Polytope of Type {6,123}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,123}*1476
if this polytope has a name.
Group : SmallGroup(1476,28)
Rank : 3
Schlafli Type : {6,123}
Number of vertices, edges, etc : 6, 369, 123
Order of s₀s₁s₂ : 246
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 : {2,123}*492
   9-fold quotients : {2,41}*164
   41-fold quotients : {6,3}*36
   123-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope