68)( 16, 64)
( 17, 66)( 18, 65)( 19, 61)( 20, 63)( 21, 62)( 22, 58)( 23, 60)( 24, 59)
( 25, 55)( 26, 57)( 27, 56)( 28, 52)( 29, 54)( 30, 53)( 31, 49)( 32, 51)
( 33, 50)( 34, 46)( 35, 48)( 36, 47)( 37, 43)( 38, 45)( 39, 44)( 80, 81)
( 82,115)( 83,117)( 84,116)( 85,112)( 86,114)( 87,113)( 88,109)( 89,111)
( 90,110)( 91,106)( 92,108)( 93,107)( 94,103)( 95,105)( 96,104)( 97,100)
( 98,102)( 99,101)(118,275)(119,274)(120,276)(121,311)(122,310)(123,312)
(124,308)(125,307)(126,309)(127,305)(128,304)(129,306)(130,302)(131,301)
(132,303)(133,299)(134,298)(135,300)(136,296)(137,295)(138,297)(139,293)
(140,292)(141,294)(142,290)(143,289)(144,291)(145,287)(146,286)(147,288)
(148,284)(149,283)(150,285)(151,281)(152,280)(153,282)(154,278)(155,277)
(156,279)(157,236)(158,235)(159,237)(160,272)(161,271)(162,273)(163,269)
(164,268)(165,270)(166,266)(167,265)(168,267)(169,263)(170,262)(171,264)
(172,260)(173,259)(174,261)(175,257)(176,256)(177,258)(178,254)(179,253)
(180,255)(181,251)(182,250)(183,252)(184,248)(185,247)(186,249)(187,245)
(188,244)(189,246)(190,242)(191,241)(192,243)(193,239)(194,238)(195,240)
(196,314)(197,313)(198,315)(199,350)(200,349)(201,351)(202,347)(203,346)
(204,348)(205,344)(206,343)(207,345)(208,341)(209,340)(210,342)(211,338)
(212,337)(213,339)(214,335)(215,334)(216,336)(217,332)(218,331)(219,333)
(220,329)(221,328)(222,330)(223,326)(224,325)(225,327)(226,323)(227,322)
(228,324)(229,320)(230,319)(231,321)(232,317)(233,316)(234,318);;
s2 := (  1,121)(  2,123)(  3,122)(  4,118)(  5,120)(  6,119)(  7,154)(  8,156)
(  9,155)( 10,151)( 11,153)( 12,152)( 13,148)( 14,150)( 15,149)( 16,145)
( 17,147)( 18,146)( 19,142)( 20,144)( 21,143)( 22,139)( 23,141)( 24,140)
( 25,136)( 26,138)( 27,137)( 28,133)( 29,135)( 30,134)( 31,130)( 32,132)
( 33,131)( 34,127)( 35,129)( 36,128)( 37,124)( 38,126)( 39,125)( 40,199)
( 41,201)( 42,200)( 43,196)( 44,198)( 45,197)( 46,232)( 47,234)( 48,233)
( 49,229)( 50,231)( 51,230)( 52,226)( 53,228)( 54,227)( 55,223)( 56,225)
( 57,224)( 58,220)( 59,222)( 60,221)( 61,217)( 62,219)( 63,218)( 64,214)
( 65,216)( 66,215)( 67,211)( 68,213)( 69,212)( 70,208)( 71,210)( 72,209)
( 73,205)( 74,207)( 75,206)( 76,202)( 77,204)( 78,203)( 79,160)( 80,162)
( 81,161)( 82,157)( 83,159)( 84,158)( 85,193)( 86,195)( 87,194)( 88,190)
( 89,192)( 90,191)( 91,187)( 92,189)( 93,188)( 94,184)( 95,186)( 96,185)
( 97,181)( 98,183)( 99,182)(100,178)(101,180)(102,179)(103,175)(104,177)
(105,176)(106,172)(107,174)(108,173)(109,169)(110,171)(111,170)(112,166)
(113,168)(114,167)(115,163)(116,165)(117,164)(235,239)(236,238)(237,240)
(241,272)(242,271)(243,273)(244,269)(245,268)(246,270)(247,266)(248,265)
(249,267)(250,263)(251,262)(252,264)(253,260)(254,259)(255,261)(256,257)
(274,317)(275,316)(276,318)(277,314)(278,313)(279,315)(280,350)(281,349)
(282,351)(283,347)(284,346)(285,348)(286,344)(287,343)(288,345)(289,341)
(290,340)(291,342)(292,338)(293,337)(294,339)(295,335)(296,334)(297,336)
(298,332)(299,331)(300,333)(301,329)(302,328)(303,330)(304,326)(305,325)
(306,327)(307,323)(308,322)(309,324)(310,320)(311,319)(312,321);;
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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope